Self-correcting Sun Compass, Spherical Geometry and Cue-transfers Predict Naïve Migratory Performance

Migratory orientation of many animals is inheritable, enabling naïve migrants to reach remote destinations independently following stepwise (often, nightly) geomagnetic or celestial cues. Which if any such “compass courses” can explain narrow-front trans-continental routes remains unresolved, and evident error-corrections by naïve migrants remain unexplained. We assessed robustness to errors among airborne compass courses and quantified inaugural 5 migration performance globally, accounting for cue transfers (e.g., sun to star compass), in-flight cue maintenance, and previously-overlooked spherical-geometry (longitude) effects. We found (i) sun-compass courses partially self-correct, making them most robust between flight-steps, (ii) within nocturnal flight-steps, geomagnetic or star-compass headings outperform cue-transferred sun-compass steps, (iii) across diverse airborne migration routes, the relative favourability of sun-compass over other courses increases with increasing goal-area, required flight steps and a spherical-geometry factor. Our results can explain enhanced naïve migrant performance, observed diversity in compass-cue hierarchies, and sun-compass orientation being key to many long-distance inaugural migrations.


Introduction
Seasonal animal migrations have evolved across taxa at spatial scales spanning meters to continents 1 . A critical factor for migratory populations is the ability to perform inaugural ("naïve") migrations without access to a navigational map 2,3 . While migratory routes are often transmitted culturally through collective and social cues 4,5 , many naïve airborne migrants reach 5 population-specific remote destinations (hereafter, goal areas) independently by following innate or inherited headings stepwise relative to proximate geophysical compass cues en route 2,6 . Naïve but migration-ready birds and insects can orient consistently in the horizontal (azimuthal) plane relative to both geomagnetic and celestial directional cues 2,6 . Birds innately distinguish between geomagnetic North and South using geomagnetic inclination (the angle between the geomagnetic 10 field and the horizontal) 2,6 , but celestial compasses accounting for (hourly) rotation in sun azimuth or star patterns need to be learned prior to migration 6,7 . Diagnosis of compass-cues used in-flight and across entire routes remains a major challenge 2,7,8 . Nonetheless, unassisted naïve migrants probably prioritize one "primary" compass system to determine (e.g., nightly) flight headings, sometimes transferred to a second, in-flight compass. Cue-conflict experiments 15 suggest various contingencies and hierarchies involving "calibration" between compasses, but often prioritization of celestial cues at twilight, particularly among North American migrants [6][7][8] .
The choice of primary compass can result in substantially different stepwise "compass courses", with five main classes proposed: 1) geographic loxodromes, following constant headings relative 20 to geographic South or North, which is identifiable either by a primary star compass 7,9 or else by averaging (more reliably available) maximum bands of polarized light at sunrise and sunset [10][11][12] ; 2) geomagnetic loxodromes, following constant headings relative to geomagnetic South or North; 3) gradually-shifting magnetoclinic course, based on maintaining a fixed (transverse) projection of proximate geomagnetic inclination en route 11,13 ; 4) fixed (menotactic) sun compass courses, following a constant heading relative to proximate sunrise or sunset azimuth, which naturally shift with date and location 11 ; 5) time-compensated sun compass (TCSC) courses, which can achieve nearly great-circle trajectories due to the "clock shift" induced by crossing longitudes, resulting in increasingly Southward headings (Northward in the Southward 5 Hemisphere) 14,15 . While imprecise stepwise loxodromes based on constant preferred headings can sufficiently explain broad-front migration 2,16,17 , migratory tracking data reveal a diverse picture featuring narrow-front and sharply direction-changing 18 . Indeed, while known birdmigration routes often resemble sun compass and magnetoclinic courses 13,15 , their relative feasibility has been debated 11,19 and robustness to stepwise errors remains untested. Even more 10 puzzling is the evidence of route-correctionsa hallmark of true navigation 2,3by some naturally and artificially displaced naïve bird migrants en route [20][21][22] .
Here, we provide a modelling framework to assess robustness of migratory compass courses across the globe, and identify key geophysical and route-geometric factors governing inaugural 15 migratory performance, here quantified as proportional arrival at goal areas. For simplicity and interpretability, we focused on inaugural airborne migration based on a single inherited or imprinted (initial) heading. We first extended current formulation of compass courses, to account for 1) imprecision within single flight-steps, including possible cue transfers to a second (inflight) compass and in-flight cue maintenance; 2) the effect of stepwise errors on subsequent 20 headings and courses, particularly for the direction-shifting sun-compass and magnetoclinic courses; 3) spherical geometry effects, in particular from the convergence of longitude bands at higher latitudes 23 , which have not been quantified for long-distance animal movements 16,18,24,25 .
To assess effects of primary cue-choice, precision and route-geometry in consort, we simulated each compass course for both a generic migrant across a broad range of global routes and magnitudes of error, and also for known routes of nine diverse long-distance airborne migratory species, incorporating dynamic geomagnetic data and in-flight error to account for wind or cuerelated drift effects. Finally, we predicted how inaugural migration performance depends on stepwise precision and route-geometric factors, by applying regression and model selection using 5 route-optimized geomagnetic and sun-compass courses among species. Table 1 lists terms relating to stepwise movement and geophysical cues, as described in the 10 Methods (equations . For stepwise compass movement to a goal (equations 1-4, Fig. 1a), the probability of successful arrival (performance) will increase with increasing angular concentration in stepwise headings (von Mises concentration κ, equation 7), goal-area breadth (ratio of goal radius to migration distance, / ) and, following the many-wrongs principle 4,26 , with increasing number of steps, N. As a first approximation, assuming independent 15 steps on a plane with a high angular concentration, i.e., small "effective standard error", = 1/√ , migratory performance will follow a cumulative normal distribution as a function of the length-adjusted goal-area breadth, = √ 0 ⁄ , where 0 is the minimum (errorfree) number of flight steps (equations 11-13).
Step length, Stepwise flight distance (radians), here constant per species (Table 2). As in fixed sun compass but offset due to longitudinal clock-shift relative to internal clock 14 , affecting perceived sunset azimuth (equation [20][21][22]. We also quantify how the TCSC offset varies with a migrant's reference (step) for sun azimuth rotation (equation 24), which could be local or when clocks are reset during extended stopover (equation 23, Figure 6). Performance -related factors (for independent steps on a plane)

