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AlphaEarth distinguishes cities but compresses urban variation

Andrew Renninger

arXiv:2609.30356v1cs.CVphysics.soc-ph

TL;DR

Global urban comparison remains underexamined because prior representation studies have not sampled cities worldwide. We audit AlphaEarth embeddings across 1,000 cities, finding geographically structured city signatures but unequal retained variation and annual movement not yet validated as urban change.

  • Problem

    Prior representation studies had not tested how a planetary field allocates capacity across cities worldwide, limiting evidence for global urban comparison.

    Source: Introduction · p. 3

  • Method

    We probe AlphaEarth’s geometry, geographic structure, and retained variation using samples from 1,000 urban areas in 162 countries.

    Source: Introduction · p. 3, Data and processing · p. 4

  • Results

    Continent and climate predict 24.3% of variation among urban-centre mean directions, urbanisation degrees account for 8.9% of embedding variation, and annual movement is nearly eight times within-year redrawing.

    Source: Introduction · pp. 3–4

  • Takeaways & Limitations

    AlphaEarth supports comparison across regions, but cities retain unequal amounts of variation and annual layers are not calibrated for temporal comparison.

    Source: Discussion · pp. 21–22

    Annual movement includes sampling and observing-condition effects, so it is not established as change in the urban fabric.

    Source: Discussion · p. 22

Abstract

from arXiv · show

Cities differ in built form, land cover and development history, complicating comparison across places and time. Satellite foundation models map Earth's surface onto common numerical representations. Yet the tasks and targets used to shape them typically do not focus on cities: globally consistent labels for urban function do not exist, and many datasets - especially land cover and land use classifications - collapse the built environment into few classes. Here we audit the representation, focusing on AlphaEarth but with broader applicability to other Earth embeddings, by probing the geometry and geography of embeddings for 1,000 urban areas in 162 countries. We find that cities occupy a shifted but overlapping region on the hypersphere, 62.7{\deg} from the global mean direction, and continent and climate predict 24.3% of variation among the mean directions of urban centres in excluded countries. Inside cities, degrees of urbanisation carry 8.9% of the variation, and what they leave holds shared directions whose local orientation varies, not one universal axis of urbanisation. Retained variation is itself unequal: dispersion within urban centres is 14.1% greater per standard deviation of national development, even after adjusting for population, land area and continent. Further controls suggest cities in developing countries present less contrast in vegetation and texture, and dispersion follows that contrast: full adjustment for it leaves at most 6.4% of the gradient. Annually, a city's representation moves nearly eight times more than redrawing its own pixels explains, and contracts where the 2022 loss of Sentinel-1B removed a pass direction. AlphaEarth's representations therefore support comparison across regions, while the differences between its annual layers are not yet validated for comparison over time.

Introduction

Earth embeddings offer a shared space for comparing cities, but existing labels and evaluations do not adequately measure urban function or variation within and between cities. This study audits AlphaEarth’s urban representations across global contexts and years, finding geographic structure alongside uneven retained variation and substantial annual motion.

Introduction

  • Earth embeddings make comparison possible in a shared coordinate system without predefined classifications, but their finite capacity must serve all terrestrial surfaces and training targets.Urban land occupies under 1% of total land area.Source: Introduction · pp. 1–2
  • Globally consistent labels for urban function are unavailable, and existing evaluations either collapse built environments into few classes or cover differentiated urban categories only in Europe.No evaluation measures how much variation a representation retains within a class or between cities.Source: Introduction · p. 2
  • 24.3% of variation among urban-centre mean directions in excluded countries is predicted by continent and climate, while urbanisation classes account for 8.9% of within-city embedding variation.The remaining variation within classes retains shared directions whose local orientation varies rather than following one universal urbanisation axis.Source: Introduction · p. 3
  • Nearly eight times more adjacent-year movement in a city’s mean direction occurs than redrawing its pixels within one year explains.The loss of Sentinel-1B coincided with an additional 4.1-percentage-point contraction in dispersion for cities that lost one viewing direction.Source: Introduction · p. 4

Materials and Methods

The methods preserve AlphaEarth’s spherical geometry while comparing city means, within-city dispersion, geographic structure, development context and annual movement. The study uses a broad global city sample, with restricted subsets for analyses requiring equal class samples or complete temporal comparisons.

Data and processing

  • Each AlphaEarth observation is treated as a unit vector, and city or class mean directions and dispersion summarize position and variation on the sphere.Comparisons of local variation map tangent vectors to a common reference while preserving lengths and mutual angles.Source: Data and processing · p. 4, Spherical comparisons · p. 5

Study sample and AlphaEarth observations

  • The sample comprises 1,000 Functional Urban Areas with at least 250,000 residents in 2015, spanning 162 countries; four-class analyses use 544 cities with 250 observations per class.The temporal city-path analysis retains all 1,000 cities across eight annual layers; comparisons within each degree require 538 cities with all four degrees in every year.Source: Study sample and AlphaEarth observations · pp. 4–5

Reference and contextual data

  • The global reference uses 247,565 labeled observations over land, while city context is described by continent, climate, population, land area and national HDI.Physical-form products describe measured form and product coverage, not contemporaneous ground measurements for 2024.Source: Reference and contextual data · p. 5

Urban position and geographic structure

  • Urban position is compared with the global mean, and geographic structure is tested through clustering, population-matched context comparisons and prediction in excluded countries.The analyses also compare city identifiability using split-sample nearest-mean assignments.Source: Urban position and geographic structure · p. 6

