AlphaEarth distinguishes cities but compresses urban variation

Andrew Renninger

Figure 1: Urban observations occupy a shifted but overlapping part of the global field. Three released coordinates carry the global field into the texture of Mexico City, on separate colour stretches (A,B), and similarity is read as an angle on

S^{63}

—80.61° between Mexico City and London, larger meaning less alike (C). Seen from above the urban mean, city mean directions fill one restricted region, coloured in turn by continent, climate and population and grouped only imperfectly by each, halo area giving each city’s internal dispersion (D). On the same view, half the cities lie within 54.69° of the urban mean and the farthest within 80.23°, against the 62.66° between urban and global means (E). Means of urban centres cluster most tightly and means of peri-urban areas most broadly (F), and the share of pixels beyond each angle from the global mean falls later for every degree of urbanisation than for rural and global samples (G). Radial distance is exact and azimuth schematic (Supplementary Sections S1 and S3 ).

Figure 2: City similarity is continuous but geographically structured. The hierarchy over all 1,000 city mean directions descends in a smooth cascade rather than into separated groups: branch height is angular distance, continent colours the leaves and 50 cities are named, and the best cut manages a silhouette of only 0.266 at 29 groups against a cophenetic correlation of 0.793 (A). Each city’s distance from the urban mean, against the variance inside it, shows that where a city sits in that continuum barely predicts how much variation it holds, and the weak decline drawn here is not robust to the catalogue’s sampling design (B; Supplementary Section S3 ). Geographic context does order position. Bars give the mean angular distance from an anchor to its population-matched partner under each pairing of shared or different continent and climate: among 948 anchors in 160 countries, cities sharing both lie 30.8% closer than cities sharing neither, and the same ordering returns in each of the four degrees of urbanisation (C).

Figure 3: Urban classes share one structure and leave most variation inside them. The four degrees of urbanisation spread their variance across the same coordinates in nearly the same proportions (A), and false-colour views of three coordinates show the texture cities share (B). Centred on its own mean, each city becomes a short path from peri-urban land to urban centre; aligned, the paths share one ordered trajectory but differ in curvature and scale (C). Variation between classes (darker) and within them (lighter) shows the between share averaging 8.88% and staying a sliver from Mumbai to Turin (D), and widening with total variation (E). Pooled across cities, the classes need 28 to 32 components for 90% of variance, fewer toward the urban centre, while a single city stays near six (F); the count depends on which covariance is asked about, and the city-centred paths have few dominant directions but need 26 components against 13 for a typical city (G).

Block	$\Delta\beta$ averaged over orderings	Share of total	Ladder’s own step	Range over orderings
	(log units, 95% CI)	(%, 95% CI)	( $\div$ the average over orderings)	(log units)
Vegetation	0.0415 ( $0.011$ to $0.078$ )	33.1 (11.9 to 56.8)	0.0248 ( $0.60\times$ )	0.016 to 0.061
Radar surface structure	0.0372 ( $0.012$ to $0.063$ )	29.6 (10.8 to 54.2)	0.0298 ( $0.80\times$ )	0.012 to 0.060
Built form	0.0359 ( $0.010$ to $0.055$ )	28.6 (11.1 to 46.4)	0.0579 ( $1.61\times$ )	0.013 to 0.073
Settlement vintage	0.0128 ( $-0.002$ to $0.025$ )	10.2 ( $-2.3$ to 24.7)	0.0117 ( $0.91\times$ )	0.006 to 0.019
Climate zones	0.0014 ( $-0.011$ to $0.014$ )	1.1 ( $-12.3$ to 9.7)	0.0013 ( $0.91\times$ )	$-0.002$ to 0.004

Table 1: Vegetation, radar surface structure and built form account for the development gradient. Each block’s contribution is averaged over all 720 orderings on the same 942 cities in 155 countries that Figure 4 B draws, and the six sum exactly to the total reduction of 0.125 log units. The fourth column gives the step the drawn ladder assigns each block and the fifth the range that step takes over the other orderings: built form collects 1.61 times its contribution averaged over orderings by entering third, vegetation 0.60 times its own by entering last. Percentage points per standard deviation are not tabulated because

\exp(\beta)-1

is not additive.

