Development and optimization of a morphodiversity model for mountainous areas using supervised classification and artificial neural networks

Abstract

Morphodiversity assessments in mountainous landscapes represent an important tool for geoconservation. Particularly useful are the Raster Continuous Morphodiversity (RCM) models, which are based on continuous variables and measures of pixel variability within statistical zones. Traditional RCM models of the Aggregating Ratings (AR) type define morphodiversity as the sum of standardized partial criteria; however, their drawback lies in the redundant aggregation of variability, which leads to evaluation errors. To mitigate this issue, a new RCM model was developed, based on Supervised Classification (SC) and Artificial Neural Networks (ANN), which accounts for interdependencies between variables and eliminates low-informative ones. This study presents the results of RCM modeling for the Pieniny Mountains (southern Poland) using the RCM_SC−ANN and the improved RCM_SC−ANN−M models. Variable reduction was performed using the Global Sensitivity Analysis (GSA) method in its Backward Analysis (BA) and Backward Stepwise Analysis (BSA) variants, resulting in simplified models. Comparisons with the SC-ANN_VSR−9−BS and RCM_SDcm models showed that RCM_SC−ANN−M produces more reliable evaluations and can be recommended for testing in other mountainous areas. The high correlation coefficients (0.96–0.98) between full and reduced sets of criteria confirm the effectiveness and efficiency of the BA and BSA procedures.

Introduction

Nature conservation is commonly associated with efforts to preserve biological diversity as well as the protection of species and habitats. This approach, although extremely important, often overlooks the fact that the conditions for the functioning of living nature are shaped by the abiotic components of the environment^{1,2,3,4,5,6,7,8}. They form the foundation upon which life develops and which determines habitat diversity and ecosystem dynamics. Therefore, in line with a holistic understanding of nature, it is necessary to establish in the scientific discourse the paradigm of geodiversity as a discipline concerned with assessing the diversity of geological structure, soil cover, landforms, hydrosphere features, and climate^{8,9,10,11,12,13,14} and to incorporate it into nature conservation within the framework of geoheritage protection^{13,15,16,17,18,19,20,21}.

Geodiversity assessments are now becoming important tools in nature conservation planning^{22,23,24,25,26}. A particularly important role in the diversity of abiotic elements is played by morphodiversity^{26,27,28,29,30,31,32,33,34,35,36,37}. A varied landforms foster the diversification of habitat conditions and the creation of ecological niches. As a result, it influences species distribution, plays an important role in shaping ecosystem structure, and increases the landscape’s resilience to environmental changes. In mountainous areas, where landform dynamics have the greatest impact on biodiversity, labor-intensive geodiversity assessments can be effectively approximated through morphodiversity evaluations²⁶. This is why efforts to improve the quality of morphodiversity models, to refine the selection of partial criteria, and, in particular, to enhance the objectivity and automation of analyses are so important.

Geodiversity assessments are typically carried out within the cells of artificial analytical grids (hereafter referred to as statistical zones). In the most commonly used indicator-based approach, the evaluation procedure employs morphometric tools to quantify the variability of abiotic elements. Selected elements of nature (e.g., geological structure, landforms, and soil cover) are represented using landscape features (e.g., hypsometry, aspect, slope, and curvatures)³⁸. These defined features are then described using a chosen set of criteria (landscape metrics), which may include measures of heterogeneity, variability, and spatial configuration^39,40,41. The metrics are calculated independently for each statistical zone, providing a localized assessment of the variability of abiotic elements. According to the definition of geodiversity^{12,42,43,44,45,46,47}, these values are subsequently standardized and aggregated within each statistical zone. The aggregated values indicate the degree (or class) of diversity of abiotic components, providing a quantifiable measure of geo- and morphodiversity. Mastej & Bartuś³⁶ refer to this method of defining geo- and morphodiversity as the Aggregating Ratings (AR) Model.

Although the AR method is widely used^{24,31,48,49,50}, its application is associated with several significant limitations. First and foremost, subjectivity in selecting landscape elements, features, and partial criteria^{24,32,37,48,49,51} leads to differences in interpretation and complicates comparisons. Another issue is the discretization of landscape features derived from raster data^{24,34,48,50,52,53,54}, which inevitably results in some information loss. The most serious challenge, however, remains data redundancy. As reported by Bartuś²⁴, Mastej & Bartuś³⁶, and Bartuś & Mastej⁵⁵, strong correlations exist between many partial criteria in geodiversity analyses, and their summation during aggregation leads to multiple counting of the same information. Consequently, the resulting assessments may be overestimated or distorted, making interpretation and comparison between different areas problematic.

Previous attempts to overcome these difficulties^26,36,55 indicate the need for a more effective methodology for assessing morphodiversity, one that accounts for nonlinear interdependencies among partial criteria and allows for the elimination of low-informative variables. The present study addresses this issue by developing and testing new morphodiversity models based on the Raster Continuous Morphodiversity (RCM) concept⁵⁵. These models enable the description of partial diversity using regionally continuous variables as well as scalar and angular measures of pixel variability within statistical zones, while their integration with Supervised Classification (SC), Artificial Neural Networks (ANN), and Global Sensitivity Analysis (GSA)^56,57 allows for the reduction of data redundancy and enhances the objectivity of the analyses.

The Pieniny Mountains, selected as the study area for morphodiversity analyses, represent one of the best-studied regions in Poland in this context. The massif has previously been investigated in geodiversity research within the broader Carpathian framework^53,58,59, as well as in detailed assessments of specific subregions⁶⁰. Moreover, the area has played a key role in the development of morphodiversity modeling methodologies, ranging from classical AR models²⁶ and early applications of Supervised Classification (SC) and Artificial Neural Networks (ANN)³⁶ to the implementation of the Raster Continuous Morphodiversity (RCM) concept, which, as shown by Bartuś & Mastej⁵⁵, outperforms previous approaches.

The use of ANN is increasingly widespread in studies of geo- and morphodiversity, as well as in related disciplines. For example, Bartuś²⁴ applied ANN to evaluate the informativeness of partial criteria in the Ojców National Park (southern Poland); Wesley & Matisziw⁶¹ employed them for geodiversity assessment based on high-resolution aerial imagery; Wolniewicz⁵⁰ integrated Generative Artificial Intelligence (GAI) with Geographic Information Systems (GIS) to detect geodiversity potential; and Manaouch⁶² used machine learning combined with morphometric data to identify potential geosites in southeastern Morocco.

Analyses of landform heterogeneity, together with neural network applications in this context, have also been successfully used in related fields, such as predicting landslide susceptibility^{63,64,65,66,67}.

Study area

The study area is situated in southern Poland, within the Lesser Poland Voivodeship, bordering Slovakia to the south. The region is characterized by pronounced geological and landscape diversity, which is reflected in its subdivision into distinct physical-geographical units. According to Solon (et al.)⁶⁸, the area lies along the boundary between the Outer Western Carpathians subprovince to the north and the Central Western Carpathians subprovince to the south (Fig. 1). Within this framework, the Outer Western Carpathians encompass the Western Beskids microregion⁶⁹, whereas the Central Western Carpathians comprise the Orawa–Podhale Basin^70,71.

The region’s physical-geographical diversity is further detailed through mesoregions. In the study area, the Western Beskids include the Gorce Mountains^72,73 and the Sącz Beskid Mountains, while the Orawa–Podhale Basin is subdivided into the Orawa–Nowy Targ Basin^70,74, the Pieniny Mountains^75,76, and the Magura Spiska Mountains^77,78.

Fig. 1

The most important geocomplexes of the study area, after Solon (et al.) ⁶⁸. Subprovinces: 1—Central Western Carpathians; 2—Outer Western Carpathians. Macroregions: 3—Western Beskidy Mts; 4—Orawa–Podhale Basin. Mesoregion names are indicated on the map Other map features: 5—rivers and streams, 6—water bodies, 7—national border.

In terms of landscape, the most important mesoregion of the study area is the Pieniny Mts. This relatively small but highly distinctive mountain massif is cut from west to east by the Dunajec River Gorge. Along the national border, the gorge divides the Pieniny Mts into the Polish part (to the north) and the Slovak part (to the south). The Pieniny Mts are subdivided into four microregions. To the west, up to the Dunajec River Gorge near Niedzica, lie the Pieniny Spiskie Mountains (Fig. 2). The central and most significant part of the massif is the Pieniny Właściwe Mountains (Central Pieniny Mts), which extend from the Dunajec valley section between Czorsztyn and Sromowce Wyżne to the Dunajec River Gorge separating the Pieniny Mts from the Sącz Beskid Mountains near Szczawnica. Within the western part of the Central Pieniny Mts, the Czorsztyn Pieniny Mountains are identified, extending between the villages of Czorsztyn and Kąty. To the east lie the Małe Pieniny Mountains (Small Pieniny Mts). Due to its exceptional natural values, the Pieniny Właściwe Mts were designated as Pieniny National Park (PNP) in 1932⁷⁹. The park’s most important features are its high biodiversity and outstanding landscape values. The primary conservation objective of PNP is to maintain and restore the natural landscape of the Pieniny Mts⁸⁰.

Fig. 2

Elevation map of the study area: 1—rivers and streams; 2—water bodies; 3—national border; 4—Pieniny National Park.

The landscape of the study area is heterogeneous (Figs. 1 and 3)⁷⁶. The most pronounced landscape contrast occurs at the boundary between the Western Beskids Mts and the Central Western Carpathians. This contrast results from an abrupt change in the geological structure of the adjacent units. The Central Western Carpathians belong to the Inner Carpathians—a major tectonic unit composed of crystalline basement rocks and a Mesozoic sedimentary cover, as well as, locally, flysch deposits and volcanic intrusions (Fig. 4). In contrast, the Western Beskids Mts geologically belong to the Outer Carpathians, which are mainly composed of flysch sandstones and shales ranging in age from the Upper Jurassic through the Cretaceous and Paleogene to the Miocene^72,81.

The Inner and Outer Carpathians are separated by the Pieniny Mts massif, which forms a narrow tectonic zone known as the Pieniny Klippen Belt (PKB)^{82,83,84,85,86}.

Fig. 3

Selected landscape features of the Pieniny Mts region. A—Trzy Korony (982 m a.s.l.) — the most famous peak of the Pieniny Właściwe Mts, viewed from Sromowce Wyżne towards the north, showing klippen composed of cherty limestones resistant to weathering (Pieniny Succession); B—the Sokolica Massif, viewed from the Dunajec River Gorge near Szczawnica (Pieniny Succession); C—the Gorce Mts (Magura Unit), view from the vicinity of Falsztyn (Pieniny Spiskie Mts) towards the north, with Czorsztyn Lake visible in the foreground; D—andesite quarry (PAL) on Mt Wdżar (767 m a.s.l.) near the village of Kluszkowce; E—deformed Jurassic ammonite on the surface of a steeply dipping bed of crinoidal limestone at Czorsztyn Castle (Czorsztyn Succession).

Fig. 4

Main structural units of the Polish sector of the Pieniny Klippen Belt and adjacent regions, after Borecka (et al.)⁸⁷. Outer Carpathian units: 1—Grajcarek Unit; 2—Magura Unit. PKB units: 3—Czorsztyn Unit; 4—Niedzica Unit; 5—Czertezik Unit; 6—Pieniny Unit; 7—Paleogene cover; 8—andesites of the Pieniny Andesite Line. Inner Carpathian units: 9—Podhale Flysch. Other map features: 10—major faults; 11—rivers and streams; 12—water bodies; 13—Pieniny National Park; 14—national border. Note: Some map elements (Paleogene cover, Podhale Flysch, and andesites) represent lithological components occurring within or above the structural units.

The PKB is a belt of outcrops primarily composed of sedimentary rocks (Fig. 4), formed from the Triassic–Jurassic transition to the Paleogene within the northern part of the Tethys Ocean, known as the Pieniny Basin (PKB Basin), and the southern part of the Magura Basin^83,88. Some authors suggest that the Pieniny Basin represented part of a larger Zlatna Basin⁸⁵. During the period of the basin’s greatest expansion, at the Middle–Upper Jurassic transition, a deep-oceanic environment developed in its central zone, with water depths exceeding the calcite compensation depth⁸². In this area, deep-marine sediments containing cherts and radiolarians were deposited (Pieniny Succession; Fig. 3A, B). At the same time, shallow-marine—mainly carbonate—sedimentation occurred on the submarine ridges (Czorsztyn Succession; Fig. 3E). Transitional zones occurred between the deep-marine and shallow-marine depositional environments: the northern zone (Czertezik Succession) and the southern zones (Niedzica and Branisko Successions).