Variation within standard urban classes

  • Four urbanisation classes are compared against finer-scale embedding variation, and subdivisions learned in some countries are evaluated in others to test residual reproducibility.The class labels are native at 1 km while AlphaEarth is nominally 10 m, so the partition does not itself assign meaning to residual variation.Source: Variation within standard urban classes · p. 6

National development and urban dispersion

  • The principal outcome is a population reduction over every valid cell in the urban centres a city contains, rather than a sample statistic; on the same cities, the 250-pixel estimate is 16.0% (9.0 to 23.5%).Source: National development and urban… · p. 7

Alternative explanations

  • Development models test robustness across years and contexts, while alternative-explanation analyses add physical, environmental and observation-related covariates.The development models are associational: lower embedding dispersion does not itself show a less varied physical city, worse downstream performance or a particular training cause.Source: National development and urban… · p. 7, Alternative explanations · pp. 8–9

Temporal separation and motion

  • Annual city motion is measured from spherical path and endpoint distances, while the sampling floor estimates movement attributable to drawing different pixels within a fixed degree.Because annual samples do not retain fixed locations, the floor leaves fixed-site annual variation and other sources of movement unresolved.Source: Temporal separation and motion · pp. 7–8

Results

The results examine how AlphaEarth represents cities across geographic contexts and within-city variation, and how those representations change from year to year. They also assess how urban classes, sensor loss, weather and selected construction projects relate to these patterns and movements.

Cities occupy a distinct part of AlphaEarth space

  • 62.66° separates the urban mean direction from the global mean, while city directions overlap broadly and urban pixels remain more alike to one another than global pixels do.Half of city mean directions lie within 54.69° of the urban mean, and 90% within 67.88°. In the 2024 release, each city was identified from its held-out half in all 5,000 trials, but individual pixels returned to their own city only 52.4% of the time; urban-class labels also track an overlapping gradient rather than four clean pixel classes. Broad land cover and climate are predictable (median AUROC 0.919 and 0.929), while physical targets are predicted more strongly than population density.Source: Cities occupy a distinct part of… · pp. 9–10, Figure 1 · p. 11

Geography structures continuous differences between cities

  • 24.3% of variation among urban-centre mean directions in excluded countries is explained by continent and climate, while city differences form a continuum rather than discrete global types.Population-matched cities sharing continent and climate are 30.8% closer than those sharing neither. The weak association between centrality and internal dispersion is not stable under the catalogue’s sampling design, while mean positions and internal-variation orientations share local structure without defining a fixed global taxonomy.Source: Geography structures continuous… · pp. 10, 12

Most variation within cities remains inside broad urban classes

  • In the largest exactly balanced sample of 544 cities across 117 countries, differences among the four urban-class means account for 8.88% of within-city tangent variation, leaving 91.12% within classes.Splitting variation by WorldCover classes leaves 71.8% within classes, and splitting by both schemas leaves 65.9%. The average class path is nearly straight but accounts for only 31.9% of variation in cities’ own paths; the remaining 68.1% is city-specific and spreads over about twenty directions.Source: Most variation within cities remains… · p. 12

Dispersion within cities is associated with national development

  • 14.1% greater dispersion within urban centres is associated with a one-standard-deviation increase in national HDI, adjusted for population, land area and continent.The association is descriptive, and its direction is broadband across the representation. Removing African cities lowers the association, and finer geographic controls widen its interval to include zero.Source: Dispersion within cities is associated… · p. 14

Vegetation, radar structure and built form account for the gradient in nearly equal parts

  • 33.1% of the HDI-gradient reduction is attributed to vegetation, 29.6% to radar surface structure and 28.6% to built form when contributions are averaged over all orderings.The table’s averaged attribution sums to a 0.125-log-unit reduction; population contributes −2.7%. The fully adjusted residual is 1.4% with raw radar backscatter but 6.4% under an optical–radar index specification, so the residual depends on how surface structure is measured.Source: Vegetation, radar structure and built… · p. 16, Table 1 · p. 17

Separation between cities contracts while city paths repeatedly reverse

  • 1.61° is the 2017–2024 decline in median separation after pooling urban observations within city, while the median pair reverses direction four times across seven annual changes.In the balanced panel, median separation falls by 1.08° in peri-urban areas, 1.31° in semi-dense, 1.75° in dense urban and 1.95° in urban centres. The overall contraction is a panel tendency rather than a path followed by every pair.Source: Separation between cities contracts… · pp. 17–18
  • Within-city sampling explained under 2% of squared movement for whole cities and about 5% within degrees of urbanisation.The two sampled halves moved means by 1.57° for a whole city and 2.99° within one degree, versus 12.46° and 12.83° of observed adjacent-year movement; documented model-release changes could also move a layer without moving a city.Source: Separation between cities contracts… · p. 18

Sentinel-1B loss changes local dispersion but leaves the global variance hierarchy intact

  • 4.08 percentage points greater contraction affected the 159 cities losing a Sentinel-1 pass direction than the 841 comparison cities from 2021 to 2022.The 46 cities with a fixed point receiving no Sentinel-1 scene in 2022 showed a −19.93-point difference; they had already contracted 2.92 points more in 2017–2018, and no other placebo contrast excluded zero.Source: Sentinel-1B loss changes local… · pp. 18, 20