Figure 4: Dispersion rises with national development, and between-city separation contracts while individual city paths keep reversing. Each point is a city’s adjusted dispersion within its urban centre against HDI (A), and adding covariate blocks walks the association from 15.0% at context down to 1.4% once radar surface enters, the equal-country fit dashed (B). Over time, median separation between the same 144,453 city pairs narrows in every degree of urbanisation, by 1.08° in peri-urban areas and 1.95° in urban centres (C), yet almost no pair follows it steadily: each pair’s seven annual steps change sign a median four times, the exchangeable expectation (dashed line) (D). From each city’s own 2017 origin, the yearly ellipses of displacement widen in no dominant direction and the mean path (red) barely moves (E,F), and yearly distance travelled against distance netted puts every city far below straight travel, the median (red) moving 11.45° a year and netting 1.91° (G).

A. Predicting the degree of urbanisation with complete countries excluded (408 cities)
Model	Balanced accuracy	Mean F1 over classes	Mean error in class steps	Within one class
Absolute unit vector	0.366 (0.360–0.371)	0.322 (0.314–0.330)	1.038 (1.018–1.055)	0.699 (0.691–0.708)
City-relative tangent	0.375 (0.369–0.383)	0.342 (0.334–0.352)	0.999 (0.981–1.017)	0.719 (0.710–0.728)

Table S1 : The four degrees of urbanisation resist recovery from individual embeddings, while continuous settlement characteristics transfer well. Prediction collapses in the two middle classes and survives only at the ends of the scale, whether complete countries are excluded from fitting (panel A) or each city’s own observations are split in half (panel B). Among the continuous targets, predicted as means for each city and class (panel C), building height and volume transfer several times better than population density.

ESA WorldCover
Class	Points	Angle (°)
Mangroves	332	76.03
Snow and ice	4,595	70.78
Built-up	1,827	66.44
Bare or sparse vegetation	42,330	56.63
Moss and lichen	6,151	53.04

Table S2 : Every major class sits tens of degrees from the global barycentre, which puts the 63° urban separation on a scale. Angles run from 28.33° for grassland to 76.03° for mangroves, and a single reference pixel sits a median 72.00° away, so distance from the global mean direction is the ordinary condition and not a mark of distinctiveness. Only very-low-density rural land sits close, at 4.64°. Angles are measured on the 247,565 reference points themselves, every one of which carries a WorldCover land label, and they weight pixels rather than cities, so they calibrate the 62.66° figure, which weights cities equally, rather than coinciding with it.

A. Identification accuracy over five random splits (1,000 cities, 162 countries)
Query	Nearest one (%)	Nearest five (%)	Nearest twenty (%)
Mean direction of a city’s held-out half	100.0 (100.0–100.0)	100.0 (100.0–100.0)	100.0 (100.0–100.0)
A single pixel from that half	52.4 (48.1–56.8)	82.4 (79.1–85.8)	96.6 (95.8–97.5)
Chance	0.1	0.5	2.0

Table S3 : Every city is identified from half of its own pixels, a single pixel finds its city half the time, and the pairs the representation cannot separate are neighbours. Averaging carries the identity: a city’s held-out half is never misassigned, a single pixel is misassigned half the time, and twenty pixels are enough to name the city in 98% of trials. Where pixels err they err locally, and the only pairs within the noise of a 250-pixel mean direction are neighbouring cities in the same conurbation.

Figure S1 : The catalogue reaches 162 countries, and the samples drawn from it are balanced by design rather than by area or population. All 1,000 catalogued urban areas are mapped with their climate and their 2015 population (A,B). What each analysis then sees differs by design: the 2024 sample stratified by degree draws 840,776 observations from all 1,000 cities and leaves continents with uneven shares of their sampled pixels, whereas the WorldCover comparison with equal samples in every degree holds 544 cities in 117 countries to exactly 250 observations per class, 544,000 in all, across which cover moves from tree and crop towards built surface between peri-urban (PU) and urban centre (UC), through semi-dense (SD) and dense urban (DU) (C,D). Beyond the cities, 307,273 of 960,000 equal-area candidates return an embedding, and the opportunity to observe from public satellites, measured at fixed points in 529 cities in 162 countries, varies with instrument and continent, abbreviated AF, AM, AS, EU and OC (E,F).