Migratory performance, pArr
Probability of successfully reaching destination, i.e., arriving within the goal area. Goal area Migratory destination, modelled by goal radius, (radians). Migration distance, Distance (radians) from initial step (e.g., natal site) to centre of goal area. Goal-area breadth, Goal-area radius divided by migration distance, . Length-adjusted goalarea breadth, Goal-area breadth, , multiplied by square root of minimum (error-free) number of steps, 0 . Governs performance in the normal planar limit (Fig. 1, equations [11][12][13] If we unwrap a single flight-step, stepwise precision will itself depend on (initial) cue detection, cue maintenance (i.e., in-flight cue redetermination; Fig. 1b), and any cue transfer (Fig. 1c). For flight-steps based on a single cue (Fig. 1b), cue maintenance will reduce expected stepwise errors (Supplementary Fig. 1b-c, equation 8a) at the expense of stepwise flight distance 31 . However, for flight-steps involving cue transfer to a second compass (Fig. 1c), cue maintenance cannot make 5 up for initial cue detection and transfer errors ( Supplementary Fig. 1b, equation 8b). Therefore, within a single nocturnal flight-step, non-transferred geomagnetic or star-compass headings are relatively more precise compared with headings transferred to a second compass (assuming equivalent precision among compasses in cue detection and maintenance).

Compass course formulations and sensitivity
In the Methods, we formulated stepwise compass headings for each compass courses (Table 1, equations 14-17, 19, 22). For interpretability across global scales, we formulated magnetoclinic courses assuming a geomagnetic dipole model, in which magnetoclinic headings vary solely with geomagnetic latitude (equation 17). We further extended "classic" TCSC courses sensu 15 Alerstam 14 , to quantify how resetting of a migrant's inner clock and, additionally, possible use of proximate sun-azimuth rotation affect its "time-compensated" offset relative to any "clock-shift" caused by crossing longitudes equations (22)(23)(24).
The heuristics of TCSC migration and self-correction are illustrated in Fig. 2. Following errorfree headings, a migrant's subsequent heading will shift oppositely to its clock shift, creating an 20 increasingly Southward trajectory (Northward in the Southern Hemisphere). Following an imprecise heading, the error-induced "time-compensation" offset (equation 21) will therefore naturally tend to counteract any (erroneous) difference in clock-shift.