Separation between cities contracts while city paths repeatedly reverse

  • 11.45° median annual path versus 1.91° endpoint net speed (straightness 0.17); pair separation narrows 1.08° in peri-urban areas and 1.95° in urban centres, while paths repeatedly reverse.Each pair’s seven annual steps change sign a median four times, matching the exchangeable expectation.Source: Separation between cities contracts… · p. 18, Figure 4 · p. 19

Sentinel-1B loss changes local dispersion but leaves the global variance hierarchy intact

  • The Sentinel-1-loss direction had discovery–holdout cosine 0.162, and projecting it removed 1.26% of total trace variance without changing any global variance share by more than 0.10 percentage points.The held-out coefficient was not distinguishable from zero, so the candidate nuisance direction did not organise the global variance hierarchy; this does not rule out sensor contributions to the HDI association.Source: Sentinel-1B loss changes local… · p. 20

In the embeddings, construction appears but weather confounds

  • Weather anomalies reduced prediction error for annual embedding changes by 12.7%; adding seasonal vegetation raised the reduction to 15.6%, and retrieval quality raised it to 17.0%.The final increment spans zero (−0.4 to 3.2 points); snow-cover change reduced error by 12.0% in 379 cities with at least 1% peak climatological monthly cover, versus 0.04% in the other 621. These predictive scores do not identify ground change as the cause of unexplained error.Source: In the embeddings, construction… · p. 20
  • At eleven construction projects in nine countries, median excess displacement beyond matched controls was positive in every case, ranging from 7.8° in Paris to 57.5° in Phoenix and reaching 20.2° across projects.The controls are not certified unchanged, and the selected projects cannot establish how much movement across all 1,000 cities is physical change.Source: In the embeddings, construction… · p. 20

Discussion

AlphaEarth distinguishes urban environments and supports comparison across regions, but its common geometry is geographically patterned, uneven in retained variation, and not yet validated for temporal comparison. Its representations are useful as features for analysis, not as self-interpreting or neutral measures of urban concepts.

Discussion

  • AlphaEarth supports comparison across regions, but its representations are not self-interpreting, geographically neutral or temporally faithful.The discussion distinguishes city-level discrimination from unequal representation and unvalidated annual differences.Source: Discussion · p. 21
  • Continent and climate explain a quarter of variation among city means, so embedding neighbours need not be functional urban analogues.Analysts must decide whether geographic context is part of the intended similarity or a nuisance to control.Source: Discussion · p. 21
  • The coarse urban schema leaves most retained variation unresolved, while finer subdivisions distinguish environments treated as equivalent by that schema.The subdivisions transfer across countries and separate independently measured vegetation, population and built form; finer land cover recovers about a fifth of what urbanisation degrees leave.Source: Discussion · p. 21
  • A shared urban gradient explains under a third of variation in cities’ paths, whose leading directions sit a median 33° from the pooled set.The common direction captures “urban” only partly; each city marks the gradient with properties unique to it.Source: Discussion · p. 21
  • Dispersion varies with national development, while vegetation, radar surface and built form account for nine tenths of the variation removed by adjustment.The discussion says landscape contrasts, rather than development itself, best explain this pattern; population accounts for none.Source: Discussion · p. 21
  • AlphaEarth suits exploratory comparison, retrieval, stratification and feature construction, and supervised models with externally defined targets and geographic holdouts.It is less defensible as an off-the-shelf measure of form, function, development or inequality.Source: Discussion · pp. 21–22
  • Annual paths move far more than they retain as displacement, and sampling, weather and sensor effects remain distinct from change on the ground.Until a temporal noise floor is calibrated, comparison over time remains unvalidated; the Sentinel-1B outage also changed dispersion where a pass direction was lost.Source: Discussion · p. 22
  • The sample excludes smaller settlements, observations are not area- or population-weighted, and fixed 2020 urbanisation classes are coarse.The labels derive from census inputs whose quality tracks development, and the direction of that bias remains unresolved.Source: Discussion · p. 22

Supplementary Information

The supplementary methods define the city sample, external covariates and global reference, then use spherical geometry to compare mean directions and within-city variation. Sampling is stratified by urbanisation degree, while the geometric summaries retain the unit-sphere structure of AlphaEarth embeddings.

S1. Sampling and data processing

  • The sample comprises 1,000 functional urban areas with at least 250,000 residents in 2015, and assigns locations to four fixed 2020 urbanisation degrees.Those degree labels were delineated at 1 km, so sampling at 30 m does not increase their resolution.Source: S1. Sampling and data processing · p. 26
  • Up to 250 valid land observations per city and urbanisation degree yield 840,776 observations across 162 countries.The equal-sample analyses use 544 cities with 250 observations in each degree; each analysis uses the largest sample its inputs allow.Source: S1. Sampling and data processing · p. 26
  • Analyses use distinct frames for the Sentinel-1B outage, and each city has equal weight in comparisons among cities.The outage analysis uses fixed global-grid points and city-years; comparisons describe the stratified sample rather than footprint area or population.Source: S1. Sampling and data processing · p. 26
  • The global reference begins with 40,000 candidates in each of 24 equal-area cells, retaining land observations with equal weight.The reference samples locations once each, and its urbanisation-degree composition describes the sample rather than selecting it.Source: S1. Sampling and data processing · p. 26
  • External city covariates include climate family, population, land area, national HDI, population density and building fields.The primary development model uses 977 cities in 157 countries; the outcome is measured over urban-centre cells while physical covariates use sampled locations.Source: S1. Sampling and data processing · p. 26