Figure S2 : Contextual groups sit at displaced mean directions and still overlap heavily, so no one context sorts the representation. Continent and climate densities pull their means apart yet share most of their range (A,B). Population density moves outward in an orderly way, from 19.8° off the global mean in the lowest positive decile to 60.4° in the highest, whereas the four degrees of urbanisation crowd into a narrow band of similarly displaced radii, 58.8 to 66.5° (C,D). Panels A to C rest on the 247,565-pixel global land sample and panel D on all 1,000 cities. Distance from the centre of each disc and the inset arrows give exact angles.

Figure S3 : Urban separation aligns with measured surface and environmental variables but stays high-dimensional. Pixels in urban centres lie 6.09° farther from the global mean than rural pixels, and broad land cover and climate transfer to equal-area cells excluded from fitting at a median AUROC, the chance that a positive location scores above a negative one, of 0.919 and 0.929 (A,B). In excluded countries, vegetation, water and radar targets transfer better than built form or intensity (C). None of that collapses into a few directions. After context adjustment a model confined to a few directions reaches

Q^{2}

of 0.192 and 0.296 against 0.267 and 0.458 for one using all of them, and holding its score in principal components fitted at the mean direction of all 1,000 cities takes a median 32 components—48 in one group of excluded countries—a count that shifts with the basis while

Q^{2}

does not (D,E). Panels C to E use 728 cities in 162 countries and a further 270 extension cities in 46 countries.

Figure S4 : Variance is spread across the released coordinates rather than concentrated in a few of them. Each cell gives one coordinate’s share of within-city variance after city centring and unit normalisation. A handful of coordinates stand out and recur across continents, but no row rests on one of them, and the urban rows share a profile that the tree and crop rows, and the rows for the brightest and densest quarter of cities, only partly follow. Rows for the degrees of urbanisation balance 544 cities in 117 countries at 250 pixels per class, giving 136,000 pixels per class row, while tree and crop rows retain 539 and 490 eligible cities under equal-city weight and Oceania crops rest on four cities. The profiles describe the released basis, not principal components.

Figure S5 : Built-form and vegetation associations spread across the released coordinates rather than settling on any one of them. Colour gives the mean association inside a city, each city weighted equally, across eligible members of the 1,000 cities sampled in 2024, reaching

-0.84

and

+0.73

on a fixed

\pm 0.85

scale. Every target draws on broad bands of coordinates rather than one apiece, the built-form rows share a profile that vegetation greenness reverses at several coordinates, and similar bands recur across continents. A row recording whether a target is mapped cannot separate physical absence from missing source coverage, and the settlement and building layers are nominally 2019 while settlement age covers 1985–2015.

Figure S6 : Geographic context transfers across countries, while the degrees of urbanisation leave much of the structure inside a city unresolved. The 1,000-city hierarchy reproduces its own pairwise distances well and still cuts badly: the tree matches the distances it was built from at a correlation of 0.793, yet the best cut separates its groups at a silhouette of only 0.266, at 29 groups, where 1 would be clean (A). Shared continent and climate keep their advantage whichever population caliper is used, the reduction holding between 30.5 and 30.9% (B), and context predicts mean directions in wholly excluded countries at a

Q^{2}

of 0.243 (C). Inside cities the picture reverses: only 8.88% of variation separates the four class means (D), and on those same cities the class ordering inverts across countries excluded from fitting (E). Validated on cities held out from fitting, splitting the degrees further reduces the spread of population and vegetation most (F).

Figure S7 : Mean position and variance orientation share neighbourhoods but not major branches. Across all 1,000 cities the two pairwise distance rankings correlate at

\rho=0.703

and ten-nearest-neighbour sets overlap by 53.9%, so a city’s near neighbours are largely the same under both constructions. The trees built from them are not: cophenetic correlation falls to 0.336, and only five exact clades of ten or more cities are shared, the largest holding 26. Read the two against one another rather than down either one—the lower tree is mirrored at 484 of its 999 binary nodes to bring identical leaves alongside, leaving topology and merge heights untouched, so the displayed Kendall

\tau=0.440

, the rank agreement between the two leaf orders, diagnoses that alignment rather than any fit, and is

-0.065

before it. Terminal colours mark continent and the 50 names match Fig. 2 A.