At the Cretaceous–Paleogene boundary, as a result of the collision between continental blocks along the boundary of the Inner and Outer Carpathians, the PKB Basin became closed^82,89,90. This process was preceded by the deposition of flysch sediments^72,81. The deposits of the Pieniny Basin, which had formed in its various parts, were folded, detached from the basement, and thrust northward onto the autochthon (Czertezik and Czorsztyn successions) or onto one another, forming the Pieniny, Branisko, Niedzica, and Czertezik nappes (Fig. 5). Consequently, rocks representing shallow-marine, deep-marine, and transitional sedimentary series came into direct contact, forming distinct tectonic units (Czorsztyn Unit, Czertezik Unit, Niedzica Unit, Pieniny Unit) (Fig. 4). While the intense tectonic deformation of the Pieniny Mts is undisputed, it is now considered that the previously suggested tectonic position of some shallow-marine rock blocks within the deep-marine sedimentary series may instead be of olistolithic origin^85,91,92.

Fig. 5

Geological cross-section of the Pieniny Klippen Belt along the Kraków–Zakopane transect (located approximately 10 km west of the study area). After Birkenmajer⁸², simplified. Outer Carpathian units: 1—Grajcarek Unit (Jurassic–Maastrichtian); 2—Cenomanian–Maastrichtian; 3—Paleogene. Pieniny Klippen Belt units: 4—Czorsztyn Unit (Jurassic–Lower Cretaceous); 5—Czorsztyn and Niedzica units (?) (incompetent Upper Cretaceous marls and flysch-like rocks); 6—Branisko and Pieniny units; 7—Jarmuta Formation (Maastrichtian molasse and flysch); 8—Maruszyna Thrust Sheet (Upper Santonian–Middle Eocene). Inner Carpathian units: 9—basal conglomerate (Sulov Conglomerate) and nummulitic limestone (Paleogene); 10—Zakopane Formation (Paleogene flysch). Orava–Nowy Targ Basin: 11—Neogene and Quaternary deposits.

Simultaneously with the formation of the PKB orogen, a post-orogenic sedimentary cover developed (Fig. 5 – Jarmuta Formation). From the southern part of the Magura Basin, the Grajcarek Unit strata were backthrust onto the PKB orogenic belt (Fig. 4 – Grajcarek Unit)⁸⁸. During the Paleocene, the PKB orogen underwent significant denudation and was subsequently submerged by a sea expanding southward from the Magura Basin. This resulted in the formation of a Paleogene flysch sedimentary cover (Fig. 4 – Paleogene cover)⁸¹. At the same time, flysch deposits were also forming south of the denuded PKB orogen (Fig. 4 – Podhale Flysch)⁷⁷. In the Miocene, intense compression of the PKB orogen occurred, accompanied by tectonic deformation and both horizontal and vertical displacements of rock masses. These intense tectonic movements resulted in a reduction of the PKB’s width from its original 100–150 km to just a few kilometers^{82,84,86,89,92}. In the southern part of the Magura Unit, andesitic magma intruded into the Magura Basin strata from deep magmatic sources (Figs. 3D and 4 – Pieniny Andesite Line – PAL)^82,83. These stages of PKB development had the greatest influence on the present-day dynamic relief and landscape of the Pieniny Mts and adjacent areas.

The most distinctive landscape feature of the Pieniny Mts region, and at the same time its main tourist attraction, is the Dunajec River Gorge^83,93,94,95 (Fig. 3B). The gorge begins near Czorsztyn, where the Dunajec River turns southward to transversely cut through the Pieniny Mts range near Niedzica, separating the Pieniny Spiskie Mts (to the west) from the Czorsztyn and Pieniny Właściwe Mts (to the east) (Fig. 2). This section is known as the Pieniny Gorge. Farther east, the valley meanders along the southern margin of the Pieniny Mts and the Polish–Slovak border. This is the most scenically attractive part of the entire gorge, where the river flows through a deep ravine flanked by steep rocky walls often exceeding 300 m in height. Near Szczawnica, the Dunajec turns northward to once again cut across the latitudinally elongated ridges of the Pieniny Właściwe Mts and the Małe Pieniny Mts. After passing Krościenko nad Dunajcem, the river continues northward, dissecting the Gorce Mts (to the west) and the Sącz Beskids (to the east). This section is referred to as the Small Dunajec River Gorge.

The Pieniny Właściwe Mts are largely composed of hard siliceous rocks of the Pieniny Unit (Fig. 4). This results in a relief characterized by steep, almost vertical rock walls, numerous rock towers (so-called “skalice”), and deeply incised valleys (Fig. 3A, B). In contrast, the Pieniny Spiskie, Czorsztyn, and Małe Pieniny Mts are dominated by more weathering-prone rocks of the Czorsztyn, Czertezik, Niedzica, and Grajcarek Units. Consequently, the relief of these ranges features gentler ridges, isolated limestone crags, and picturesque gorges with small waterfalls (Fig. 3C).

To the north, the PKB borders the flysch of the Outer Carpathians, which in this region is represented by deposits of the Magura Unit^72,81 (Fig. 4). These deposits formed within the Magura Basin, located northwest of the main structural high of the Pieniny Basin – the Czorsztyn Ridge^85,96. As previously mentioned, during the Miocene, intermediate magmas intruded into the flysch rocks of the southern part of the Magura Unit, forming the Pieniny Andesite Line (PAL)⁹⁷ (Fig. 3D).

In the western part of the study area, between the Western Beskids Mts and the Pieniny Mts, lies the Orawa–Nowy Targ Basin (Fig. 1). This Neogene tectonic depression is filled with Neogene and Quaternary clastic sediments⁷⁰. Its basement consists of Eocene and Oligocene Podhale Flysch deposits⁷⁷. The landscape of the Orawa–Nowy Targ Basin has been significantly transformed by human activity, particularly through the construction of the Czorsztyn and Sromowce artificial reservoirs^71,74 (Figs. 2 and 3C).

To the south of the Pieniny Mts lies a section of the Magura Spiska Mts⁷⁸. This ridge reaches an average elevation of about 1000 m a.s.l., with most of its area located within Slovakia. The portion of the Magura Spiska Mts within the study area is composed of Podhale Flysch⁷⁷ (Fig. 4).

Material and method

Data acquisition and Preparation

Morphodiversity modeling based on the RCM approach and SC-ANN methodology builds upon the findings presented in Bartuś & Mastej ^26,55 and Mastej & Bartuś ³⁶. To ensure that the evaluations obtained using this method are comparable with previous morphodiversity analyses of the Pieniny Mts and their surroundings, the study area was defined to cover the same spatial extent. This area corresponds to a rectangular domain with coordinates expressed in the National Geodetic Reference System PUWG 1992 (cartographic projection; Well-Known ID – WKID: 2180): X_min = 590250.0; X_max = 607550.0; Y_min = 170850.0; Y_max = 179250.0.

In the conducted analyses, a Digital Elevation Model (DEM) from 2022 was used, obtained from the National Geodetic and Cartographic Resource (Państwowy Zasób Geodezyjny i Kartograficzny). The raster, with a resolution of 5 m × 5 m, employed the National Geodetic Reference System PUWG 1992 and the European Vertical Reference Frame PL-EVRF2007-NH.

The description of the landform was reduced to seven landscape features: altitude, aspect, slope, plan and profile curvature, local denivelation, and the presence of rock formations. Their selection was motivated by the intention to continue previous work on the archival morphodiversity model RCM_SDcm⁵⁵. The listed landscape features were characterized using the DEM, primary and secondary topographic attributes, and feature class data. The analyses were carried out using ArcGIS Pro software.

For the analysis of local denivelation, the secondary topographic attribute Topographic Position Index (TPI) was applied^98,99. The study showed that, for the analyzed landscape type and the specified DEM resolution, the appropriate observation scale corresponds to a circular neighborhood with a radius of 80 m.

Analytical grid

The analyzed dataset comprised 3,876 statistical zones with hexagon-shaped cells (see Fig. 6, stage 2). Calculations were performed within these zones to illustrate the spatial variability of abiotic landscape features. The shape and size of the analytical grid cells were determined to ensure methodological consistency with previous evaluations^26,36,55.

It should be noted that the original selection of the analytical grid was inspired by the findings of Parysek¹⁰⁰, who indicated that the most favorable shapes for artificial analytical networks depend on the spatial compactness of the units and the optimal arrangement of cells relative to each other. Using the minimum boundary length of units as a measure of compactness and the minimum centroid distance between adjacent objects as a measure of optimal positioning, Parysek concluded that the most compact analytical network is a structure composed of regular hexagons, while the most advantageous spatial arrangement is a structure of squares arranged in a “brick-like” pattern. This latter layout was adopted by the author for the statistical zones.

The size of the statistical zones was determined based on the autocorrelation ranges of the continuous regionalized variables used^101,102,103. Suchożebrski¹⁰⁴ demonstrated that statistical zones are homogeneous when their size is 3–5 times smaller than the autocorrelation radius of the analyzed variable. Preliminary tests revealed that when multiple variables are used, each characterized by a different autocorrelation radius, a compromise must be reached by adopting an intermediate autocorrelation radius to determine the grid cell size. Based on this, the optimal size of the hexagonal cells, measured along their shorter diameters, was defined as 200 m. Consequently, during the analytical stage, each grid cell contained approximately 1,385 raster pixels representing the variability of individual landscape features in the Pieniny Mts region. The procedure for selecting the analytical network was described in detail by Bartuś & Mastej²⁶.

Partial criteria

The analyses were based on a set of indicators hereinafter referred to as partial criteria (see Fig. 6, stage 2). These are specific measures describing the properties of the natural environment, used for the classification and evaluation of physiogeographical phenomena³⁸. The application of the RCM_SDcm model concept⁵⁵ required the use of a specific set of partial criteria.

For data derived from the DEM (Altitude, Aspect, Slope, Plan curvature, Profile curvature, and Local denivelation), the variability of landscape features was calculated using scalar and circular standard deviations (SD, SDc). As indicated by Bartuś & Mastej⁵⁵, the scalar standard deviation (SD) can be applied to angular features with low variability that do not simultaneously span the first and fourth quadrants of the full circle, whereas the circular standard deviation (SDc) is suitable for parameters with a full range of directional variability (e.g., Aspect).

In the present analyses, the modified circular standard deviation (SDcm) introduced by Bartuś & Mastej⁵⁵was also applied. This measure eliminates random fluctuations in the values of features within flat or gently inclined areas, where the variability of slope-related parameters is negligible.

Additionally, in the present study, the RCM_SDcm model was extended by introducing an additional criterion — the presence of rock-outcrop forms. Vector data for this feature were obtained by digitizing archival topographic maps at a scale of 1:5000. Within each statistical zone, the variability of this categorical feature was calculated using the Shannon–Weaver diversity index (SHDI) (1)¹⁰⁵.

$$\:{SHDI=-\sum\:_{i=1}^{k}({P}_{i}\cdot\:\text{ln}{P}_{i})\:[-]}_{}$$

(1)

Explanations: i—landscape feature (patch) category, k—total number of landscape feature categories, P_i—proportion of the landscape occupied by category i (i.e., the probability of occurrence of a patch of category i in the landscape).

Standardization

To ensure equal weighting of individual partial criteria in the applied morphodiversity models, all variables were scaled to a common range of variability (0–1). Standardization was performed using the min–max method (2) (Table 1). This approach was selected due to its computational simplicity and the robustness of DEM-derived data against outliers.

$$\:{x}_{i}^{{\prime\:}}=\frac{({x}_{i}-{x}_{min})}{{x}_{max}-{x}_{min}}$$

(2)

Explanations: x_i'—feature value after standardization; x_i—feature value before standardization; i – i^th statistical zone; x_min, x_max—minimum and maximum value of the feature set before standardization.

Table 1 Used partial criteria.

All morphodiversity evaluations considered were also subjected to the standardization procedure.

Morphodiversity modeling using the RCM model and artificial neural networks

The assessment of the morphodiversity of the Pieniny Mts landscape and its surroundings using the RCM model and Artificial Neural Networks (ANN) consisted of six stages (Fig. 6).

Fig. 6

Flowchart of the main steps of the analysis.