S2. Spherical geometry

  • Comparisons use angular separation between unit-length 64-component vectors and normalized resultants, or barycentres, to summarize directions.Urban barycentres weight cities equally, whereas the global reference barycentre weights observations equally; these are extrinsic directional means.Source: S2. Spherical geometry · p. 27
  • Within-city dispersion is measured after mapping observations to a tangent plane, and vectors from different cities are parallel-transported before comparing directions.Parallel transport preserves lengths and mutual angles, while unit vector norms make dispersion a share of a bounded total second moment.Source: S2. Spherical geometry · p. 27

S3. Semantic, geographic and urban comparisons

This section compares what urban classes and embedding directions capture, how geographic context structures city similarities, and how cities can be distinguished from one another. Its comparisons also expose the limits of available labels and sampling for testing finer urban schemas.

S3. Semantic, geographic and urban comparisons

  • Semantic transfer is tested by predicting 18 WorldCover, Köppen–Geiger and GHSL contrasts from directions fitted with one equal-area cell held out at a time.The contrasts include two independent of urban labels, while GHSL settlement serves as a positive control; targets describe surface, environment, built form and intensity, not human function.Source: S3. Semantic, geographic and urban… · pp. 27–28
  • The geographic analysis tests whether city similarities transfer to excluded countries using continent and climate models, compared with population quintiles and whole-country splits.The analysis also matches cities on population across continent and climate combinations before comparing their embedding distances.Source: S3. Semantic, geographic and urban… · p. 28
  • City identification assigns held-out city means and pixels to the nearest of 1,000 city means computed from separate pixel halves.The analysis also measures errors by their geographic proximity and estimates how many city pairs are separated by less than twice the median angular standard error.Source: S3. Semantic, geographic and urban… · p. 28
  • The variance partition separates displacement among four urbanisation-degree means from variation within each degree, then tests whether subdivisions learned in discovery countries persist in confirmation countries.The subdivisions use 20 principal directions and four clusters per degree; the 16-group refinement is illustrative rather than selected.Source: S3. Semantic, geographic and urban… · p. 29
  • WorldCover leaves 71.81% of within-city tangent variation within its classes, versus 91.12% for the four degrees of urbanisation; crossing the schemas leaves 65.89%.Pooling classes with fewer than ten pixels changes the WorldCover share by 0.03 percentage points; the finer schema recovers about a fifth of what the degrees leave unresolved.Source: S3. Semantic, geographic and urban… · p. 29
  • Local climate zones remain untested because the 544,000-pixel sample lacks coordinates needed to join pixels to an external map.The coordinate-bearing table has only 1,160 urban-centre points across 21 equal-area cells, insufficient for within-city partitioning or country-resampled intervals.Source: S3. Semantic, geographic and urban… · p. 29

S4. National development models

This section models the relationship between national development and urban embedding dispersion, accounting for city and geographic context. It also tests whether the observed gradient travels to excluded countries and how measured covariates account for it.

S4. National development models

  • HDI coefficients are scaled as percentage differences per standard deviation, with population, land area, collection stage and continent represented in the models.HDI standard deviations vary from 0.110 to 0.168 across subsamples, so coefficients from different subsamples are on different scales.Source: S4. National development models · p. 30
  • 14.1% (6.7 to 22.0%) is the dispersion difference associated with one standard deviation higher HDI for every valid urban-centre cell, after controlling for cell count.The same association on 250 sampled urban-centre pixels is 16.0% (9.0 to 23.5%).Source: S4. National development models · p. 31
  • The wild cluster bootstrap gives a p-value of 0.0003 against zero for the principal coefficient, while prediction is also tested in countries excluded from fitting.Each bootstrap replicate sign-flips a country's restricted residuals under the null; excluded-country prediction resamples countries and refits the repeated split plan.Source: S4. National development models · p. 31
  • The covariate adjustment uses both a fixed-order ladder and an order-invariant decomposition that averages each block's marginal contribution across all 720 orderings.The six blocks' contributions sum to the difference between the base and fully adjusted HDI coefficients; paired comparisons of the ladder and surface index separate their two measured differences.Source: S4. National development models · p. 31
  • Cross-validated variance analyses fit common references and principal directions on other countries, then project excluded cities onto those directions.They compare absolute variance and variance shares across individual directions and spans, with continent and collection-stage fixed effects.Source: S4. National development models · pp. 31–32

S5. Physical form and observation checks

This section measures physical form and observation opportunity as potential accounts of development-related dispersion, and tests whether AlphaEarth changes alongside Sentinel-1B coverage loss. The tests use city-level physical summaries and observation contrasts at multiple spatial scales.