Figure S8 : Geographic distance organises both where a city’s urban core sits and how its internal variation is oriented. Cities near one another on the ground have closer urban-core mean directions (A) and the leading ten directions of their internal covariance overlap more (B). Neither relation decays to indifference: overlap settles well above the reference line, the 0.159 expected of two random ten-dimensional subspaces, rather than at zero. Five folds that exclude cities from fitting give spline

R^{2}=0.413

for mean angular distance and 0.297 for the overlap of the leading ten directions, and the curves are descriptive rather than inferential. The comparison keeps 992 of the 1,000 cities, in 162 countries, each with 250 valid observations from its urban centres; the eight excluded have none, and are named with the deposited values.

Figure S9 : Dimensionality depends on which covariance on the sphere is asked about, and shared covariance transfers far better than the directions of a single city. Participation ratio, in circles, and the rank reaching 90% of variance, in squares, sort by which covariance is asked about, not by any fixed property of the representation: the aggregates sit high and an individual city far lower (A,B). Transfer separates the constructions. A basis of ten directions from the training countries reaches 94.2% of what an oracle of the same rank, ten directions fitted to the excluded cities themselves, manages on those cities (C), yet an excluded city’s own directions lie a median 33.0° from that basis where two pooled folds lie 11.1° apart (D), and capture falls from 86.1% on bases fitted inside a city to 24.4% on ten fixed released axes (F). Classification is cheaper, 15 components reaching 95% of the rank-63 mean on both scores (E), and no released coordinate carries the leading component’s weight—a median 3.11% across 500 random bases against 18.20% (G).

Figure S10 : Copied to non-urban land, a city’s footprint pools to higher dimension than the city itself; what cities keep is structure that transfers. Both the city and its copy are locally simple (A), yet once footprints are pooled the copies span more directions than the cities, the two bootstrap clouds sitting well apart (B). Transfer runs the other way. A basis learned in other countries’ cities recovers most of an excluded city’s variance and a basis learned in their copies noticeably less, the dark diagonal blocks against the paler off-diagonal ones (C), with the same gap at every rank (D); pooled cities also reach 90% of variance sooner (E). Within a footprint the leading directions are stable across spatial halves in both sets (F), so the contrast is not sampling noise.

A. Every dispersion outcome, computed both ways
Outcome	Reduced over	Cities	% per HDI SD	95% CI
Urban centre (principal)	every valid cell	977	14.13	6.50 to 22.32
Urban centre (principal)	250 sampled pixels	977	16.04	9.00 to 23.53
All available urban	every valid cell	985	16.72	9.30 to 24.64
All available urban	250 sampled pixels	985	18.75	11.69 to 26.26
Balanced four degrees	every valid cell	532	17.52	9.95 to 25.62

Table S4 : The association with national development survives every route to the outcome, every control for the extent of the urban centre and all three ways of forming intervals. Every cell is the association between one standard deviation of national HDI and log embedding dispersion. Across the three panels it ranges from 13.9 to 18.8% per standard deviation, and no interval reaches zero.

A. Landscape coefficients between countries and within them (859 cities, 94 countries)
Covariate	Between countries (% per SD, 95% CI)		Within countries (% per SD, 95% CI)		Within / between
Vegetation contrast	12.7	8.7 to 16.9	10.8	7.2 to 14.4	0.86 (0.60 to 1.25)
Radar contrast	9.9	5.7 to 14.3	8.5	5.0 to 12.0	0.86 (0.59 to 1.32)
Relief	8.6	5.3 to 12.0	7.9	5.9 to 9.8	0.92 (0.71 to 1.50)

Table S5 : Landscape predicts dispersion as strongly within countries, where development is held fixed, as between them. Fixed effects for country leave vegetation contrast, radar contrast and relief at 86 to 92% of their pooled coefficients (A), country means of the same covariates take the HDI coefficient from 14.9 to 4.0% (B), pairs across countries follow development less once vegetation contrast is matched (C), relief rises weakly with HDI (D), and a tenfold increase in relief buys the same dispersion in the lowest third of HDI as in the highest (E).

