Table 2 Fractal dimensions of waviness and unevenness of rock joint sample S2 at varying sample sizes.

From: Unveiling the mystery of scale dependence of surface roughness of natural rock joints

Resolution

(mm)

Fractal

estimation

Sample size (\(\mathrm mm \times mm\))

\(100 \times 100\)

\(200 \times 200\)

\(300 \times 300\)

\(400 \times 400\)

\(500 \times 500\)

\(600 \times 600\)

\(700 \times 700\)

\(800 \times 800\)

\(900 \times 900\)

\(1000 \times 1000\)

0.5

\(D_w\)

2.1663

2.2253

2.2264

2.2501

2.2412

2.2200

2.2330

2.2508

2.447

2.2242

\(A_w\)

− 3.248

− 3.114

− 3.091

− 3.004

− 2.988

− 3.070

− 3.029

− 2.978

− 2.972

− 3.059

\(R^2_w\)

0.9978

0.9959

0.9952

0.9944

0.9960

0.9985

0.9974

0.9968

0.9962

0.9976

\(D_u\)

2.6690

2.7391

2.6969

2.7573

2.6501

2.7402

2.7288

2.7443

2.7335

2.7737

\(A_u\)

− 2.142

− 1.900

− 1.972

− 1.763

− 2.171

− 1.857

− 1.881

− 1.852

− 1.855

− 1.579

\(R^2_u\)

0.9898

0.9393

0.9000

0.9745

0.9928

0.9175

0.9305

0.9193

0.9119

0.9395

1.0

\(D_w\)

2.2448

2.3214

2.2892

2.3257

2.2884

2.2586

2.2754

2.2944

2.2852

2.2664

\(A_w\)

− 2.471

− 2.252

− 2.317

− 2.212

− 2.259

− 2.364

− 2.314

− 2.269

− 2.271

− 2.335

\(R^2_w\)

1.0000

0.9978

0.9950

0.9941

0.9993

0.9993

0.9989

0.9991

0.9973

0.9988

\(D_u\)

2.6250

2.6325

2.8642

2.9069

2.6473

2.9132

2.8751

2.9210

2.9117

2.9195

\(A_u\)

− 2.062

− 2.057

− 1.315

− 1.199

− 1.989

− 1.180

− 1.281

− 1.148

− 1.153

− 0.976

\(R^2_w\)

1.0000

0.9977

0.9363

0.9761

0.9962

0.9496

0.9585

0.9941

0.9230

0.9277

2.0

\(D_w\)

2.2194

2.3634

2.3395

2.3433

2.3378

2.2842

2.2892

2.3141

2.3313

2.2840

\(A_w\)

− 2.037

− 1.627

− 1.666

− 1.646

− 1.645

− 1.759

− 1.764

− 1.701

− 1.617

− 1.762

\(R^2_w\)

1.0000

0.9911

0.9906

0.9930

0.9969

0.9992

0.9978

0.9982

0.9958

0.9975

\(D_u\)

2.7017

2.8733

2.8909

2.9162

2.9193

2.9588

2.9654

2.9606

2.9593

2.9542

\(A_u\)

− 1.750

− 1.384

− 1.297

− 1.267

− 1.263

− 1.118

− 1.092

− 1.113

− 1.095

− 0.936

\(R^2_u\)

1.0000

1.0000

0.9550

0.9194

0.9226

0.9224

0.9181

0.9912

0.9358

0.9598

  1. \(D_w\) and \(D_u\) are fractal dimensions of waviness and unevenness, respectively. \(A_w\) and \(A_u\) are the coefficients during linear correlation for estimating \(D_w\) and \(D_u\), respectively. \(R^2_w\) and \(R^2_u\) represent the coefficients of determination during linear correlations for estimating \(D_w\) and \(D_u\), respectively.
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