S5. Physical form and observation checks

  • Physical form is summarized with Shannon entropies of population and building fields, while the surface index combines standardized Sentinel-2 NDVI and DpRVI summaries.The entropy measures use ten globally fixed bins of positive values and, where applicable, a physical-zero class; the index enters models through city medians and interquartile ranges.Source: S5. Physical form and observation checks · p. 32
  • Terrain adjustment summarizes elevation spread, local relief and topographic position, while the analysis excludes unprojected atlas slope summaries because they correlate 0.22 with projected-surface slope.Terrain enters as an eighth ladder rung and seventh order-invariant block on the 937 cities covered by the model.Source: S5. Physical form and observation checks · p. 32
  • The fixed 942-city ladder's final rung is 0.1% (−4.1 to 4.6%) with raw Köppen zone fractions, versus 4.6% (−0.9 to 10.4%) with the documented index.The ladder uses raw Sentinel-1 VH summaries rather than the previously described radar index; the nine documented climate groups lower every rung by 0.8 to 1.8 percentage points and reorder none.Source: S5. Physical form and observation checks · pp. 32–33
  • Observation opportunity is measured from Sentinel-1, Sentinel-2 and Landsat acquisition counts, clear fractions, orbit geometry, pass imbalance and Sentinel-1B loss.Counts are taken at eight fixed urban-centre points for adjustment models in 214 cities and mapped across 529 cities.Source: S5. Physical form and observation checks · p. 33
  • The Sentinel-1B analysis tests 2021–2022 changes against other annual transitions at fixed points, across whole cities and within urbanisation degrees, then tests for a transferable shared direction.At city scale, a global rather than continent-specific background leaves a +0.89° contrast for cities that lost a pass direction, with an interval crossing zero.Source: S5. Physical form and observation checks · pp. 33–34

S6. The sampling floor beneath annual movement

This section estimates how much annual movement in city representations could arise from sampling different pixels rather than from changes in the representations. It distinguishes that estimable component from other sources of movement and states what would be needed to validate temporal comparisons.

S6. The sampling floor beneath annual movement

  • The temporal noise floor bounds displacement expected when the underlying place is unchanged, but only the component from sampling different places within a degree is estimable from the original city samples.The floor also includes fixed-site year-to-year and acquisition variation and change across the whole product.Source: S6. The sampling floor beneath annual… · p. 34
  • Split-half samples estimate spatial sampling error, and the expected component in an adjacent-year separation is the sum of the two annual estimates.The analysis partitions each city-degree-year sample into non-overlapping halves and reports the share of mean squared annual separation remaining after subtracting the estimated spatial component.Source: S6. The sampling floor beneath annual… · p. 34
  • The reported excess ratio is not a reliability ratio: other noise-floor components and real landscape change remain in its numerator.It therefore cannot be read as the share of movement that is physical or reproducible.Source: S6. The sampling floor beneath annual… · p. 34
  • Rows are sampled afresh without retaining cell geometry, so recurring row identifiers do not represent repeated locations across years.Only 0.0064 to 0.0090% of identifier-matched rows agree exactly on all five fields expected to remain fixed; the data are not treated as a fixed-site panel.Source: S6. The sampling floor beneath annual… · pp. 34–35
  • A claim to monitor change requires movement to exceed the full floor and persist in locations and countries excluded from fitting, with observation opportunity accounted for.Fixed-location extracts alone do not establish unchanged surfaces; stable-surface negative controls and dated positive controls are separately required, and the construction comparisons do not establish the full floor.Source: S6. The sampling floor beneath annual… · p. 35

S7. Weather and construction as tests of annual movement

The section tests whether annual embedding movement reflects observable conditions or documented construction change. It models weather and observation proxies across city-year transitions and tracks fixed pixels at eleven projects against matched city controls.

Predicting movement from weather and surface state.

  • The 1,000-city weather analysis covers 8,000 city-years and 7,000 adjacent transitions, using cross-country and cross-year tests to predict transported tangent change.Predictors include ERA5-Land weather, MODIS vegetation and quality measures, and Sentinel-2 acquisition and visibility proxies; the imagery AlphaEarth accepted is undisclosed.Source: S7. Weather and construction as tests… · p. 35
  • The weather analysis separates predictor gain from a year-aware intercept baseline, with paired intervals resampling countries but not refitting models or sampling different years.Preprocessing, imputation and penalty selection use training folds alone.Source: S7. Weather and construction as tests… · p. 35

Fixed pixels at construction projects.

  • At eleven construction projects, fixed 10 m pixels are followed through eight annual layers and compared with named places and matched locations elsewhere in each city.Paris tracks 7,847 pixels within 500 m of the Olympic village centre; that circle covers the site and surroundings, not the development boundary.Source: S7. Weather and construction as tests… · pp. 35–36
  • Project excess displacement subtracts matched-control movement, while a second residual tracks whether targets move closer to or farther from the same-year local place vocabulary.Positive residuals mean targets move farther outside that vocabulary; medians and interquartile ranges describe pixels, not uncertainty across independent projects.Source: S7. Weather and construction as tests… · p. 36
  • The fixed-pixel cases span nine countries and 88 extracted city-years, with project intervals ranging from Paris’s 2019–2024 period to eight cases covering 2017–2024.The original city tables cannot follow locations through time, so these analyses use separate fixed pixel sets rather than a common eight-year city panel.Source: S7. Weather and construction as tests… · pp. 35–36

S8. Supplementary Results

The supplementary results show that urbanisation degrees are only partly recoverable from individual embeddings, while continuous built-form measures transfer better than population density. The analyses use samples whose size and balance vary by question.