Fig. 2. Time-compensated sun compass (TCSC) headings and self-correction.
A TCSC migrant clock-synchronized to local conditions (above) maintains its preferred direction (solid black arrow) by adjusting its heading relative to the daily clockwise rotation in sun azimuth (here at sunset, solid red arrow). Following an error-free flight-step (lower left), the longitudinally (here, 5 Westward) displaced migrant will be clock-shifted (here, clock-accelerated) relative to local time. This results in an "over-compensation" to proximate sun azimuth, i.e., counter-clockwise TCSC offset (dashed red arrow), hence more Southward (here, less Westward) heading (dashed black arrow). If the migrant's initial heading is imprecise (dot-dashed grey line), its stepwise longitudinal displacement will lead to a contrasting clock-shift (here, clock-lag). The now clock-lagged migrant (lower right) will "under- 10 compensate" relative to proximate sun azimuth, resulting in a clockwise offset (dashed red arrow) and hence self-corrected heading (dashed black line). Between-step shifts in proximate sunset azimuth (not shown) become biologically relevant at multi-day and multi-step scales (Fig. 5).
We quantified sensitivity to stepwise error algebraically as iterative (proportional) growth in errors of stepwise headings, revealing contrasting latitudinal and directional patterns, with large ranges in iterative growth in errors including partial self-correction (Fig. 3). Preferred geographic loxodrome headings (equation 14) will per definition not depend on previous headings, resulting in "zero" growth or correction in error as long as cue-detection errors are stepwise independent 5 (Fig. 3a). This also holds for geomagnetic loxodrome headings in a dipole field (relative to geomagnetic axes, equation 15). Contrastingly, the latitude-dependence of magnetoclinic headings (equation 17) renders them stepwise inter-dependent, and leads to extremely high sensitivity for virtually any non-Southerly heading at both high and low latitudes (Fig. 3b, equation 25). Errors in fixed sun compass courses remain largely stepwise independent (close to 10 "zero" growth), but will iteratively grow or self-correct at high latitudes, depending on whether East or West oriented, and before or after the fall equinox (equation 26, Fig. 3c-d). Sensitivity in TCSC headings is similarly East-West antisymmetric about the equinox (Fig. 3e-f), but their self-correcting nature (Fig. 2) renders them relatively insensitive, with 5-25% stepwise selfcorrection over a broad range of directions (equation 27), into which headings (blue arrows) 15 moreover tend to "converge". While the degree of stepwise TCSC correction remains small away from polar latitudes (as shown in Fig. 2, roughly to scale), subsequent steps will also partially self-correct for any discrepancy in longitude as long as inner clocks are not reset.

Fig. 3. Stepwise sensitivity varies strongly with heading and among compass courses.
Stepwise sensitivity , i.e., iterative growth of small errors in heading (%, with colour scales on right), as a function of current heading (clockwise from South) and latitude (geomagnetic South and geomagnetic latitude for geomagnetic courses), for (a) constant-heading geographic loxodromes, or equivalently geomagnetic loxodromes in a geomagnetic dipole Earth, (b) magnetoclinic courses in a geomagnetic

Simulation of migration routes
For each species and compass course, route-optimized trajectories, i.e., with headings maximizing performance (probability of successful arrival), are illustrated in Fig Diversity in compass-cue favourability for Marsh Warbler (Acrocephalus palustris) migration 20 over a range of (component-wise) errors is illustrated in Fig. 5, including greater drift tolerance among TCSC courses ( Fig. 5i-j) and a slight advantage of geomagnetic over geographic loxodromes, particularly when the latter are based on polarized light, e.g., when the star compass is unavailable on departures ( Fig. 5a-b, e-f). Table 2. Model parameters of the species compass course simulations. Species and routes, ordered by migration distance, used in model simulations to assess compass course performance. Routes and migration pace were based on tracking and other studies, including initial departure dates ± standard deviation (and maximum arrival date), great-circle (followed by loxodrome) distances and headings, flight (ground) speed, travel (migration) speeds, and migration schedule, the latter modelled as a (fixed) sequence of consecutive flight steps followed by an extended stopover (mean ± standard deviation). Length-adjusted goal breadth, (equation 13), governs performance in the normal planar limit (equation 11, Fig. 3a). 5 All migrants except the Monarch Butterfly are principally night-migratory.