The procedure of modeling morphodiversity using RCM criteria and the SC-ANN methodology refers only partially to the geodiversity definition presented in the introductory chapter, which postulates the aggregation of partial criteria ratings (so-called AR models). Studies by Bartuś²⁴, Mastej & Bartuś³⁶, and Bartuś & Mastej⁵⁵ indicate that in assessments conducted using classical methods, a significant portion of the variability in geo- and morphodiversity is redundant. This is evidenced by the typically high correlation coefficients between partial criteria, as well as by results from informativeness and redundancy analyses. Redundancy in morphodiversity analyses is likely related to the use of the same data source for calculating basic and secondary topographic attributes (DEM derivatives). This is an unfavorable phenomenon that should be assessed and minimized during analyses.

Effective reduction of the influence of redundant or strongly correlated variables is possible through supervised classification (SC) methods. ANN are particularly useful in this context, as they automatically assign lower weights to less informative variables during the optimization of the learning process. Consequently, ANNs limit the contribution of such variables in the models, thereby producing more accurate predictive models.

Artificial neural networks

ANN are computational models inspired by the structure and functioning of the human brain. They consist of many interconnected units – artificial neurons. Each neuron receives incoming signals through its inputs, often referred to as dendrites in analogy to biology, weights these signals, sums (aggregates) them, transforms the result using an activation function, and then passes it forward through its output (axon) to subsequent neurons (Fig. 7).

Fig. 7

Artificial neuron model.

The aggregation of signals inside artificial neurons (denoted by the symbol Σ in Fig. 7) in so-called linear neurons consists of summing the inputs multiplied by their assigned weights (w₀, w₁, w₂,…, w_n), which reflect the importance of each connection. The aggregated signal is then transformed by a usually nonlinear activation function, which determines whether and to what extent the signal will be passed on further. Commonly used activation functions include exponential, Gaussian, identity, sigmoid, sine, softmax (particularly useful in multi-class classification), tanh, and sometimes linear functions.

In this study, one of the most widely used neural network architectures – the Multilayer Perceptron (MLP)¹⁰⁶ – was applied. It consists of at least three layers of neurons: input, hidden, and output layers (Fig. 8). The input layer receives the analysis input variables – in this case, the standardized values of seven partial criteria (Table 1). The number of input signals determines the number of input-layer neurons. At the start of the analysis, each connection is assigned a random weight, which is subsequently adjusted during the iterative network training process (see Trainingpatterns).

Fig. 8

Example of an RCM_SC-ANN analysis model (MLP-7-4-2) using a multilayer perceptron with 7 partial criteria inputs in the input layer, 4 neurons in a single hidden layer, designed for two-class classification.

The presented analyses were conducted using Statistica 14.1.0.4 software. The user has limited control over the structure of the hidden layer, with the possibility to specify only the minimum and maximum number of hidden-layer neurons. In this study, this range was set from 4 to 12 neurons.

The number of neurons in the output layer depends on the nature of the output variable. For continuous variables, it corresponds to the number of these variables, whereas for discrete variables, it equals the number of possible categories. Although geodiversity (and morphodiversity within it) is usually classified into more than two classes (e.g., very low, low, medium, high, and very high diversity), this study applied a simplified two-class division into morphodiverse (MD) and non-morphodiverse (NMD) objects. Therefore, the output layer of each analyzed ANN always contained two neurons (Fig. 8).

The outputs of the activation functions for these neurons are continuous values, which can be interpreted as the probability of an object belonging to the MD class (first neuron) or the NMD class (second neuron). If necessary, after standardization, these outputs can be discretized to assign objects to a specific morphodiversity class. Continuous outputs provide the most information⁵⁵, which is why all statistical analyses in this study used continuous activation function outputs. Discretization was applied solely for visualization purposes on cartograms.

The architecture of the analyzed ANN is reflected in its name. For example, the code MLP-7-4-2 denotes an MLP-type network with seven neurons in the input layer, four hidden-layer neurons, and two output-layer neurons. For clarity, the presented ANN codes were additionally preceded by a numerical identifier characteristic for each analysis.

As mentioned earlier, ANN training proceeds iteratively in so-called epochs. Traditionally, the Error Backpropagation (EBP) algorithm is used, which involves backward calculation of gradients from the output to the input layer and updating the weights. However, in the Statistica environment, a more advanced optimization algorithm, Broyden–Fletcher–Goldfarb–Shanno (BFGS)¹⁰⁷, was utilized, providing an efficient alternative to traditional gradient-based methods.

Training patterns

The learning process of ANN can follow two main approaches: supervised learning or unsupervised learning. For classification tasks, supervised learning is most commonly used. In this approach, the neural network learns to formulate classification rules based on provided examples. This approach was also adopted in the present study. It required defining so-called training sets – collections of data consisting of objects described by input variable vectors, along with assigned expected class memberships. In our case, the objects were selected statistical zones, and the dependent variable was binary (MD and NMD) (see Fig. 6, stage 4a).

Proper construction of training sets requires that patterns be defined independently of the input data. In practice, they should be based on well-defined independent criteria or expert knowledge. It is also important that training sets be as large as possible. The larger the training set, the better the classification results. Moreover, large sets allow for the inclusion of a greater number of input variables in the model. In practice, however, researchers rarely have access to extensive training datasets, which often necessitates various compromises.

An important rule in creating training sets is to ensure clear inter-class differentiation while ensuring adequate intra-class variability. As experience shows, using overly homogeneous patterns within a class can lead to too restrictive and less accurate classification results.

The set of objects with known class membership is traditionally divided into three parts: the training set (T), the test set (U), and the validation set (V). Each has a distinct purpose. The training set is used during the network’s learning phase. It is repeatedly presented to the neural network over successive epochs to update weights and build a model of internal relationships. The test set is used concurrently with training to monitor model generalization and to prevent overfitting – the excessive adaptation of weights to the training data. The validation set contains completely unseen objects by the classifier and is used to assess classification performance after training is complete.

Typically, networks perform well in recognizing the T and U sets (especially T) – this reflects approximation capability – but struggle more with recognizing unknown objects from the V set, which reflects generalization capability. Additionally, generalization may deteriorate due to overfitting.

In the present analyses, classification quality was measured using the Area Under the Curve (AUC), that is, the area under the Receiver Operating Characteristics (ROC) curve¹⁰⁸, and the number of correctly classified objects in the T, U, and V sets (confusion matrices).

Given the special importance of the training set, it is standard practice to allocate 70% of the patterns to T, and 15% each to the U and V sets. There are also various more or less conservative rules for determining the required size of ANN training sets. One such rule states that the training set size should be at least 10 times the number of independent variables¹⁰⁹. In our case, with seven independent variables, the training set should therefore include approximately 70 objects. This number of T patterns – and accordingly of U and V sets – was adopted in the present analysis.

The study was conducted in two variants: RCM_SC−ANN and RCM_SC−ANN−M. Separate training sets were prepared for each variant, mostly based on different reference objects (see Fig. 6, stage 4a). Each set included 210 reference objects (approximately 5.5% of the total number of objects), with 105 assigned to the MD_{RCM−SC−ANN} class and 105 to the NMD_{RCM−SC−ANN} class (and analogously, for the MD_{RCM−SC−ANN−M} and NMD_{RCM−SC−ANN−M} classes). As previously mentioned, each of the T, U, and V subsets contained 70 objects – 35 from the MD class and 35 from the NMD class. In both experiments, the input data consisted of the same partial criteria of analysis (Table 1).

In the RCM_SC−ANN analysis variant (as well as its derivatives RCM_{SC−ANN−BA} and RCM_{SC−ANN−BSA} – see Backward Analysis,Backward Stepwise Analysis), the spatial distribution of the training set objects was identical to that used in the morphodiversity SC-ANN models reported by Bartuś & Mastej³⁶. Figure 9 presents the locations of the training patterns for the RCM_SC−ANN analysis variants overlaid on the Terrain Relief (TR) model³⁶ – an independent morphodiversity model developed based on a geomorphological sketch generalized to the level of 1:100,000-scale maps¹¹⁰.

Fig. 9

Patterns in the data used for training the neural networks in the RCM_SC-ANN variants of the analysis (pattern locations after Mastej & Bartuś³⁶. SC-ANN pattern classes: 1—non-morphodiversity class (NMD_RCMSC-ANN); 2—morphodiversity class (MD_RCMSC-ANN). Symbols within the objects indicate: T—training sample; U—test sample; V—validation sample. Terrain Relief (TR) model classes: 3—non-morphodiversity class (NMD_TR); 4—morphodiversity class (MD_TR). Peak types: 5—dome-shaped; 6—cone-shaped. Ridge types: 7—broad, rounded; 8—narrow, rounded. Valley types: 9—flat-bottomed; 10—V-shaped; 11—antecedent gorge; 12—alluvial terrace. Other landforms and features: 13—flattened surface; 14—landslide; 15—stone rubble; 16—tors and klippen; 17—rivers and streams; 18—water bodies; 19—Pieniny National Park; 20—national border.

The MD_{RCM−SC−ANN} class patterns were located in the most naturally attractive regions of the Pieniny Mts and adjacent areas. It was assumed that their attractiveness stemmed from their morphodiversity. These areas included mountain peaks, rock outcrops, cliff faces of the Dunajec River Gorge, scenic viewpoints, and similar landforms. In contrast, the NMD_{RCM−SC−ANN} class patterns were derived from built-up areas that are morphologically favorable for human activity and settlement, as well as from flat valley floors and the surfaces of the Czorsztyn and Sromowce Lakes.

Including the archival training dataset in the analyses enabled a comparison between the evaluations of RCM models computed using ANN and the assessments of the best SC-ANN model reported by Mastej & Bartuś³⁶ — SC-ANN_{VSR−9−BSA}.

The evaluations of the SC-ANN_{VSR−9−BSA} model yielded satisfactory results, as reported in the aforementioned study. However, detailed comparisons with the evaluations of the RCM_SDcm model⁵⁵ revealed that the statistical zones of the training patterns were not as internally homogeneous as initially assumed. This was reflected in the low value of the Spearman’s rank correlation coefficient between the evaluations of the RCM_SDcm and SC-ANN_{VSR−9−BSA} models. These findings prompted the author of the present study to develop a new set of training patterns.

In this case, the selection of patterns was based on a qualitative assessment of the diversity of the main landform types. The patterns were delineated using the previously described TR model and a shaded relief representation. Statistical zones were classified as morphodiversity patterns (MD_{RCMSC−ANN−M}) when two or more morphological forms occurred within their boundaries, e.g., mountain summits, planation surfaces, slope forms, or denudational residuals. Conversely, objects assigned to the NMD_{RCMSC−ANN−M} class were defined as morphologically homogeneous areas. Qualitative interpretation of the shaded relief also aided the assessment process: areas exhibiting abrupt tonal contrasts were considered more morphodiverse than those with uniform shading. It should be emphasized, however, that this approach was used only as a supplementary method.

The selected training objects in the RCM_SC−ANN−M model underwent qualitative validation using two geomorphometric models: Slope Position Classification (SPC)⁹⁸ and Geomorphon¹¹¹, and, in part, through field verification. These procedures confirmed the representativeness of the training objects and their suitability as reference patterns for supervised classification. In the SPC model, as in the case of the TPI index, an observation scale of 80 m was applied, while for the Geomorphon method a search distance of 2000 m was used. Objects belonging to the morphodiversity class (MD_{RCMSC−ANN−M}) were positively verified when both validation models indicated high variability of landforms within the statistical zones. Conversely, objects of the non-morphodiversity class (NMD_{RCMSC−ANN−M}) were considered correctly verified when both validation models showed little or no variation in landform types. Figure 10 presents selected examples of the verification of the RCM_SC−ANN−M model training patterns.

Fig. 10

Validation of ANN training patterns in the RC_MSC-ANN-M model. Rows present selected examples of the NMD_RCMSC-ANN-M class training patterns from the southern slopes of the Gorce Mts (A) and from the Small Dunajec River Gorge (D), as well as selected examples of the MD_RCMSC-ANN-M class from the Pieniny Spiskie Mts near Czorsztyn Lake (B), the Trzy Korony Mt area (982 m a.s.l.) (C), and from deeply incised valleys and ridges of the Sącz Beskid Mts (E) (see Fig. 11). Columns: I—TR model³⁶; II—SPC validation model⁹⁸; III—Geomorphon validation model¹¹¹. SC-ANN pattern classes: 1—non-morphodiversity class (NMD_RCMSC-ANN-M), 2—morphodiversity class (MD_RCMSC-ANN-M); TR model classes: 3—non-morphodiversity class (NMD_TR), 4—morphodiversity class (MD_TR); other TR landforms and features as in Fig. 9: 5—ridge, 6—upper slope, 7—middle slope, 8—flat slope, 9—lower slope, 10—valley, 11—flat, 12—peak, 13—ridge, 14—shoulder, 15—spur, 16—slope, 17—hollow, 18—footslope, 19—valley, 20—pit.