S8. Supplementary Results

  • Balanced accuracy for recovering four urbanisation degrees is 0.366 from absolute-position vectors and 0.375 from city-centred vectors, against 0.25 by chance.Recall is about 0.60 at the peri-urban and urban-centre ends, while the middle classes remain weak even when centroids are computed within each city.Source: S8. Supplementary Results · p. 36
  • Splitting each city’s observations in half reproduces the endpoint-versus-middle pattern, showing that the classification shape persists when cities cannot differ across the split.Among 544 equally sampled cities, endpoint scores are 0.542 and 0.567, versus 0.392 and 0.334 in the middle classes.Source: S8. Supplementary Results · p. 36
  • Building height and volume transfer to unseen countries better than population density, with Q2 values of 0.495 and 0.390 versus 0.090, and 0.566 and 0.556 versus 0.250 in 270 further cities.These values measure prediction beyond urban class; the further-city models are applied without refitting.Source: S8. Supplementary Results · p. 36, S8.1. Samples and coverage · p. 37

S8.1. Samples and coverage

  • The catalogue contains 1,000 urban areas in 162 countries above a 2015 population of 250,000, and is a sample rather than a census of all qualifying cities.The 2024 table samples 840,776 locations, stratified by four 2020 urbanisation degrees and capped at 250 observations per city and class.Source: S8.1. Samples and coverage · p. 36
  • Analyses requiring equal representation in all four classes use 544 cities in 117 countries, while other analyses use broader but variably classed samples.Across the balanced sample, built-up share rises from 19.5% in peri-urban locations to 52.0% in urban centres; this is sample composition, not physical area or population.Source: S8.1. Samples and coverage · p. 36
  • Table S1’s panels classify pixels in A and B but predict cell means in C, so their metrics are not directly comparable.The table contrasts degree-class recovery with transfer of continuous settlement characteristics.Source: S8.1. Samples and coverage · p. 37, Table S1 · p. 37

S8.2. Planetary context of the urban representation

Urban mean directions are shifted from the global reference but overlap across contexts, and the embedding distinguishes cities more reliably when their pixels are averaged. Environmental targets describe physical correlates, not human function.

S8.2. Planetary context of the urban representation

  • 62.66° separates the equally weighted urban barycentre from the global mean direction, which lies close to very-low-density rural land rather than representing a typical Earth surface.The reference is dominated by very-low-density rural land, whose barycentre is 4.64° from the global one; major land-cover classes are also tens of degrees away.Source: S8.2. Planetary context of the urban… · p. 38, Table S2 · p. 38

S8.3. Physical and environmental correlates of urban separation

  • Urban-centre pixels lie 6.09° farther from the global mean than rural pixels, and excluded equal-area cells yield median AUROC values of 0.919 across 11 land-cover concepts and 0.929 across five climate families.The AUROC comparison concerns independently mapped concepts; GHSL settlement contrasts serve as a positive control.Source: S8.3. Physical and environmental… · p. 38
  • The available targets cover environment, surface composition, built form and intensity, but not human function, and fixed-basis displays do not establish intrinsic axes or causal interpretation.Night light is an intensity proxy, and independent activity or land-use labels were unavailable.Source: S8.3. Physical and environmental… · p. 38

S8.4. Identifying cities from their own pixels

  • A city’s held-out half is nearest its own first-half mean in all 5,000 trials, while individual pixels identify their city only 52.4% of the time.Averaging raises city identification to 86.3% for five pixels, 97.9% for twenty and 99.8% for sixty.Source: S8.4. Identifying cities from their… · p. 39
  • Of misassigned pixels, 54.4% go to a city in the same country versus 3.6% under a uniform draw, and the median wrong-city distance is 323 km versus 6,780 km under the null.Pixel accuracy also falls from 93.3% among cities whose nearest neighbour is over 30° away to 18.6% among those within 6°.Source: S8.4. Identifying cities from their… · p. 39
  • At 250 sampled pixels per city, only two city pairs lie closer than twice the median angular standard error; at the atlas’s median 1,000 pixels, no pair remains within its noise scale.The two closer pairs are Leeds–Sheffield and Sheffield–Nottingham; the closest pair at 1,000 pixels is 1.50 times its own noise scale apart.Source: S8.4. Identifying cities from their… · p. 39

S8.5. Geographic context and the degree of urbanisation

Geographic context orders city mean directions across countries, but neither city means nor the four degrees of urbanisation form clean global groups. Inside cities, the degrees capture little variation, while further subdivisions still distinguish measured urban conditions.

S8.5. Geographic context and the degree of urbanisation

  • The city-mean tree reproduces its input distances at 0.793 cophenetic correlation, yet its best cut reaches only 0.266 silhouette at 29 groups.The gap statistic selects no division, so the tree is a descriptive ordering rather than a global city taxonomy.Source: S8.5. Geographic context and the… · p. 39
  • 24.3% of variation among urban-centre mean directions in excluded countries is predicted by continent and climate; matched cities sharing both contexts sit 30.8% closer than those sharing neither.The 30.8% reduction remains 30.5–30.9% across tested population calipers; excluding same-country matches leaves 26.9%.Source: S8.5. Geographic context and the… · pp. 39–40
  • Averaging pixels carries city identity: half-city samples are never misassigned, while single pixels identify their city in half of trials and 20 pixels succeed in 98%.Errors occur locally, and pairs within the noise of 250-pixel mean directions are neighbouring cities in the same conurbation.Source: Table S3 · p. 40
  • 8.88% of within-city variation separates the four degree means, and only 70.8% of cities preserve their expected ordering across all four classes.Peri-urban and urban-centre pixels reverse order in 21.6% of randomly drawn within-city pairs.Source: S8.5. Geographic context and the… · p. 40
  • Four cross-fitted subdivisions per degree reduce residual variance most for vegetation (46.6%) and population (44.5%), with smaller reductions for other measured targets.The subdivisions are illustrative, not optimal, and sampled locations lack coordinates to establish ground contiguity or map coherence.Source: S8.5. Geographic context and the… · p. 41

S8.6. City position and the orientation of internal variation

City mean position and the orientation of within-city variation share local neighbourhood structure, but not large-scale city groupings. Shared covariance directions transfer across countries, although individual cities’ internal directions vary and released coordinates do not have independent meaning.