Fig 4. Diverse compass course performance among species and migration routes. (a)
Compass-route performance, assuming 20° total stepwise equivalent error, vs. length-adjusted goal breadth, which governs expected performance (dashed line) in the normal planar limit (equations 11-13), for 9 species (Table 2) Performance (%) and, where applicable, also cue-transferred courses ("T") are depicted above each panel.   Fig. 6) further revealed narrower longitudinal ranges for high-latitude fixed compass courses, and heterogeneity in the performance gain of both sun-compass courses relative to non-transferred loxodrome courses, with TCSC courses losing their self-correcting advantage (cf. Fig. 4a) with effective within-step errors exceeding ~30° Supplementary Fig. 6).
The effects of inner-clock resetting and time-compensation across continental scales are 10 illustrated in Fig. 6 for simulated Gray-cheeked Thrush migration (with known routes in inset 39,41 ). "Classic" TCSC trajectories (Fig. 6a) resemble both great circles and known routes but rely on stepwise (nightly) headings always being adjusted according to sun-azimuth rotation rates as experienced on departure from the natal grounds. Contrastingly, when adjusting nightly headings to proximate sun-azimuth rotation (Fig. 6b), trajectories deviate strongly from great 15 circles, unless (Fig. 6c) inner-clocks are reset and headings retained during extended stopovers.
Finally, trajectories vary more strongly and contrastingly with departure date when migrants inherit sun compass headings (Fig. 6d) as opposed to geographic headings (Fig. 6a-c).

Factors governing compass-course performance
We used regression and model selection to diagnose and fit the extent to which spherical geometry and compass-course sensitivity effects modulate effective stepwise error (Fig. 1b), including how performance increases with increasing number of steps, N (introducing a generalized exponent g, comparing equations 11 to 34). We also accounted for seasonal 5 constraints on performance via a population-specific maximum number of steps (   Table 2), with baseline performance of TCSC 5 courses predicted to increase faster with number of steps compared with loxodrome courses, but also "decaying" nearly twice as rapidly with increasing stepwise effective error. Erroraugmentation due to the spherical geometry factor was also three times larger along geomagnetic loxodrome courses compared with geographic loxodrome or TCSC courses, reflecting heightened sensitivity to crossing bands of geomagnetic longitude (e.g., 42 ).   Fig. 7).