In practice, in the new training dataset, the MD_{RCMSC−ANN−M} class patterns encompassed the most geomorphologically and visually distinctive areas of the Pieniny Mts, including mountain peaks and ridges, the highest and most rugged rocky sections of the range, parts of the Dunajec River Gorge, deeply incised V-shaped valleys encompassing opposite slopes, landslide zones, and areas showing high light–shadow contrast on the shaded relief (Fig. 11). The NMD_{RCMSC−ANN−M} class patterns included lower, gently sloping or flat areas on the southern slopes of the Gorce Mts and the northern slopes of the Pieniny Mts, ridge sections with planation surfaces, flat portions of major valleys, alluvial terraces, floodplains, and the surfaces of the Czorsztyn and Sromowce lakes.

Fig. 11

Patterns in the data used for training the neural networks in the RCM_SC-ANN-M variants of the analysis. RCM_SC-ANN-M pattern classes: 1—non-morphodiversity class (NMD_RCMSC-ANN-M); 2—morphodiversity class (MD_RCMSC-ANN-M). Other symbols as in Fig. 9.

In all analysis variants, the procedure for determining the best morphodiversity model involved the automated testing of 100 artificial neural networks (ANNs) and the selection of the five best-performing models. The quality of each network was evaluated using learning error functions: either the Sum of Squares Error (E_SOS) (3) or the Cross Entropy Error (E_CE) (4). Networks that produced higher prediction errors were discarded, while the top-performing ones, generating outputs with the lowest error values, were retained for further analysis.

$$\:{E}_{SOS}=\sum\:_{i=1}^{N}{\left({y}_{i}-{t}_{i}\right)}^{2}$$

(3)

Explanations: N—number of training data samples, y_i—ANN prediction (output) for the i-th case, t_i—target value for the i-th case.

$$\:{E}_{CE}=-\sum\:_{i=1}^{N}{t}_{i}\text{ln}\left(\frac{{y}_{i}}{{t}_{i}}\right)$$

(4)

Explanations: Symbols as in 3 equation.

The five ANNs that demonstrated the best training performance were subsequently subjected to individual evaluation based on the consistency between the predicted class membership of the statistical zones and the expected memberships defined in the T, U, and V datasets (confusion matrices). The greatest emphasis was placed on the prediction quality obtained for the validation set (V). The best-performing ANNs (hereafter referred to as morphodiversity models) were then used to generate evaluation datasets for all statistical zones, which were visualized in the form of evaluation maps. These datasets were further compared with the evaluation outputs of other morphodiversity models. Additionally, for the five best ANNs from both the RCM_SC−ANN and RCM_SC−ANN−M variants, an analysis of the contribution and quality of the independent variables was conducted (see Global Sensitivity Analysis).

Global sensitivity analysis

Apart from the training set size, the quality of supervised classification is closely linked to the quality of the independent variables. Artificial neural networks (ANNs) handle this issue effectively by reducing the influence or setting the weights of input variables close to zero when these variables exhibit low informational value or data redundancy. As previously mentioned, the findings presented in Bartuś²⁴, Mastej & Bartuś³⁶, and Bartuś & Mastej⁵⁵ indicate that such situations frequently occur in geo- and morphodiversity analyses. Variables of this kind can be excluded from the model, which helps reduce computational cost.

One useful method for assessing the informational contribution of independent variables is the Global Sensitivity Analysis (GSA). This approach enables evaluation of the appropriateness of variable selection and determination of how the potential removal of specific variables affects model performance. The sensitivity analysis algorithm operates iteratively: for each input variable, its values in the analyzed dataset are replaced with the mean value from the training population. In this way, the tested variable ceases to influence the model.

The sensitivity of the network to the removal of a given input variable is measured as the ratio of model prediction errors after and before the removal of that variable (5). If this ratio is less than 1, the variable can be safely removed. Removing factors that contribute relatively large amounts of information results in a higher ratio, whereas removing less relevant variables has little or no effect, and in some cases may even improve prediction accuracy. To facilitate interpretation, independent variables are ranked in descending order of their informational contribution.

$$\:{\sigma\:}_{wj}=\frac{{\sigma\:}_{j}}{\sigma\:}$$

(5)

Explanations: σ_wj—the prediction error ratio, σ_j—the prediction error of the ANN after removing variable j, σ—the prediction error of the ANN including all variables.

Backward analysis

The procedure of removing low-informativeness variables, referred to here as Backward Analysis (BA), involved a one-step elimination of input variables occurring at the point of a sudden decrease in informativeness, rather than – as previously mentioned – at the point where the error ratio σ_wj fell below 1. As a result, several variables were eliminated from the model simultaneously. Contrary to intuitive expectations, this outcome is not necessarily undesirable. ANNs, when stripped of variables contributing only minimal information, are often able to compensate for such losses effectively.

However, experience shows that when the ANN’s output variable is categorical, and the number of input variables is relatively small, the elimination of multiple input variables may lead to a transformation of the originally continuous distributions of class membership probabilities (e.g., MD and NMD) into a binary distribution. This occurs because ANNs are capable of performing classifications even with a very limited number of input variables. When the goal is to maintain a continuous distribution of output probabilities, this effect becomes undesirable. In the present study, it was counteracted by limiting the number of eliminated variables to 3 (out of 7).

When analyzing a larger number of ANNs, some input variables consistently carry important information. These can be considered key variables within the model. Other variables consistently prove to be uninformative in sensitivity analyses and can usually be safely removed without degrading model performance. There are also intermediate variables – those that show high importance in some models and low importance in others. These are most likely redundant variables. Input variables identified for elimination should exhibit similar characteristics across all analyzed networks.

Nonetheless, the BA approach may occasionally lead to significant errors. Numerous experiments¹¹² have demonstrated that removing variables with low standalone informativeness – but correlated with other valuable variables – may reduce the apparent importance of otherwise valuable features, causing them to drop in the ranking.

Backward Stepwise analysis

An extension of the BA method that addresses the aforementioned drawback is Backward Stepwise Analysis (BSA). This procedure involves gradually removing input variables occupying the lowest positions in the informativeness ranking – one variable at a time in successive steps of the analysis. At each step, after eliminating the least informative variable, the entire process of building and training the ANN must be repeated, followed by another GSA. Although this procedure can sometimes yield good results, it is very labor-intensive and does not guarantee the identification of a single optimal subset of informative variables. An ideal algorithm would need to test all possible subsets of the input variable set, which is practically infeasible for larger datasets. For this reason, the BA approach was retained, which later allowed for a direct comparison of the outcomes produced by both procedures.

Comparison of the morphodiversity models

Morphodiversity evaluations face a significant challenge due to the lack of independent assessments. Attempts have been made to create comparative models based on morphological sketches, such as the TR model referenced in Mastej & Bartuś³⁶ and Bartuś & Mastej⁵⁵. Unfortunately, mismatches between their resolution and that of the presented analyses, as well as the binary distribution of evaluations (MD and NMD), limit their usefulness. In this context, the degree of convergence among model assessments can serve as a measure of model quality.

In the present study, the evaluation populations obtained from morphodiversity models – RCM_SC−ANN, RCM_SC−ANN−M, RC_{MSC−ANN−BA}, RCM_{SC−ANN−M−BA}, RCM_{SC−ANN−BSA}, and RCM_{SC−ANN−M−BSA} – were supplemented with two evaluation datasets known from the literature for the Pieniny Mts and adjacent areas: SC-ANN_{VSR−9−BSA}³⁶ and RCM_SDcm⁵⁵. The selection of archival models was motivated by their high performance and their similarity to RCM evaluations obtained using ANN. For the SC-ANN_{VSR−9−BSA} model, the analogy lay in the application of the same ANN methodology, while the difference was the use of a completely different set of partial criteria. For the RCM_SDcm model, the discrepancy was in the computational methodology, whereas the similarity lay in the use of comparable partial criteria.

The evaluation populations were subjected to statistical description. Evaluation effectiveness was analyzed by examining the convergence of the obtained results (see Fig. 6, stage 5). Comparison tools included Spearman’s rank correlation and spatial analyses of similarities and discrepancies. The latter were performed using bivariate choropleth cartograms¹¹³, with a default 3-class discretization based on the quantile method. An exception was made for the SC-ANN_{VSR−9−BSA} evaluation, where the bimodal distribution of assessments required the use of the natural breaks classification method¹¹⁴.

SC-ANN _VSR-9-BSA model

SC-ANN _{VSR−9−BSA}³⁶ belongs to the SC group, with the classification performed using ANN. The model development methodology was consistent with that presented in Sect.Morphodiversity Modeling Using the RCM Modeland Artificial Neural Networks. By applying the BSA procedure, nine top-performing input variables were extracted from an initially broad set of 55 partial criteria, primarily based on vector data (V), the SHDI index (S), and raster data (R). These included: Npc_Klippen, Nc_Klippen, ΔZ, ΔSlope, Nc_Slope, Npc_Aspect, Npc_ProfileCurv, Ne_Aspect, and Nc_PlanCurv. Here, Nc denotes the number of homogeneous element categories in vector datasets; Ne, the total number of elements in vector datasets; Npc, the number of pixel categories in raster datasets; ΔSlope, the difference between maximum and minimum slope; and ΔZ, the elevation range. The prediction dataset, containing probability values of object membership in the MD_SC−ANN class, was standardized using the min–max method (2).

RCM _SDcm model

RCM _SDcm⁵⁵ is an example of an AR-type model, in which the final assessment of geo- and morphodiversity depends on the arithmetic sum of partial criteria²⁶. RCM_SDcm differs from conventional AR models by employing raster datasets that describe the variability of landscape elements, as well as partial criteria based on scalar or angular measures of variability. Excluding the Klippen_SHDI criterion, the complete set of applied criteria is presented in Table 1. The evaluation dataset was standardized using the min–max method (2).

Results

Partial criteria of the morphodiversity analysis (see Table 1) were subjected to statistical description and their correlations were examined (see Fig. 6, stage 3). As a result of morphodiversity modeling using ANNs and two independent training datasets (see Fig. 6, stage 4a), two primary morphodiversity models were established – RCM_SC−ANN and RCM_SC−ANN−M (see Fig. 6, stage 4b). The study then focused on their optimization to eliminate data redundancy and streamline the models (see Fig. 6, stage 4c).

Partial criteria

As demonstrated by numerous studies^24,26,36, the partial criteria used in geo- and morphodiversity analyses often exhibit very strong correlations. This is likely related to common sources of certain abiotic landscape features, such as soil cover and geological structure, or DEM and slope, or aspect. It can be suspected that these correlations may have a cause-and-effect nature, and the partial criteria describing these landscape features may share some of their variability with other criteria.

The statistical description of the partial criteria aimed to characterize the variable populations and select an appropriate method for analyzing the relationships among them.

Descriptive statistics

The partial criteria exhibit high coefficients of variation (Table 2). This is a desirable feature because it translates into a high differentiation of values across individual statistical zones. The distributions of the analyzed variables deviate from the normal distribution – they are slightly to strongly right-skewed and, except for Altitude_SD, Aspect_SDc, and Slope_SD, are leptokurtic. For CurvVert_SD and Klippen_SHDI, greater asymmetry (right skewness) and more platykurtic distributions can be observed compared to the distributions of the other criteria. Deviations from normality, indicated by skewness and kurtosis coefficients, are also reflected in statistical significance tests not reported here. Due to the large sample size (3,876 objects), such tests detect statistical significance even for minor deviations from normality, making them of limited use here. These deviations from normality necessitated abandoning correlation analysis using the classical Pearson correlation coefficient and instead applying Spearman’s rank correlation coefficient¹¹⁵.

Table 2 Descriptive statistics of the partial criteria (before standardization). Min—minimum value of the variable; Max—maximum value of the variable; SD—standard deviation; SD_c—circular standard deviation. Other abbreviations as in Table 1.