S8.6. City position and the orientation of internal variation

  • Mean-direction distance and internal-covariance orientation distance correlate at ρ=0.703 across city pairs, with 53.9% average overlap among their ten nearest neighbours.Their complete trees agree much less: cophenetic correlation is 0.336, with only five shared clades containing at least ten cities.Source: S8.6. City position and the… · p. 48, Figure S7 · p. 49
  • Similarity decays mainly within the first 5,000 km: held-out-city splines give R2=0.413 for mean angular distance and 0.297 for leading-direction overlap.The curves are descriptive, and cities recur across many pairs, so pair observations are not independent.Source: S8.6. City position and the… · p. 48, Figure S8 · p. 50

S8.7. Principal directions on the sphere and their transfer

  • A typical city’s covariance has participation ratio 5.95 and needs 13 components for 90% of variance, unlike pooled-pixel ratio 19.05 and rank 31.Dimension depends on the covariance being measured; degree-specific 90% ranks range from 32 in peri-urban areas to 28 in urban centres.Source: S8.7. Principal directions on the… · p. 48, Figure S9 · p. 51
  • A rank-ten basis trained in other countries captures 94.2% of the variance an excluded-country rank-ten oracle captures, while excluded-city directions lie a median 33.0° from that basis.Pooled training and test subspaces differ by 11.1° on average.Source: S8.7. Principal directions on the… · p. 48, Figure S9 · p. 51
  • Fifteen components retain 95% of full-rank mean balanced accuracy and macro-F1 when classifying the four degree means.The task classifies degree-mean directions, not the path a city traces across degrees.Source: S8.7. Principal directions on the… · p. 48, Figure S9 · p. 51
  • No released coordinate has independent loading meaning: a single coordinate carries a median 3.11% under random bases, versus 18.20% for the rotation-invariant leading component.Shared-city bases capture 58.6% of within-city covariance, compared with 86.1% for a city’s own leading directions and 24.4% for ten fixed released axes.Source: S8.7. Principal directions on the… · p. 48, Figure S9 · p. 51

S8.8. Placebo footprints

Non-urban footprint copies reproduce the increase in dimensionality seen when landscapes are pooled, but city-trained directions transfer better to cities. Across urban centres, dispersion is associated with national development; measured landscape structure accounts for much of that gradient, while terrain and sampling choices qualify its interpretation.

S8.8. Placebo footprints

  • City-trained bases recover 69.3% of excluded-city variance across ranks one to thirty, versus 53.9% for bases trained on non-urban footprint copies.Pooled copies span more directions than pooled cities (26.76 versus 21.04), although the reported interval for their difference includes zero.Source: S8.8. Placebo footprints · p. 52, Figure S10 · p. 53

S8.9. National development, measured form and the opportunity to observe

  • 14.1% greater dispersion is associated with one standard deviation higher HDI after adjustment for population, land area, collection stage and continent, across 977 cities.The estimate remains between 12.1 and 14.5% in every annual layer from 2017 to 2024; urban-centre extent suppresses rather than produces the association.Source: S8.9. National development, measured… · p. 52, Figure S11 · p. 58
  • Geographic weighting changes the estimate: equal-country weighting leaves 10.2%, UN-subregion adjustment 8.4% with an interval including zero, and omitting Africa 7.2%.On a common increment of 0.1 HDI, the full sample estimate is 9.8% and the sample without Africa is 6.6%.Source: S8.9. National development, measured… · p. 52
  • Adding built form reduces the HDI association from 15.0% to 8.5%, and further environmental and vintage controls reduce it to 1.4% in the fixed 942-city ladder.An optical–radar contrast index instead leaves 6.4%; the difference from the raw-backscatter specification reflects both covariate choice and summary statistic.Source: S8.9. National development, measured… · p. 55
  • Released-site proxies and satellite-observation measures do not explain away the association, but the unpublished v2.1 training sample and source mixture cannot be reconstructed.Site-density proxies leave 11.9–14.3% across 516 cities; sensor counts move 19.6% to 18.6% in a separate 214-city subset.Source: S8.9. National development, measured… · pp. 55–56
  • A tenfold increase in local relief is associated with 37% greater dispersion, yet terrain leaves 3.3% of the HDI gradient when added as a control.Relief predicts dispersion while remaining nearly unrelated to HDI; rugged cities and flat, high-vegetation-contrast cities both occur among the least coherent centres.Source: S8.9. National development, measured… · p. 56
  • Vegetation and radar contrast accompany the development gradient: their spreads correlate with HDI at 0.48 and 0.42, while relief correlates at 0.17.Country fixed effects leave landscape coefficients near pooled values, but country means of the three landscape covariates reduce the HDI coefficient from 14.9% to 4.0%.Source: S8.9. National development, measured… · p. 56, Table S5 · p. 57

S8.10. Where Sentinel-1B coverage was lost

The Sentinel-1B outage coincides with greater movement and contraction in exposed cities, with responses increasing where coverage loss is more severe. These associations are bounded by limited within-country comparisons and do not isolate a causal effect of the satellite’s loss.