Discussion
Our extended formulation of compass courses has facilitated a first global assessment of factors governing robustness of inaugural migration routes, and provides an explanation for enhanced performance by naïve migrants. We propose that unassisted inaugural migratory performance is 5 mediated by the required goal breadth and minimum number of flight steps (Figs. 1, 4), and relative favourability among compass-courses by a readily-derived spherical geometry factor (equation 32, Supplementary Fig. 7), the minimum number of steps, and stepwise flight distance ( Fig. 7). While cue transfers compound effective stepwise errors to reduce overall performance, unbiased in-flight cue maintenance will improve performance with little penalty in stepwise 10 distance ( Supplementary Fig. 1c-d). Our simulation results based on hourly cue-maintenance indicate, consistently with radar measurements of nocturnally migrating birds, that magnitudes of expected cue detection, transfer and within-flight errors should remain below about 30° (circular lengths < 0.85) 31 . Motion and cue related effects presumably limit the effectiveness of higherfrequency cue maintenance. The contrasting stepwise sensitivity of the compass courses (Fig. 3) have strong implications regarding their adaptive value to migratory populations. We propose that magnetoclinic migration routes are highly unlikely to have evolved given their general high sensitivity and poor performance along strongly direction-changing routes (for which they were envisaged 13 ), with loxodrome or sun compass courses performing equivalently well or better along nearly 5 Southward routes (Fig. 4, Supplementary Figs. 3, 5-6). Contrastingly, as an emergent "many slightly-corrected wrongs" phenomenon, TCSC courses are ubiquitously more robust compared with fixed sun-compass courses, even outweighing the penalty of cue-transfers and more closely matching known routes in comparison to non-transferred loxodrome courses for the longestdistance and most high-latitude night-migratory populations tested. For most other night-10 migratory routes, geomagnetic loxodrome courses and star-compass courses performed best in biological scenarios (Fig. 4, Supplementary Fig. 3), at least assuming equivalent cue precision and availability, with slight advantages for geomagnetic courses among simulated migrants in Europe and for celestial courses in North America. When stars are not visible on departure, starcompass courses transferred from more ubiquitously available polarized light cues performed 15 less well (Figs. 4-5), and moreover rely on averaging of cues at dusk and dawn (often from different locations) in order to diagnose geographic South 8,10 . A further advantage of nocturnal TCSC courses is in similarly being achievable using polarized light cues but only at either dusk or dawn, avoiding the need to average cues. Our results support that continental-scale TCSC courses can be robust to variable schedules and to updating of hourly rates of time-compensation 20 en route (Fig. 6, equation 26), at least if inner clocks are updated and headings maintained during stopovers. Indeed, flight directions of high-latitude bird migrants 14,44 and some migratory insects 2 most closely resemble TCSC headings, and stepwise-calibration using twilight cues may 25 be most prevalent among longer-distance migrants 7, 8 . An important caveat to TCSC courses in pre-breeding (spring) migrations is that self-correction will not work for poleward movement, at least without integration with additional cues.
The finding that TCSC courses are self-correcting provides a novel explanation for naïve migrants overcoming otherwise prohibitive errors 19 , and observed route-corrections following 5 displacement. Previous studies 45 proposed naïve migrant self-correction relies on a timecompensating star compass, now generally regarded as not supported 9,29 . For the self-correction in TCSC courses based on sun azimuth described here, subsequent headings following displacement would be offset by equation (21), based on the displaced migrant's reference latitude (equation 24). Experimental evidence of self-correction following displacement by naïve 10 migrants is inconsistent, and has also often been confounded by polar or equatorial cue effects 42,46 , and probably by inner-clock updates 9,47 . For the perhaps clearest and most convincing case for naïve self-correction, by GPS-tagged Eurasian Cuckoos (Cuculus canorus) following a 28° longitude displacement at 55°N 20 , the estimated shift in headings compared with non-displaced "control" individuals (21°) is intriguingly close to as predicted (23°) using 15 equation (21).
Our results support observed diversity in migratory compass-cue hierarchy 6-8 , consistent with cue-conflict experiments, with celestial cues dominating among the most extreme routes and daytime migrants, and geomagnetic-calibrated orientation among most migrants in Europe 7,8 .
Naturally, cue favourability is also contingent upon appropriate biological cue mechanisms 2,6 , 20 and further modulated by relative cue precision and availability (particularly in polar regions or when crossing the equator 6,11,18 ), as well as topography, habitat quality and weather factors 2,48 .
For example, analysis of light-level geolocation data of Common Rosefinches migrating along the modelled route (Fig. 4c) revealed largely wind-driven movement with a detour around the Iranian Desert 34 , and it is not clear whether naïve Nathusius Bats display innate migratory directions 49 . Our models can be readily extended to consider geographical and meteorological factors, as well as regarding the extent to which spatial variability in inherited headings 50 can maintain routes given long-term spatiotemporal variability in the Earth's geomagnetic field 27,51 5 or, for sun compass courses, seasonal migration schedules 32,52 . More generally from a movement ecology perspective, our study highlights that care must be taken when assessing movement without accounting for cue precision, and that, even in the age of big data and tracking 53 , models of simple responses in simplified environments can still reveal novel emergent effects with potentially profound life-history implications.

Stepwise directed movement on a sphere
Terms defining stepwise movement, precision and geophysical orientation cues are listed in Table 1. Since seasonal migration nearly ubiquitously proceeds from higher to lower latitudes, it 5 is convenient to define headings clockwise from geographic South (counter-clockwise from geographic North for migration initiated in the Southern Hemisphere). Given stepwise headings on a sphere, , with i = 0, ..., N-1, stepwise latitudes, ∅ +1 and longitudes, +1 , can be calculated using the Haversine equation 54 , which can be approximated by stepwise planar movement: Here, the stepwise distance where ,ℎ are hourly in-flight headings relative to geographic South. In the absence of drift effects (see below), migrants were assumed to retain their preferred (i.e., expected) headings 20 from stepwise departures 17,48 , either by accounting for (hourly) sun or star rotation, or else relative to a geomagnetic axis 6,7 . Accordingly, for a geomagnetic in-flight compass, expected headings, ̅ ,ℎ , are modulated by changes in the magnetic declination, , ,ℎ , i.e., the clockwise difference between geographic and geomagnetic South 6 : ̅ ,ℎ = { ̅ ,0 , celestial compass ̅ ,0 + , ,ℎ − , ,0 , geomagnetic compass (5)
Stepwise and in-flight errors were simulated using a von Mises distribution, defined by an 15 angular "concentration" parameter, κ, analogous to the reciprocal of variance in headings: where is the modified Bessel function of the first kind and order j 30 . For sufficiently small concentrations, κ , von Mises samples are similar to normally sampled variables with "effective standard error", = 1 √κ ⁄ 30 . However, unlike sums of normal variables, circular random errors do not sum in a scale-free way, or necessarily even follow the same distribution as their components 30,58 . Therefore, to assess compass courses, it is convenient to first consider the case of independent stepwise normal movement on a plane 16,24 , and then extend this to account for circular error 5,25,59 , spherical geometry effects 23 and, for non-loxodrome courses, We can analogously estimate effective standard error after N steps for a single individual, , or within a migratory population, considering both within-individual effective error following the expected number of steps, ̂, and between-individual variability in preferred (inherited) headings, :