Correlations

Some of the Spearman’s rank correlations between the morphodiversity analysis criteria had already been reported in the description of the RCM_SDcm model⁵⁵. Here, they have been supplemented with correlations involving the Klippen_SHDI criterion (Fig. 12). Strong correlations between certain pairs of criteria in the current analysis (e.g., CurvPlan_SD vs. CurvVert_SD and CurvPlan_SD vs. TPI_SD) indicate probable data redundancy. The Klippen_SHDI partial criterion shows weak correlations with the other variables, which is most likely due to the spatially patchy distribution of klippe-type rock formations in the studied landscape.

Fig. 12

Spearman’s rank correlation coefficient (R_S) [–] for the continuous partial criteria used (after standardization). Abbreviations as in Table 1.

The suspected redundancy can be eliminated from the model by removing one of the strongly correlated variables. However, if this is done in an uncontrolled manner, it may lead to the loss of an important part of the variable’s unique variability. The use of ANNs solves this problem. As previously mentioned, neural networks are capable of reducing or even eliminating the influence of correlated or low-informative variables through the weighting of input signals. To gain a deeper understanding of the mechanisms operating between variables, two GSA procedures were added to the standard ANN-based morphodiversity evaluation workflow (see Fig. 6, stage 4c; Sect. Elimination of Low-Informative Variables).

Selection of the best ANN models

Using the training datasets of RCM_SC−ANN and subsequently RCM_SC−ANN−M (see Fig. 6, stage 4a), 100 artificial neural networks were trained for each case (see Fig. 6, stage 4b). Comparison of the values of the learning error functions — E_SOS (3) and E_CE (4) — enabled the automatic selection of the five best-performing networks for each analysis variant.

RCM _SC-ANN model

The best-performing ANNs trained on the RCM_SC-ANN training dataset — presented in the study by Mastej & Bartuś ³⁶ (see Fig. 9) — used between 4 and 12 neurons in the hidden layer and were trained for between 4 and 11 epochs (Table 3). The activation functions applied to the hidden layer neurons included sigmoid, exponential, sine, and tanh, while the output layer neurons used identity, tanh, and sigmoid functions. The prediction accuracy of the ANNs, as measured by the E_SOS learning error function, ranged from 91.43 to 97.13 (for the T set), 91.43 to 95.71 (for the U set), and 92.86 to 98.57 (for the V set).

Table 3 Summary of the training results for the five best ANNs in the RCM_SC-ANN variant of the analysis. See text for details.

The most important and positive outcome of the training process was the generally higher prediction accuracy achieved on data not previously presented to the classifier (V), indicating a high generalization ability of the analyzed ANNs. The best-performing network of the RCM_SC−ANN model was identified as 2 MLP 7-7-2 (Table 3). It is characterized by a moderate number of neurons in the hidden layer and a relatively long training process but achieved the highest prediction accuracy across all three datasets (T, U, and V).

Analysis of the confusion matrices (not reported here) indicates that this network correctly classified 99.0% of the NMD_RCMSC−ANN class cases (only one object misclassified) and 95.2% of the MD_RCMSC−ANN class cases (five objects misclassified). The average classification accuracy across both classes was 97.1%, which was the highest among all analyzed ANNs. The 2 MLP 7-7-2 network was therefore selected for the morphodiversity evaluation in the RCM_SC−ANN model (see Sect.RCM_SC-ANN-Mmodel evaluations).

RCM _SC-ANN-M model

The five best-performing ANNs trained on the new RCM_SC−ANN−M training dataset (see Fig. 11) employed between 4 and 10 neurons in the hidden layer, and their training lasted from 3 to 18 epochs (Table 4). The activation functions applied to the hidden layer neurons included sigmoid, sine, and identity, while the output layer neurons used softmax, exponential, and identity functions. The prediction accuracy of these ANNs was higher than that of the RCM_SC−ANN model (Table 3), ranging from 97.15% to 100% for the test set (U) and reaching a perfect 100% for the training (T) and validation (V) sets. The maximum accuracy achieved on the validation set confirms the high generalization capability of the ANNs trained within the RCM_SC−ANN−M model.

Table 4 Summary of the training results for the five best ANNs in the RCM_SC-ANN-M variant of the analysis. See text for details.

The best-performing RCM_SC−ANN−M network was 1 MLP 7–10-2 (Table 4). It was characterized by a high number of neurons in the hidden layer and a long training process lasting 18 epochs. However, it achieved perfect prediction accuracy for all three datasets: training (T), test (U), and validation (V). This means that the network made no classification errors when predicting the class membership of NMD_{RCMSC−ANN−M} and MD_{RCMSC−ANN−M} objects across the combined T, U, and V sets. The 1 MLP 7–10-2 network was used for morphodiversity evaluation prediction in the RCM_SC−ANN−M model (see Sect.RCM_SC-ANN-M model).

Spatial distribution of the morphodiversity evaluations obtained using the best ANN models

The best-performing neural morphodiversity models selected in the previous step — RCM_SC−ANN and RCM_SC−ANN−M — were used to generate morphodiversity predictions for the statistical zones. These predictions were then used to create choropleth maps illustrating the spatial variation in evaluation scores. For ease of interpretation, the continuous evaluation values were discretized into five bonitation classes using the natural breaks method.

RCM _SC-ANN model evaluations

The distribution of standardized continuous scores generated by the RCM_SC−ANN model was distinctly positively skewed, closely approximating an exponential distribution. The calculated skewness coefficient was 0.37, while the kurtosis reached − 0.97. The score population exhibited high variability, amounting to 62.4%.

The largest proportion of statistical zones (33%) was characterized by very low morphodiversity (Fig. 13). Such areas are mainly located in the lower parts of the southern slopes of the Gorce Mts, within the morphologically less diverse sections of the Pieniny Spiskie Mts, at both the northern and southern foothills of the Pieniny Właściwe Mts, and at the southern foothills of the Sącz Beskid Mts. They also occasionally occur along the lower slope zones of the aforementioned massifs and within broad, U-shaped, flat-bottomed segments of the main valleys.

Within the PNP, areas of very low morphodiversity were identified around Polana Majerz — between Czorsztyn and Hałuszowa — in the mountain clearings southeast of the village of Tylka, and in the area situated north of the main ridge of the Pieniny Właściwe Mts. A balanced distribution of low, medium, and high morphodiversity scores was also observed, with approximately 20% of statistical zones in each class.

Fig. 13

RCM_SC-ANN morphodiversity model evaluation for the study area, developed using ANN 2 MLP 7-7-2. The map presents a five-class discretization of the standardized probability of object affiliation with the MD_RCMSC-ANN class, based on the natural breaks method. Morphodiversity levels: 1—very low; 2—low; 3—medium; 4—high; 5—very high. Other map features: 6—klippen; 7—rivers and streams; 8—water bodies; 9—Pieniny National Park; 10—national border.

Statistical zones characterized by low morphodiversity are mainly located on slopes along the more diversified portions of the lower and upper slopes across all massifs of the region, as well as within weakly incised valley segments in the lower parts of the Pieniny Spiskie and Gorce Mts.

Medium morphodiversity occurs in higher or deeply incised segments of V-shaped valleys, middle-slope zones, and steeper sections of both lower and upper slopes. It is also observed at the foothills of the most morphologically diverse parts of the Pieniny Mts and within morphologically less varied areas that nonetheless display high diversity in one of the analytical criteria, such as the presence of rock formations.

In the analyzed model, high morphodiversity characterizes the upper valley sections situated within upper slopes, as well as the higher but rock-free portions of the Pieniny Właściwe, Sącz Beskid, and Gorce Mts massifs.

Very high morphodiversity was identified in approximately 9% of statistical zones within the PNP, including the highest parts of the Pieniny Właściwe Mts, the most scenically attractive sections of the Dunajec River Gorge, and the PNP enclaves located west of the park’s main area — particularly near the villages of Czorsztyn and Falsztyn. These zones are typically characterized by the presence of rock formations. Outside the PNP, statistical zones with the highest morphodiversity occur in the summit regions of the Sącz Beskid Mts, within higher and deeply incised V-shaped valleys, and occasionally in the highest parts of the Gorce Mts. A few such zones are also found in the Pieniny Spiskie Mts.

RCM _SC-ANN-M model evaluations

The population of standardized morphodiversity evaluations for the Pieniny Mts region, calculated using the RCM_SC−ANN−M model, exhibits a positively skewed distribution, approximating a geometric model. The skewness coefficient is 0.23, and the kurtosis is −1.2. The evaluations show high variability (74.6%).

Discretization using the natural breaks method¹¹⁴ classified 21% of statistical zones as very low morphodiversity (Fig. 14). These zones are predominantly located in flat or gently sloping areas within main river and stream valleys, on alluvial terraces, and on lower-lying flattened surfaces. Most are found at the southern foothills of the Gorce Mts, the base of the northern slopes of the Pieniny Właściwe Mts, wider sections of the Dunajec River valley, southern foothills of the Sącz Beskid Mts, and various parts of the Pieniny Spiskie Mts.

Fig. 14

RCM_SC-ANN-M morphodiversity model evaluation for the study area, developed using ANN 1 MLP 7–10-2. The map presents a five-class discretization of the standardized probability of object affiliation with the MD_RCMSC-ANN-M class, based on the natural breaks method. Map features as in Fig. 13.

A low morphodiversity level characterizes 31% of the statistical zones, making it the most frequent class in the dataset. These zones are widely distributed across the study area, forming small clusters, and occur outside incised valleys across lower, middle, and upper slopes. They are particularly numerous along the ridges of the Pieniny Spiskie Mts, while the fewest are located in U-shaped valleys and in the highest parts of the Pieniny Mts and Sącz Beskid Mts massifs.

Medium morphodiversity was observed in 29% of zones. These are located in higher parts of all massifs, commonly within V-shaped mountain valleys on the slopes but outside the valley bottoms. Fewer medium-level zones occur in main U-shaped valleys and on lower slopes.

High morphodiversity characterizes 16% of statistical zones. These zones are mainly found in higher parts of the Sącz Beskid, Pieniny Właściwe, and Gorce Mts, including valley bottoms, ridges, and middle and upper slopes. In lower parts of the region, they are confined to deeply incised valleys and generally occur in areas lacking rock formations.

Very high morphodiversity is present in only 3% of zones, mostly within the PNP, particularly in the highest parts of the Pieniny Właściwe Mts. Its occurrence is strongly associated with rock formations. Outside the park, larger concentrations occur only in the Sącz Beskid Mts, forming elongated clusters along deeply incised V-shaped valleys, and occasionally along ridges. In the Gorce Mts, very high zones mostly appear as isolated units in higher parts of the massif, mainly along valleys. Single zones also occur along the rocky ridges of the Pieniny Spiskie Mts and in valleys. Several zones are located along the Krośnica stream valley, separating the Gorce and Pieniny massifs.

Elimination of Low-Informative variables

The observed strong correlations between pairs of partial criteria in the morphodiversity analysis (see Fig. 12) justified the application of GSA (see Fig. 6, stage 4c), which allows for the removal of low-informative and redundant variables from the models. This procedure could both improve model performance and simplify the models. GSA was subsequently applied to both morphodiversity models (RCM_SC−ANN and RCM_SC−ANN−M) using two independent computational procedures: BA and BSA (see Sect. Global sensitivity analysis).

GSA, RCM _SC-ANN-M model

GSA of the five best-performing ANNs of the RCM_SC−ANN variant (see Table 3) enabled the calculation of average prediction error ratios (σ_wj) and the construction of a ranking of input variable informativeness (Table 5). Typically, each partial criterion in geodiversity modeling contributes some, usually small, amount of information to the model^24,36, and the same situation was observed here.

Table 5 Results of the global sensitivity analysis based on the combined T, U, and V datasets and the five best ANNs of the RCM_SC-ANN model. See text for details.

The GSA conducted on the combined T, U, and V sample sets of the RCM_SC−ANN model indicated that none of the partial criteria was completely uninformative. Variables of slightly higher significance included TPI_SD, Altitude_SD, and Slope_SD. The prediction error ratios for these criteria consistently demonstrated their high usefulness for the analyzed ANNs.

The variable Aspect_SDcm exhibited variable behavior: some networks (e.g., 2 MLP 7-7-2) indicated it as highly useful, whereas others (e.g., 5 MLP 7–9-2) showed improved prediction performance when it was removed. This is typical for redundant variables correlated with other predictors, such as Slope_SD or TPI_SD (see Fig. 12).