S8.10. Where Sentinel-1B coverage was lost

  • 3.79° of excess movement at fixed points losing every scene, versus 1.04° after losing one pass, identifies a possible input pathway but not AlphaEarth’s actual inputs.Restricted-view replay shifted 2021 radar medians by 0.50 dB and 0.82 dB in the two restricted groups.Source: S8.10. Where Sentinel-1B coverage was… · p. 59
  • 3.57° farther translation followed loss of every scene, compared with 1.17° farther translation after losing one direction, relative to retained cities.These contrasts remove background motion estimated within each continent.Source: S8.10. Where Sentinel-1B coverage was… · p. 59
  • Only 31 countries contain both one-pass-loss and retained cities, while the earlier placebo is also nonzero; public scene counts proxy observation opportunity, not model inputs.The analysis does not isolate a causal Sentinel-1B effect, and its pattern does not establish a universal radar direction.Source: S8.10. Where Sentinel-1B coverage was… · pp. 59–60
  • −4.08 percentage points for losing one direction and −19.93 points for losing every scene were the associated changes in within-city dispersion.All four urbanisation degrees contracted; the largest contractions occurred in cities that lost every scene.Source: S8.10. Where Sentinel-1B coverage was… · p. 59
  • 37% higher adjusted dispersion per tenfold increase in local relief marks terrain as a second channel of within-city dispersion.Among the ten least coherent centres, some are rugged and others are flat deltas with strong vegetation contrast; the ten most coherent score low on both.Source: Figure S12 · p. 60

S8.11. Annual movement against the floor set by sampling

Annual movement in city mean directions substantially exceeds the estimated floor from resampling pixels within a year. Weather and surface-state variables predict part of that movement, but observing-condition information does not transfer consistently across years.

S8.11. Annual movement against the floor set by sampling

  • The sampling floor measures uncertainty from drawing different places, not repeated model runs or physical change at fixed sites.The estimated city-level sampling separation has a 1.573° root mean square; the annual excess does not establish physical change.Source: S8.11. Annual movement against the… · p. 60
  • 12.456° annual separation versus 1.573° expected from spatial sampling gives a 7.92 ratio at city level; observed movement exceeds sampling on every city transition.The corresponding within-degree ratio is 4.29, and 98.41% of pooled squared city-level movement lies beyond the estimated sampling component.Source: S8.11. Annual movement against the… · p. 60, Figure S13 · p. 61

S8.12. Weather and annual wobbles

  • 12.7% lower held-out squared prediction error from monthly weather variables shows that weather predicts part of annual movement beyond year effects alone.Movement rises from 11.3° in the lowest decile of annual weather change to 13.4° in the highest.Source: S8.12. Weather and annual wobbles · pp. 60, 62, Figure S14 · p. 63
  • Snow-cover change is especially informative: its standalone error reduction is 12.0% in 379 cities with climatological snow cover and 0.04% in the other 621.Both gains and losses of snow cover accompany greater movement; 5,483 of 7,000 transitions are near zero change.Source: S8.12. Weather and annual wobbles · p. 62
  • 17.0% total held-out error reduction combines 5.8 points unique to weather, 3.7 to vegetation and 1.4 to quality, with 6.1 shared.Vegetation raises the gain to 15.6%; adding quality raises it to 17.0%, although the 1.4-point increment spans zero and turns negative in the last cohort.Source: S8.12. Weather and annual wobbles · p. 62, Figure S14 · p. 63
  • Sentinel-2 opportunity lowers the joint prediction gain to 16.4%, and observing-condition information fails to transfer across years, worst in 2017–2018.January’s weather is the most costly to remove in six of seven cohorts; these prediction gains cannot be added to the sampling or outage contrasts.Source: S8.12. Weather and annual wobbles · p. 62, Figure S14 · p. 63

S8.13. Temporal signal at construction sites

At eleven dated construction projects, target pixels move farther in embedding space than matched controls elsewhere in their cities. The direction of change in relation to named places varies, and the selected projects cannot establish how much movement across all cities reflects physical change.

S8.13. Temporal signal at construction sites

  • The Paris Olympic village moved 14.0° closer to Les Halles and 12.2° closer to Gare du Nord from 2019 to 2024, while moving 8.2° farther from Père Lachaise.Its largest pixel share nearest any one place fell from 34.2% to 18.6%, spreading across more of the city’s places.Source: S8.13. Temporal signal at construction… · p. 64, Figure S15 · p. 65
  • 20.2° median excess displacement over matched controls was positive at all eleven construction projects, ranging from 7.8° in Paris to 57.5° in Phoenix.Cairo’s excess remains when controls are restricted to a narrower functional boundary, rising from 31.2° to 38.2°.Source: S8.13. Temporal signal at construction… · p. 64, Figure S16 · p. 66
  • Excess change in distance to the nearest named place runs both ways: Paris ends 1.7° closer to its local vocabulary than controls, while Toronto ends 10.0° farther.At Mexico City’s new airport, 89% of target pixels end nearest the city’s existing international airport.Source: S8.13. Temporal signal at construction… · p. 64, Figure S16 · p. 66
  • Eleven selected projects and controls not certified unchanged cannot establish what fraction of movement across 1,000 cities is physical change.Sensitivity to matching and the Sentinel-1B discontinuity also remains to be established.Source: S8.13. Temporal signal at construction… · p. 64
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