Migratory performance on a plane
Performance (arrival probability) of independent stepwise planar movement to a (circular) goal area of radius will approximate a cumulative normal distribution (erf function), based on the breadth of successful angles and overall effective error, which is modulated by the expected number of steps. For long-distance migration, successful angles follow the goal-area breadth 10 ( Fig. 1, Table 1), since = ⁄ ≅ tan −1 ( ⁄ ). Assuming uniform population headings and applying equation (9) and the Central Limit Theorem for large numbers of steps, a first planar approximation to sufficiently directionally accurate migration is where ̂= 0 • 1 (κ ) 0 (κ ) ⁄ is the expected number of steps, κ ≅ −2 5,30,59 , and is the minimum (error-free) number of steps to reach the closest edge of the goal area. From equation (11) we see that within the planar and normal limit, i.e., high stepwise concentrations, κ , performance roughly follows the "length-adjusted goal breadth", 20

Formulation of compass course headings
Since sun compass headings vary with date, to ensure temporally consistent flight directions from the initial (natal) site with sun compass courses, we assumed that preferred headings were imprinted from inherited geographic or geomagnetic headings 2,6,7 .
Loxodrome headings 5 Expected stepwise geographic headings remain unchanged en route, i.e., Expected stepwise geomagnetic headings remain unchanged relative to proximate geomagnetic South, i.e., are offset by stepwise declination en route 10

Magnetoclinic compass headings
As described and illustrated in detail in 13 , the magnetoclinic compass was hypothesized to explain the prevalence of "curved" migratory bird routes, i.e., for which local geographic

Sunrise and sunset azimuth
In Supplementary

Fixed sun compass headings
Fixed sun-compass headings represent a uniform (clockwise) offset, α ̅ relative to the spatiotemporally-shifting sun azimuth, , , where, to ensure consistent initial flight directions at the initial (natal) site, the preferred heading 5 α ̅ = α ̅ 0 − ,0 is presumed to be imprinted using an innate geographic or geomagnetic heading.

Time-compensated sun compass (TCSC)
Even outside the realm of migration, many insects 58,61 , and birds 28,57 are known to use a timecompensated sun compass to maintain preferred directions locally, by accounting for the daily rotation in sun azimuth. In a pioneering work addressing migration, Alerstam and Pettersson 14 10 made the link between the "clock-shift" induced by crossing bands of longitude (meridians), ∆h = 24•∆λ/2π, and its effect on a migrant adjusting its heading to the (hourly) rotation of the sun's azimuth, resulting in an offset to their interpretation of sun azimuth, and therefore to their "time- 15 compensated" offset on departure at sunset: Equation ( and , specifying the reference step for "time-compensated" adjustments to sun azimuth

Sensitivity of compass course headings
Sensitivity was assessed by the marginal change in expected heading from previous headings, ; when this is positive, small errors in headings, and therefore migratory trajectories, will grow iteratively. Geographic and geomagnetic loxodromes are per definition constant 15 relative to their respective axes so that, as long as stepwise errors are stochastically independent, have "zero" sensitivity.
All three terms in the denominator indicate, as illustrated in Fig. 3b, that magnetoclinic courses become unstably sensitive at both high and low latitudes, and any heading with a significantly 5 East-West component.  10 The sine factor on the right-hand side in equation (26)  sensitivity-reducing, resulting in a broad range in latitude and headings with self-correcting 5 headings ( Fig. 3c-f). The third bracketed terms in equation (27b) with proximate TCSC is also negative, and in fact increasingly so until clocks are reset, bur remains small in magnitude compared to the second term.