Finally, the group of variables consisting of CurvPlan_SD, Klippen_SHDI, and CurvVert_SD consistently showed low informativeness, with prediction error ratios ranging from 1.03 to 1.08. In the subsequent BA analysis (see Sect. RCM_SC-ANN-M model), these variables were treated as low-informative and removed. Using the reduced set of partial criteria – TPI_SD, Altitude_SD, Slope_SD, and Aspect_SDcm – a new model, RCM_{SC−ANN−BA}, was created and evaluated following the standard procedure.

GSA, RCM _SC-ANN-M model

Analogously to the RCM_SC−ANN variant, the best-performing networks of the RCM_SC−ANN−M variant (see Table 4) were subjected to GSA. Table 6 presents the ranking of input variable informativeness calculated for the combined T, U, and V sample sets (see Fig. 11). The resulting hierarchy of partial criteria informativeness indicates a generally higher contribution of the variables compared to the previously presented RCM_SC−ANN analysis.

Table 6 Results of the global sensitivity analysis based on the combined T, U, and V datasets and the five best ANNs of the RCM_SC-ANN-M model. See text for details.

Among the input variables, Aspect_SDcm, Klippen_SHDI, CurvPlan_SD and CurvVert_SD showed a clear relative increase in informativeness. Conversely, the remaining criteria – TPI_SD, Altitude_SD, and Slope_SD – exhibited a slight relative decrease in informativeness. Following the principle of minimizing the number of criteria, the variables selected for the BA analysis were Aspect_SDcm, Klippen_SHDI, TPI_SD, and Altitude_SD (see Sect. RCM_SC-ANN-M model).

Backward analysis

For each of the training datasets (RCM_SC−ANN and RCM_SC−ANN−M), the BA procedure was conducted (see Fig. 6, stage 4c). As a result, two additional morphodiversity models were developed: RCM_{SC−ANN−BA} and RCM_{SC−ANN−M−BA}.

RCM _SC-ANN-BA model

As a result of the BA procedure, the CurvPlan_SD, Klippen_SHDI, and CurvVert_SD criteria were simultaneously removed from the RCM_SC−ANN morphodiversity model (see Table 5), after which the ANN training process was repeated (Table 7).

Table 7 Summary of the training results for the five best ANNs in the RCM_SC-ANN-BA variant of the analysis (after the BA procedure). See text for details.

The tested ANNs required a longer training period; however, despite the reduced set of partial criteria, they still achieved very good learning performance. This confirms the redundancy of information within the RCM_SC−ANN analysis variant (cf. Table 3). The best modeling results were obtained using the 1 MLP 4–7-2 network. In the combined T, U, and V datasets, this network made only four errors in predicting NMD_RCMSC−ANN class objects and just two errors in identifying MD_RCMSC−ANN class objects. The 1 MLP 4–7-2 ANN was subsequently used for morphodiversity evaluation prediction in the RCM_{SC−ANN−BA} model.

RCM _SC-ANN-M-BA model

Based on the BA results, three morphodiversity criteria—Slope_SD, CurvPlan_SD, and CurvVert_S – were simultaneously removed from the RCM_SC−ANN−M morphodiversity model (see Table 6), after which the ANN training process was repeated (Table 8).

Table 8 Summary of the training results for the five best ANNs in the RCM_SC-ANN-M-BA variant of the analysis (after the BA procedure). See text for details.

The tested ANNs required a similar number of epochs as in the RCM_{SC−ANN−BA} analyses. Despite the removal of three variables from the dataset, the analyzed ANNs still achieved high prediction accuracy for the training objects. This confirms the previously indicated hypothesis regarding the high level of data redundancy in the RCM_SC−ANN−M analysis variant (cf. Table 4). The best modeling performance was obtained using the 1 MLP 4–6-2 network. During prediction on the combined T, U, and V datasets, this network made no classification errors. The same performance was achieved by the 5 MLP 4–9-2 network, although it used a greater number of neurons in the hidden layer. Ultimately, the 1 MLP 4–6-2 ANN was selected for predicting morphodiversity evaluation in the RCM_{SC−ANN−M−BA} model.

Backward Stepwise analysis

Similar to the BA procedure, BSA was also conducted for both training datasets (RCM_SC−ANN and RCM_SC−ANN−M) (see Fig. 6, stage 4c). In both cases, the procedure followed the same steps (Fig. 15). The starting point was again the GSA results obtained for the complete datasets, including all morphodiversity analysis criteria.

In the first step of the BSA, based on the GSA rankings (see Tables 5 and 6), the variable contributing the least information to the model was removed from each analysis variant. For both models, this variable was CurvVert_SD.

In the second step, 100 ANNs were recalculated for the reduced sets of input variables, and the best-performing network was selected from each. Morphodiversity predictions obtained from these ANNs for the combined T, U, and V sample sets were again subjected to GSA. This time, in the RCM_SC−ANN variant, the least informative variable was Aspect_SDcm, while in the RCM_SC−ANN−M variant it was Slope_SD. These variables were removed, and another 100 ANNs were calculated, from which the best networks were again selected.

In the third step of the BSA procedure, the selected ANNs were once more subjected to GSA. For both analysis variants, the least informative variable was CurvPlan_SD. This criterion was removed from both models, and to maintain the continuous nature of the output neurons’ activation functions, the BSA procedure was concluded at this stage.

For the remaining input variables – Altitude_SD, Slope_SD, TPI_SD, and Klippen_SHDI (RCM_SC−ANN) and Aspect_SDcm, Klippen_SHDI, TPI_SD, and Altitude_SD (RCM_SC−ANN−M) – ANN models were calculated, and the best-performing networks were selected. The chosen networks were 4 MLP 4–10-2 for the RCM_SC−ANN variant and 3 MLP 4-3-2 for the RCM_SC−ANN−M variant (Fig. 15).

Analysis of confusion matrices showed that, during test predictions on the combined T, U, and V sample sets, the best RCM_SC−ANN network made 3 errors for NMD_RCMSC−ANN objects and 2 errors for MD_RCMSC−ANN objects (97.61% accuracy), whereas the best RCM_SC−ANN−M network made 0 errors for NMD_{RCMSC−ANN−M} and 1 error for MD_{RCMSC−ANN−M} (99.52% accuracy). These networks were then subjected to a final GSA (Fig. 15). The neural networks 4 MLP 4–10-2 and 3 MLP 4-3-2 were used to generate the final morphodiversity models, RCM_{SC−ANN−BSA} and RMC_{SC−ANN−M−BSA}, respectively.

Fig. 15

Summary of the BSA procedure in the RCM_SC-ANN-BSA (A) and RCM_SC-ANN-M-BSA (B) variants of the analysis. The orange, dark green, and blue elements (BSA steps 1–3) represent the first three stages of the BSA process, involving the successive elimination of the least important (i.e., least informative) input variables from the ANN morphodiversity models. The numbers shown below indicate the ratios of prediction errors after removing each variable (σ_wj). The purple element illustrates the architecture of the best-performing ANN after eliminating three input variables. The light green element (GSA) presents the result of the GSA conducted after the same three input variables were removed from the model.

Discussion

Predictions obtained from models based on the full set of partial criteria (RCM_SC−ANN and RCM_SC−ANN−M), as well as those derived from the BA procedure (RCM_{SC−ANN−BA} and RCM_{SC−ANN−M−BA}), the BSA procedure (RCM_{SC−ANN−BSA} and RCM_{SC−ANN−M−BSA}), and archival models (RCM_SDcm and SC-ANN_{VSR−9−BSA}), were subjected to correlation analyses and spatial comparisons. This approach aimed to evaluate the degree of similarity among the modeling outputs, assess the quality of the models, and identify recurring patterns.

Descriptive statistics of the morphodiversity evaluations

Morphodiversity predictions from all analysis variants (RCM_SC−ANN, RCM_SC−ANN−M, RCM_{SC−ANN−BA}, RCM_{SC−ANN−M−BA}, RCM_{SC−ANN−BSA}, and RCM_{SC−ANN−M−BSA}), as well as evaluation datasets obtained from the comparative models SC-ANN_{VSR−9−BSA}³⁶ and RCM_SDcm⁵⁵, were standardized and subsequently subjected to statistical description and distribution analysis (see Fig. 6, stage 5).

Continuous morphodiversity evaluation datasets calculated using ANN exhibit similar mean values (Fig. 16). Their average evaluation ranges from 0.37 to 0.53, approximately twice as high as the values from the RCM_SDcm model. The analyzed populations show lower variability than the SC-ANN_{VSR−9−BSA} model and roughly double the variability compared to the RCM_SDcm model. Slightly higher variability levels were observed in datasets produced by networks after the BA and BSA procedures.

Fig. 16

Comparison of the distributions of morphodiversity models. SC-ANN_VSR-9-BSA—standardized activation function of the output-layer neuron assigned to the MD_SC-ANN class in the SC-ANN morphodiversity model, after Mastej & Bartuś³⁶. RCM_SDcm—standardized sum of modifiable standardized circular SD partial criteria in the RCM_SDcm morphodiversity model, after Bartuś & Mastej⁵⁵. RCM_SC-ANN—standardized activation function of the output-layer neuron assigned to the MD_RCMSC-ANN class in the RCM_SC-ANN model. RCM_SC-ANN-M—as above, for the MD_RCMSC-ANN-M class in the RCM_SC-ANN-M model. RCM_SC-ANN-BA—as above, for the MD_RCMSC-ANN class in the BA variant. RCM_SC-ANN-M-BA—as above, for the MD_RCMSC-ANN-M class in the BA variant. RCM_SC-ANN-BSA—as above, for the MD_RCMSC-ANN class in the BSA variant. RCM_SC-ANN-M-BSA—as above, for the MD_RCMSC-ANN-M class in the BSA variant. Other abbreviations as in Table 2.

The predictions display distributions with varying degrees of similarity to the normal distribution. Recurring deviations from normality include positive skewness (RCM_SC−ANN, RCM_SC−ANN−M, and RCM_{SC−ANN−M−BSA}) and bimodality (RCM_{SC−ANN−BA}, RCM_{SC−ANN−M−BA}, and RCM_{SC−ANN−BSA}). Statistical tests confirmed that these distributions cannot be considered normal. Consequently, further analyses required a correlation procedure robust to departures from normality—Spearman rank correlation¹¹⁵.

The bimodal distributions and higher variability in the ANN-based models result from the networks’ tendency to classify objects with high probability into either the MD class (values close to 1) or the NMD class (values close to 0), assigning them real numbers near the lower and upper bounds of the activation function’s range.

Relationships between partial criteria and morphodiversity evaluations, as well as between different morphodiversity evaluations

Correlations Beetwen partial criteria and morphodiversity evaluations

High correlations between morphodiversity analysis criteria (see Sect. Correlations) sometimes correspond to strong correlations with landform diversity evaluation datasets (Fig. 17). Notably, TPI_SD exhibits consistently very high or high correlations across all morphodiversity models, indicating that it is one of the most important criteria in such analyses.

The study also revealed very high correlations of the Altitude_SD criterion with the evaluation outputs of the RCM_SC−ANN, RCM_{SC−ANN−BA}, RCM_{SC−ANN−BSA}, and SC-ANN_{VSR−9−BSA} models, all trained using the T, U, and V samples of the first analysis variant (see Fig. 9). These results confirm the high significance of Altitude_SD identified in the GSA analyses (cf. Table 5). In contrast, the same criterion shows only moderate correlations with the models of the second variant (RCM_SC−ANN−M) (see Fig. 11).

An opposite pattern is observed for the Aspect_SDcm criterion, which exhibits very high correlations with the RCM_SC−ANN−M model and its derivatives, but moderate or low correlations with the RCM_SC−ANN variant. Here again, correlation analyses corroborate the previous GSA findings (cf. Table 6).

Finally, the analysis revealed that the Klippen_SHDI criterion has low correlations with the evaluation datasets of all examined morphodiversity models.

Fig. 17

Spearman rank correlation coefficient (R_S) [-] of the used continuous partial criteria and continuous morphodiversity evaluations. Explanations as in Table 1 and in Fig. 16.