Figure S11 : Dispersion rises with national development in every year and under every specification, and the two choices that move it are geographic. Released training sites concentrate where development is high (A), and the residual after context is mapped for every city (B). A city’s pixels fan out around its mean direction, and dispersion is the width of that fan: 41° for Busan, 14° for Yakutsk (C). The association stays between 12.1 and 14.5% in every annual layer (D). Equal-country weighting lowers it to 10.2% and UN-subregion adjustment to 8.4% with an interval crossing zero; built form takes it from 15.0 to 8.5% and the environmental blocks to 1.4%, a terrain block added on the 937 cities the elevation model covers leaves 3.3%, and the proxies for released training sites leave 11.9 to 14.3% on the 516 cities that carry them (E). HDI is national context, not a causal treatment, and focus maps the density of released v2.0 training sites rather than what AlphaEarth was exposed to.

Figure S12 : Terrain is a second channel for dispersion within cities. The same population-, area- and continent-adjusted dispersion that Figure 4 A relates to national development rises 37% for every tenfold increase in local relief within the centre, with country means sized by city count (A). The ten least coherent centres, those whose pixels spread widest around their mean, divide into rugged cities and flat delta cities with strong vegetation contrast, and the ten most coherent sit low on both, each city placed at its percentile of relief and of vegetation contrast among 992 centres (B).

Figure S13 : Where Sentinel-1B coverage was lost, AlphaEarth moves further and holds less internal variation. Fixed locations move most where public coverage falls to zero, and replaying the same locations and year through a restricted view shifts the annual radar medians they were built from; both rest on an independent global frame of 4,374 fixed points (A,B). Exposed city centroids then translate

1.17^{\circ}

and

3.57^{\circ}

farther than retained ones once background motion within each continent is removed (C). Dispersion contracts at the outage boundary, by 4.08 percentage points where a direction was lost and 19.93 points where every scene was, across all 1,000 cities in 162 countries, negative in every degree of urbanisation and deepest in the cities that lost most, while no other annual transition comes close (D–F). The public scenes intersected with each point measure the opportunity to observe, not AlphaEarth’s inputs.

Figure S14 : Weather and surface state predict part of the annual wobble. Mean movement rises from 11.3° in the lowest decile of annual weather change to 13.4° in the highest (A), and both gains and losses of snow cover accompany larger movement, with most of the 7,000 transitions crowded near zero change (B). Monthly weather reduces held-out prediction error by 12.7%, vegetation lifts this to 15.6% and MODIS quality to 17.0%, while adding Sentinel-2 opportunity lowers it to 16.4% (C). By transition year (D), vegetation helps in every cohort, quality turns negative in the last, opportunity fails badly in the first, and removing January’s weather costs the most in six of seven years. Of the 17.0%, 5.8 points are unique to weather, 3.7 to vegetation and 1.4 to quality, with 6.1 shared (E).

Figure S15 : The olympic village changes its relation to Paris places as it is rebuilt. Seventeen comparator places and the 500 m target circle are mapped (A,B) and followed on one sphere whose pole is the village’s 2017 direction, the red path tracing the village and thin paths the comparators (C). Every place drifts from its own 2017 state across the eight layers (D), so the village’s movement is read against same-year distances to each place, on which Île de la Cité stays its nearest place at both endpoints (E). Between 2019 and 2024 the village moves 14.0° closer to Les Halles, 12.2° closer to Gare du Nord and 8.2° farther from Père Lachaise (F).

Figure S16 : Construction sites move farther than matched pixels elsewhere in their cities. Median excess displacement over matched controls rises above zero at every one of the eleven projects, from 7.8° in Paris to 57.5° in Phoenix (A), whereas the excess change in distance to the nearest named place runs either way, Paris moving 1.7° closer to its local vocabulary and Toronto 10.0° farther (B). At the endpoints (C) the dashed line marks the 20.2° median across projects and marker area the share of pixels that switch comparator group. Which groups gain and lose those pixels differs by city (D), as do the named places that lose and gain most (E), both in percentage points. The largest movers change vocabulary almost wholesale: every Phoenix pixel and 96% of Austin’s switch group, Austin ending nearest The Domain rather than its executive airport, and the airport built at Mexico City ends nearest the city’s existing one (C,E).

AlphaEarth distinguishes cities but compresses urban variation

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