Table 1 Definitions of terms describing stepwise movement, among and within-step precision, and geophysical orientation cues.

Stepwise movement

Flight-step

Encompasses departure and (daily or nightly) flight (Fig. 1a). Identified by subscript i, also subdivided hourly to include within-step processes.

Location

Flight-step latitude, ∅_i, and longitude, λ_i, in radians (Eqs. (1) and (2). Geomagnetic-dipole simulations use geomagnetic latitude and longitude.

Step length

Flight-step distance, R_step (radians), constant or subdivided hourly.

Orientation and precision

Preferred heading

Expected heading, \({\bar{\alpha }}_{i}\), based on the primary compass. Defined clockwise from geographic South (S), counter-clockwise from N in S Hemisphere.

Flight orientation

Quantified using von Mises distribution with concentration parameter, κ⁵³. We describe orientation precision, and also between-individual variability in inherited headings, by \(\sigma =1/\sqrt{\kappa }\) (≅angular std. deviation for σ ≤ 30°, κ > 3.7).

Compass precision

Within flight-steps, affects initial cue detection, if applicable cue transfer and cue maintenance, i.e., in-flight redetermination of headings (Fig. 1).

Error scenarios

We modelled precision both among flight-steps (0°–60° precision, implicitly including all sources of errors) and considering biologically relevant within-step variability (0°–40° compass precision in cue detection, transfers and maintenance, up to 20° within-flight drift, and 2.5° default between-individual variability).

Geophysical orientation cues

Geomagnetic axis

Offset from geographic headings by magnetic declination, δ_m³² (constant in dipole model, otherwise interpolated from IGRF data⁵¹).

Geomagnetic inclination

Angle of field vector to horizontal, γ_i. Strongly latitude-dependent^33,51.

Sun azimuth

Sunrise or sunset azimuth, θ_s (Eq. (9)). Along time-compensated sun compass courses, clock-shifted until resetting of inner-clock.

Polarized light

At sunrise and sunset, maximum bands of polarized light are perpendicular to sun azimuth, and average to geographic N–S.

Stellar axis

Fixed star or centre of rotation. Not time-compensated between steps^16,17.

Compass courses

Geographic loxodrome

Constant heading relative to geographic axis, identifiable by a star compass, or by averaging polarized light cues between dawn and dusk²⁴.

Geomagnetic loxodrome

Constant heading relative to perceived geomagnetic axis.

Magnetoclinic

Geomagnetic headings based on maintaining a fixed transverse projection of proximate inclination³³.

Fixed sun compass

Constant heading vs. sunrise or sunset azimuth.

Time-compensated sun compass (TCSC)

As in fixed sun compass, but offset due to longitudinal clock-shifts, according to how migrants track sun azimuth²². We further extended the original formulation²² to allow for proximate sun-azimuth tracking, and headings to be retained from the first night of extended stopover (rather than on arrival, which is less consistent with sun azimuth headings).