Correlations Beetwen morphodiversity evaluations

The assessment of correlations between evaluation datasets essentially reflects the quality of the models used to generate them. The following sections discuss rank correlations between: (i) the outcomes of RCM_SC−ANN and RCM_SC−ANN−M modeling (see Sect.Comparison of RCM_SC-ANN and RCM_SC-ANN-M Evaluations); (ii) the evaluation datasets of these models and the results of archival assessments SC-ANN_{VSR−9−BSA} and RCM_SDcm (see Sect. 4.2.2.2); and (iii) evaluations based on full versus reduced sets of partial criteria (see Sect. Comparison of archival models (SC-ANN_VSR-9-BSA and RCM_SDcm) with RCM_SC-ANN and RCM_SC-ANN-M Evaluations).

Comparison of RCM _SC-ANN and RCM _SC-ANN-M Evaluations

The evaluations of the two main morphodiversity models (RCM_SC−ANN and RCM_SC−ANN−M), developed using ANN trained on different sets of objects, exhibit a Spearman rank correlation of R_S = 0.82 (Fig. 18). This value represents the lower bound of a very strong correlation. The result suggests that, provided the general principles of training data selection are followed, the specific choice of the training set has no decisive impact. It also highlights the remarkable robustness of ANN-based morphodiversity evaluations to potential errors in the constructed training datasets. Remaining discrepancies between the evaluation sets are further illustrated by spatial comparisons (see Sect. RCM_SC-ANN vs. RCM_SC-ANN-M).

Fig. 18

Spearman rank correlation coefficient (R_S) [-] of the used continuous morphodiversity model evaluations. Explanations as in Fig. 16.

Comparison of archival models (SC-ANN _VSR-9-BSA and RCM _SDcm ) with RCM _SC-ANN and RCM _SC-ANN-M Evaluations

Modeling results of morphodiversity using the archival SC-ANN_{VSR−9−BSA} model³⁶ show a strong correlation (Rs = 0.82) with evaluations from the RCM_SC−ANN model and a moderate correlation (Rs = 0.56) with those from the RCM_SC−ANN−M M model (Fig. 18). All compared evaluation sets were generated using the SC-ANN procedure. The origin of this discrepancy can be interpreted as follows.

From a spatial perspective, the SC-ANN_{VSR−9−BSA} and RCM_SC−ANN models were based on the same set of training objects (see Fig. 9). Hence, differences in correlation levels likely stem from the use of different sets of partial criteria. During the BSA procedure, the SC-ANN_{VSR−9−BSA} network was stripped of as many as 44 (out of 53) low-informative and redundant partial criteria, ultimately retaining input variables based on Nc, Ne, Npc, ΔSlope, and ΔZ (see Sect. SC-ANN_VSR-9-BSA model). In contrast, the RCM_SC−ANN model employed scalar and circular measures of pixel variability within statistical zones (see Table 1). The high correlation between the evaluations from these models therefore suggests that effective morphodiversity assessments can be conducted using different sets of partial criteria.

In the two previously discussed cases, high correlation coefficients were likely due either to the use of identical sets of partial criteria or to the same training dataset. A different situation arises in the correlation between SC-ANN_{VSR−9−BSA} and RCM_SC−ANN−M. Here, both models were created using different sets of partial criteria and different training datasets, which likely accounts for the moderate (or rather low) agreement between their evaluations.

The RCM_SC−ANN and RCM_SC−ANN−M models were developed to refine the archival RCM_SDcm model⁵⁵. This refinement aimed to reduce the influence of variables that amplify redundant information and to augment the set of partial criteria with Klippen_SHDI. Although the methodologies for developing the RCM and SC-ANN models differed, all models relied on very similar sets of input variables. Therefore, the very strong correlations between the evaluations from RCM_SDcm and those from RCM_SC−ANN (Rs = 0.91) and RCM_SC−ANN−M (Rs = 0.97) are unsurprising. The higher similarity of RCM_SDcm and RCM_SC−ANN−M evaluations suggests a relatively better selection of training patterns in RCM_SC−ANN−M compared to RCM_SC−ANN, supporting previous recommendations to use the RCM_SC−ANN−M model in other mountain regions.

Evaluations of model evaluations using full sets of partial criteria and after GSA-Based optimization

The final group of comparisons involved evaluation datasets obtained using morphodiversity models based on the full sets of partial criteria (RCM_SC−ANN and RCM_SC−ANN−M), as well as models employing reduced sets of partial criteria derived through the BA (RCM_{SC−ANN−BA} and RCM_{SC−ANN−M−BA}) and BSA (RCM_{SC−ANN−BSA} and RCM_{SC−ANN−M−BSA}) procedures (Fig. 18).

Fig. 19

Spearman rank correlation coefficient (R_S) [-] of the RCM_SC-ANN model vs. RCM_SC-ANN-BA and RCM_SC-ANN-BSA (A) and RCM_SC-ANN-M model vs. RCM_SC-ANN-M-BA and RCM_SC-ANN-M-BSA (B).

The evaluation datasets generated by the aforementioned morphodiversity models exhibit very strong correlations (Fig. 19). A similar pattern is observed among the derivative models of both RCM_SC−ANN and RCM_SC−ANN−M. Correlations between evaluations derived from the two GSA-based reduction procedures are slightly higher than those between these evaluations and the results obtained using the full sets of partial criteria.

These findings support the following conclusions:

• (1) In morphodiversity analyses conducted using artificial neural networks (ANNs), due to the high redundancy of the data, low-informative input variables can be safely reduced without compromising the quality of the evaluations.

• (2) Despite the known limitations of the BA method relative to the BSA procedure, it may be safely applied without a substantial decrease in predictive performance.

Spatial comparison of the selected morphodiversity models

The previous sections described the general patterns observed among the evaluation datasets generated by the tested morphodiversity models. This section presents the effects of the applied models on the spatial distribution of the evaluations (see Fig. 6, stage 5). We examine how the evaluation results of the RCM_SC−ANN model differ from those of the RCM_SC−ANN−M model (see Sect. RCM_SC-ANN vs. RCM_SC-ANN-M), and where the RCM_SC−ANN−M evaluation results converge with or deviate from those of the archival models SC-ANN_{VSR−9−BSA} (see Sect. RCM_SC-ANN-M vs. SC-ANN_VSR-9-BSA) and RCM_SDcm (see Sect. RCM_SC-ANN-M vs. RCM_SDcm). Given the very strong correlations between the evaluation datasets obtained using models based on the full sets of partial criteria and those optimized through the BA and BSA procedures (see Sect. Evaluations of model evaluations using full sets of partial criteria and after GSA-Based optimization), their bivariate choropleth cartograms are thus omitted from this section.

RCM _SC-ANN vs. RCM _SC-ANN-M

Figure 20 presents a spatial comparison of morphodiversity evaluations obtained using the RCM_SC−ANN and RCM_SC−ANN−M models. Dark purple, white, and grayish-purple colors denote areas of agreement representing, respectively, high, low, and average evaluations. Despite the use of two different ANN training datasets, there is a clear consistency in the evaluation of the main morphological elements of the study area. This is particularly evident in the highest and most morphologically diverse parts of the Pieniny Właściwe Mts and Sącz Beskid Mts, as well as in the less diverse southern foothills of the Gorce Mts, the gently undulating Pieniny Spiskie Mts, and the flat valley floors of the Dunajec River and the Niedziczanka stream.

Fig. 20

Bivariate choropleth cartogram of the RCM_SC-ANN vs. RCM_SC-ANN-M morphodiversity models; 3-class discretization using the quantile method. 1—rivers and streams; 2—water bodies; 3—Pieniny National Park; 4—national border.

From an interpretive perspective, however, the differences between the evaluations produced by the two models are more informative. On the cartograms, these discrepancies are shown in shades of pink and blue. Statistical zones marked in pink occur mainly along the mountain ridges of the Gorce Mts, Pieniny Właściwe Mts, Sącz Beskid Mts, and Pieniny Spiskie Mts. These are areas that received higher evaluations from the RCM_SC−ANN model than from the RCM_SC−ANN−M model (cf. Figure 13–Figure 14). The observed differences arise from the distinct definitions of the ANN training samples.

In the RCM_SC−ANN dataset, the NMD_RCMSC−ANN class included almost no samples located in the higher parts of the mountain massifs (Fig. 9), which led the network to classify such areas as MD_RCMSC−ANN. In contrast, the RCM_SC−ANN−M model was trained using a dataset in which the NMD_{RCMSC−ANN−M} class also contained samples from mountainous areas that did not exhibit high morphodiversity (Fig. 11). These included ridge-top surfaces and monotonous, gently inclined slope segments. Consequently, the RCM_SC−ANN−M model produced relatively lower evaluations for morphologically less diverse mountain areas.

Conversely, in the gently sloping hillside zones—particularly along the southern foothills of the Gorce Mts and Pieniny Właściwe Mts—among many areas consistently evaluated as low-morphodiversity, clusters of blue-colored zones appear. These represent locations that received lower evaluations from the RCM_SC−ANN model than from the RCM_SC−ANN−M model. The main reason lies in the more careful definition of the NMD_{RCMSC−ANN−M} training samples compared with those used for the RCM_SC−ANN model.

For the RCM_SC−ANN−M variant, the training samples were deliberately constructed to encompass entirely homogeneous morphological forms and to avoid intersections with ridges, valleys, or other major landform elements. When the ANN encountered slightly more heterogeneous neighboring zones, it classified them higher than the surrounding uniform terrain. The approach in the RCM_SC−ANN model was different. As noted earlier, the low-morphodiversity objects were defined based on built-up areas—morphologically favorable for human settlement and activity. Many of these were located in narrow mountain valleys, sometimes covering parts of opposite slopes or deeply incised stream channels. When the ANN encountered even simpler adjacent slopes with nearly uniform inclination, it automatically classified them as low-morphodiversity. This lack of sensitivity to subtle terrain features in the RCM_SC−ANN model led to oversimplified evaluations and, consequently, to more realistic results in the RCM_SC−ANN−M model.

In line with the main objective of this study—to develop a universal morphodiversity model suitable for various mountainous areas—the results confirm that the RCM_SC−ANN−M model is the more robust and should be recommended for further field validation and applied testing.

RCM _SC-ANN-M vs. SC-ANN _VSR-9-BSA

The RCM_SC−ANN−M and SC-ANN_{VSR−9−BSA} models were both developed using the SC-ANN methodology. The spatial comparison of their evaluation outputs aimed to assess the consequences of employing different types of partial criteria—scalar and angular measures of variability in the case of RCM_SC−ANN−M, versus discrete criteria such as counts or categorical variables in SC-ANN_{VSR−9−BSA}.

As revealed by the correlation analysis (Fig. 18), the relationship between the evaluation sets generated by the two models can be regarded, at best, as moderate. This is illustrated in Fig. 21, where dark purple and white colors dominate, alongside various shades of blue and pink. These patterns indicate a spatial mosaic in which zones with consistent evaluations alternate with those showing divergence. A clear tendency can be observed: blue-tinted statistical zones prevail across mountain massifs, whereas numerous pink-colored zones appear within broad valley floors and gently sloping foothills of the Gorce Mts, Pieniny Właściwe Mts, and Pieniny Spiskie Mts. Blue shades mark areas that received higher evaluations from the SC-ANN_{VSR−9−BSA} model, while pink shades denote those rated higher by the RCM_SC−ANN−M model.

The pattern visible on the cartogram reflects the refinement introduced by the RCM_SC−ANN−M model in areas where the SC-ANN_{VSR−9−BSA} model produced coarser assessments. This is well exemplified by the valleys of mountain streams—such as the Krośnica and Niedziczanka rivers, among others. Objects located along these valleys were most frequently classified as non-morphodiverse by the SC-ANN_{VSR−9−BSA} model, due to the high density of NMD_{SC−ANNVSR−9−BSA} training samples situated along valley bottoms (see Fig. 9). In contrast, the RCM_SC−ANN−M model incorporated MD_{RCMSC−ANN−M} training samples located in the upper valley sections (see Fig. 11), resulting in more cautious evaluation of the lower valley segments. Owing to local variation in elevation, slope, aspect, and curvature, these areas often received higher evaluations compared to those produced by SC-ANN_{VSR−9−BSA} (see Fig. 14).

In summary, the RCM_SC−ANN−M model refined the analysis by lowering the scores of less morphologically diverse areas situated in higher parts of the mountain ranges, while increasing the scores of more morphologically complex zones located within valleys and at the mountain foothills. This outcome represents a depiction that is undoubtedly closer to geomorphological reality.

Fig. 21

Bivariate choropleth cartogram of the RCM_SC-ANN-M vs. SC-ANN_VSR-9-BSA morphodiversity models; 3-class discretization using the quantile method. 1—rivers and streams; 2—water bodies; 3—Pieniny National Park; 4—national border.