Spatiotemporal orientation and movement model
To assess the feasibility and robustness of each compass course to spatiotemporal effects on a contrasting airborne species (hereafter species simulations) chosen for diversity among taxa, latitude and longitude ranges and goal-area breadths ( We assessed robustness of the global and species simulations in two ways: 1) for effective total stepwise standard errors of 0°-60°, i.e., ignoring schedule-related or further sources of 20 variability, and 2) for biologically-relevant scenarios incorporating within-step cue detection, transfer and maintenance errors (assuming equivalent magnitudes in standard error), variability in migratory departure and stopovers (Table 2), as well as effective standard errors of 2.5° in inherited (between-individual) headings 50 and 15° in hourly in-flight drift, presumed to be autocorrelated 51,55 with hourly (coefficient 0.75) and also between flight-steps (coefficient 0.25), but not following extended stopovers.

Accounting for seasonal constraints, spherical-geometry and self-correction effects
Seasonal migration constraints 5 In assessing performance, we also accounted for seasonal migration constraints via a populationspecific maximum number of steps, ( Table 2; this became significant for the longest-

Spherical-geometric modulation of longitude errors
On the sphere, stepwise longitude (equation 2) naturally contains a secant factor, i.e., cosine of 15 latitude in the denominator, reflecting the convergence of meridians (bands of longitude) with increasing latitude. This secant factor causes the sensitivity of stepwise longitude to stepwise headings to increase with latitude: meaning that orientation errors at higher latitudes will exert a greater influence on overall longitudinal error, for any compass course. Due to this secant factor, the effective route-mean longitudinal error will scale approximately as in a Mercator projection 23 : ln (tan ( ∅ 0 + ∅ 2 ) + 4 ) (30) where ∅ 0 and ∅ are the initial (natal) and arrival latitude, respectively. To assess total error, the 5 multiplicative factor L will be modulated by the (mean) orientation en route: = √(L sin ̅ ) 2 + cos 2 ̅, the scaling factor therefore being largest for purely Eastward or Westward headings ( = ≥ 1) and nonexistent for North-South headings ( = 1, reflecting no longitude bands being crossed). 10 We further modified the effective goal-area breadth by a fixed factor to account for a (geographically) circular goal area on the sphere, i.e., effectively modulating the longitudinal component of the goal-area breadth at the arrival latitude, ∅ : = √sin 2 ̅ + (cos ̅ / cos ∅ ) 2 (32). 15

Error sensitivity and error-correction effects
To accommodate compass-course-specific sensitivity (iterative augmentation or self-correction in stepwise errors), we generalized the "normal" inverse-square-root relation between performance and number of steps (equations 11-12), from 1/̂0 .5 , to 1/̂, with ( | , ) = (0.5 + ) − 2 , 20 where b < 0 reflects iterative augmentation of stepwise errors and b > 0 self-correction, and s represents an exponential damping factor, consistent with the limiting circular-uniform case (as κ → 0, i.e., → ∞), where no convergence of heading is expected with increasing step number (given modelled migration was terminated South of the goal area).
Assessing performance using regression and model selection 5 For each compass course, based on route-optimized simulations among all 9 species, we fitted performance as the product of sufficiently timely migration (equation 27) and sufficiently accurate migration (equation 11), with the latter updated to account for the "non-normal" effects (equations 30-32), i.e., = ∅, • ,̂. Accordingly, we used MATLAB routine fitnlm based on the route-optimized species simulations and, to fit all combinations of up to four 10 parameters for each compass course, and selected among models with parameter combinations using AICc, the Akaike information criterion corrected for small sample size 64 . Specifically, we accounted for i) a compass-route specific fitted exponent, g, to the spherical geometry factor (equation 30), i.e., , reflecting how sensitivity or self-correction in stepwise errors further 15 augments or reduces this factor, ii) a baseline offset, b0, to = 0.5, as in equation (33), iii) a fitted exponential damping factor s with respect to stepwise error (equation 33), (iv) for TCSC courses, a fitted modulation , quantifying the extent to which selfcorrection increases with increased stepwise distance , i.e., = 0 ′ in 20 equation (33), where ′ is the stepwise distance scaled by its median value among species.