RCM _SC-ANN-M vs. RCM _SDcm

From a methodological perspective, comparing the results of modeling RCM criteria using the classical AR approach (RCM_SDcm) with those obtained through the SC-ANN methodology (RCM_SC−ANN−M) offers valuable insights. The Spearman rank correlation analysis revealed a very strong relationship between the evaluation datasets of both models (see Fig. 18). This finding is supported by the cartogram shown in Fig. 22, which is dominated by dark purple, white, and grayish-purple tones, representing agreement across high, low, and average scores. Statistical zones shaded in pink and blue are rare, typically occurring in isolation or forming small clusters.

In this case, the majority of pink-shaded objects appear on the southern, morphologically less diverse slopes of the Gorce Mts. These areas received higher morphodiversity evaluations from the RCM_SC−ANN−M model compared to those produced by RCM_SDcm. Given the absence of rocky landforms in these areas, the observed differences in evaluations likely stem from the intrinsic characteristics of ANN-based versus AR-based modeling approaches.

Fig. 22

Bivariate choropleth cartogram of the RCM_SC-ANN-M vs. RCM_SDcm morphodiversity models; 3-class discretization using the quantile method. 1—rivers and streams; 2—water bodies; 3—Pieniny National Park; 4—national border.

AR models assess morphodiversity by aggregating the values of partial criteria, producing evaluations that represent a simple linear function of the independent variables. ANN models, however, operate differently. Through the weighting process and the capacity to minimize or even eliminate the influence of less informative variables, ANN evaluations become complex nonlinear functions. A similar mechanism likely applies here: in statistical zones marked in pink, the ANN increased the weights of key criteria, leading to higher evaluation scores relative to their surroundings.

Conversely, blue-shaded objects are primarily located in the upper parts of the Pieniny Właściwe Mts, Gorce Mts, and Sącz Beskid Mts. These zones received higher evaluations from the RCM_SDcm model compared to RCM_SC−ANN−M. In this case, the ANN tended to underestimate the morphodiversity relative to the AR model — the opposite of the situation described above. This difference likely resulted from the reduction in weights assigned to certain criteria relative to the linear model.

Limitations of the method

The proposed method for assessing geodiversity, based on RCM and SC models supported by artificial neural networks (ANNs), has particular applicability in morphodiversity analyses. However, it can also be applied in other cases where partial criteria are derived from continuous regionalized variables. A significant portion of geodiversity analyses, however, relies on discrete data. Such variables can be incorporated into RCM models, for instance, through the use of diversity indices such as SHDI. To ensure equal treatment of all partial criteria, the data should be standardized. For the same reason, all input variables derived from raster data should have a uniform pixel resolution.

Although the method is largely scale-independent, increasing the pixel size requires a proportional increase in the size of the statistical zones. This is necessary to maintain a sufficient number of pixels for statistically reliable estimation of variability.

Particular caution should be exercised when analyzing the variability of certain topographic attributes in flat or gently sloping areas (e.g., Aspect). In such cases, the observed variability structures in pixel values often represent artifacts and should be masked or zeroed. Additional limitations arise from the use of scalar measures of variability for angular variables, which is related to their cyclic nature. As noted by Bartuś & Mastej (2025), scalar standard deviations can only be used when the variability range of a partial criterion does not include both the first and fourth quadrants of the full angle—in such cases, the relative estimation error does not exceed 5%. However, this rule has been tested exclusively on synthetic data and therefore requires further empirical verification.

The training data used for developing morphodiversity models through SC-ANN require independent verification. In this study, validation was performed using two independent geomorphometric models – SPC⁹⁸ and Geomorphon¹¹¹ – as well as field verification. This approach confirmed the correctness of training pattern selection in the TR model. Combining morphometric methods with expert-based field evaluation appears to be an effective approach for independently assessing the quality of training datasets. However, it should be noted that the SPC and Geomorphon models, being derivatives of DEM data, cannot serve as direct sources of information for selecting ANN training samples.

A major challenge in geodiversity analyses remains the process of result validation. Previous attempts to develop independent validation datasets^36,55 can be considered moderately successful, mainly due to the difficulty of ensuring their full independence from the primary topographic data source (DEM) and achieving adequate spatial resolution. As suggested by Bartuś & Mastej⁵⁵, under such circumstances, the most realistic approach is to compare the results generated by different models with one another.

The final quality of an ANN classifier, which determines the reliability of morphodiversity evaluations, should be controlled using several independent techniques, such as analysis of the area under the ROC curve, assessment of prediction accuracy changes in the T, U, and V datasets, and evaluation of the confusion matrix reflecting the number of correctly classified objects. Otherwise, the network may become overfitted to the data, resulting in a loss of its generalization capability.

In summary, the proposed method enables a quantitative, objective, and repeatable assessment of morphodiversity. Nevertheless, it requires further empirical validation, particularly regarding independent verification and optimization of ANN training datasets.

Conclusions

The present study aimed to develop a morphodiversity model that could be applied in other mountainous regions in the future. A set of partial criteria based on raster data was tested, incorporating both scalar and circular measures of pixel variability within statistical zones, as well as the SHDI index. Using SC and ANN methods, six morphodiversity models were developed: two main models based on different training datasets (RCM_SC−ANN and RCM_SC−ANN−M), and, for each of them, two models optimized through two GSA procedures: BA (RCM_{SC−ANN−BA} and RCM_{SC−ANN−M−BA}) and BSA (RCM_{SC−ANN−BSA} and RCM_{SC−ANN−M−BSA}). The developed models were subjected to comparative testing against archival models SC-ANN_{VSR−9−BSA}³⁶ and RCM_SDcm⁵⁵, as well as against each other. The aim was to assess their performance, identify their key characteristics, and evaluate their suitability for assessing morphodiversity in other mountainous areas.

Recommended Morphodiversity Model.

The comparison of the quality of the developed RCM_SC−ANN models with the archival models (SC-ANN_{VSR−9−BSA} and RCM_SDcm) confirmed the higher precision and realism of the evaluations obtained using the new methodology. Compared to the SC-ANN_{VSR−9−BSA} model, the RCM_SC−ANN−M model more accurately represented the morphological variability of valleys and mountain foothills, avoiding the simplifications typical of models based on discrete partial criteria. In comparison with the RCM_SDcm model, the use of ANN made it possible to capture nonlinear relationships between variables and reduce the influence of redundant features, resulting in more balanced and spatially consistent morphodiversity assessments.

The strong correlation between the evaluations of the two main morphodiversity models (RCM_SC−ANN and RCM_SC−ANN−M) could suggest a secondary role of the ANN training datasets defining the classification rules, as well as a high robustness of ANN-based morphodiversity evaluation to potential errors in the constructed training sets. However, spatial comparisons of the evaluation datasets indicate that the details of the classification rules remain highly significant. It was found that the high level of correlation mainly results from the agreement in the evaluations of objects forming the principal landform elements—morphodiverse highest parts of mountain massifs and their weakly diversified foothills and flat valley floors. In this context, the potential superiority of one model over the other was determined by the finer details of evaluation.

The MD_RCMSC−ANN class objects were defined based on a criterion of tourism attractiveness, assuming its relationship with morphological diversity. Conversely, the NMD_RCMSC−ANN class objects were identified using a settlement-related criterion, under the assumption that the location of a farmstead requires favorable morphological conditions. Therefore, the adopted criteria were not overly restrictive and, as it turned out, proved inaccurate for some NMD_RCMSC−ANN class objects.

In contrast, the training objects in the RCM_SC−ANN−M model were defined differently—primarily through qualitative analysis of landform types in the TR model. The MD_{RCMSC−ANN−M} objects were defined as those containing a greater number of morphological forms, while the NMD_{RCMSC−ANN−M} objects were defined as homogeneous areas characterized by a single landform type. The quality of the training sample selection in this model was controlled using two independent geomorphometric methods and field verification.

As a result of morphodiversity modeling based on the new training dataset, NMD_{RCMSC−ANN−M} class objects appeared even within generally more morphologically diverse mountain massifs—such as ridge fragments intersected by planation surfaces and monotonous, gently inclined slope sections—while morphodiverse objects emerged among the typically less diverse foothills and valley floors. Consequently, the RCM_SC−ANN−M model provided morphodiversity evaluations that more accurately reflected the landscape of the Pieniny Mts region compared to the RCM_SC−ANN model. The presented analytical results, together with comparisons to the aforementioned archival models, allow the RCM_SC−ANN−M model to be recommended for testing in other mountainous regions.

Model Optimization.

The remaining issues concerned the possibility of optimizing the analyzed SC-ANN models to achieve the permanent elimination of low-informative and redundant criteria. To this end, Global Sensitivity Analysis (GSA) was performed on both main morphodiversity models. The analyses revealed that among the applied partial criteria, some consistently exhibited a lower level of informativeness than others. These were identified as low-informative or redundant and subsequently removed from the input datasets during the BA and BSA procedures, resulting in the optimized morphodiversity models: RCM_{SC−ANN−BA} and RCM_{SC−ANN−M−BA}, as well as RCM_{SC−ANN−BSA} and RCM_{SC−ANN−M−BSA}. These models utilized a set of the most informative partial criteria, including Altitude_SD, Aspect_SDcm, Slope_SD, TPI_SD and Klippen_SHDI. The conducted analyses support recommending these criteria for morphodiversity evaluation in other mountainous regions.

It is important to note one significant limitation of the applied optimization procedures. An overly restrictive approach to GSA—consisting of the elimination of too many input variables from the datasets (e.g., 4 out of 7)—may, during the classification of training objects into the NMD and MD classes, lead to a form of binary activation in the output neuron functions. The training data become almost perfectly classified, with the probabilities of belonging to one of the two classes approaching 0 or 1. However, if the goal is for the probability of an object’s membership in the MD or NMD class to remain a continuous function—allowing for its subsequent discretization into any number of bonitation classes (e.g., very low, low, medium, high, and very high morphodiversity)—then the number of removed variables should be limited (e.g., to 3 out of 7).

Despite this limitation, the results of the analyses indicate the advantages of applying optimization procedures even to small input variable sets, such as those consisting of seven elements. GSA enables the straightforward removal of redundant information and reduction of analytical costs, practically without any loss in evaluation quality. As demonstrated by the correlation analysis results, the BA procedure performs just as well as the more complex and computationally demanding BSA procedure. Both methods can be applied with virtually no adverse impact on the evaluation outcomes.

Implications for the Development of Morphodiversity Models.

The presented research has significant implications for the development of methods for the quantitative assessment of geo- and morphodiversity. The integration of the RCM model with SC and ANN techniques represents an important step in applying machine learning to the analysis of non-discretized raster data describing terrain heterogeneity. This approach opens new opportunities for the study of mountainous landscapes, where classical AR models are unable to faithfully capture complex, nonlinear spatial relationships. Key achievements of this work include the reduction of input data redundancy, thereby improving the quality of evaluations, and demonstrating that the removal of low-informative and redundant variables from RCM models through GSA procedures does not preclude reliable morphodiversity assessment. The proposed methodology also allows for quantitative analysis of the influence of individual partial criteria on morphodiversity evaluation, identification of low-informative variables, and optimization of criteria sets, which was one of the main objectives of this study. The method can be readily transferred to other regions of the world, adapted to different analysis scales and types of environmental data, making it a universal tool for geo-diversity research and geo-conservation.

The developed training dataset (RCM_SC−ANN−M) will be used in the future for modeling morphodiversity in other areas, potentially leading to the development of a universal morphodiversity model for mountainous regions. In this way, the present research contributes to the global effort in quantitative modeling of landscape complexity and the application of artificial intelligence in the analysis of geographical processes.

For further development, planned studies will consider the integration of remote sensing data, such as LiDAR and multispectral datasets, to enrich input variables, as well as testing more advanced deep learning algorithms for morphodiversity analysis. An important direction is the expansion of method validation across different regions and spatial scales to assess the potential for generalization, along with the continued refinement of partial criteria selection and optimization, taking into account data quality and measurement artifacts. Furthermore, future research may focus on combining quantitative data with information on geomorphic processes, enabling a more comprehensive assessment of mountainous landscapes. Such an approach ensures the continuation of efforts toward a universal and flexible tool for quantitative morphodiversity analysis, allowing the integration of new data sources and analytical techniques in future models.