Earth4D: 4D Earth Space-Time Encoder

Updated 4 July 2026

Earth4D is a 4D space-time positional encoder that transforms continuous Earth coordinates (latitude, longitude, elevation, time) into a compact 192-dimensional learnable representation.
It extends multi-resolution 3D hash encoding by incorporating time and learned hash probing, reducing collisions by 33% and improving R² by 35% over baseline models.
Within the DeepEarth framework, Earth4D fuses with modality-specific embeddings to precisely anchor events in space and time, enhancing tasks like ecological forecasting.

Earth4D is the 4D space-time positional encoder introduced within DeepEarth, a self-supervised multi-modal world model for Earth observations. Its stated purpose is to provide a planetary-scale representation of where and when an event occurs by mapping continuous coordinates in space and time into learnable embeddings that can be fused with modality-specific observations. In the DeepEarth formulation, Earth4D extends 3D multi-resolution hash encoding to include time and is described as scaling “across the planet over centuries with sub-meter, sub-second precision” (Legel et al., 7 Mar 2026). In adjacent literature, the term “Earth4D-style” is also used more loosely for systems that model time-varying environments, weather, geometry, or atmosphere over real scenes; however, Earth4D proper refers specifically to the positional encoding mechanism inside DeepEarth rather than to a general-purpose Earth simulator (Legel et al., 7 Mar 2026).

1. Definition and conceptual role

Earth4D is introduced as the positional component of DeepEarth’s multi-modal architecture. The architecture description states that “The Earth4D encoder maps continuous space-time coordinates (latitude, longitude, elevation, time) to learnable positional embeddings, which are fused with embeddings from modality-specific encoders and processed as tokens in an autoencoder context window” (Legel et al., 7 Mar 2026). In that design, modality-specific encoders represent what was observed, while Earth4D represents where and when it happened.

The motivation is explicitly tied to three simultaneous requirements: very large spatial and temporal extent, very fine precision, and learnability with bounded memory. The paper contrasts Earth4D with a naïve dense grid, noting that even a dense high-resolution 3D level can become infeasible; the appendix gives the illustrative requirement

$(2^{28})^3 = 10^{25}.$

This motivates a compressed, trainable alternative rather than a dense 4D table (Legel et al., 7 Mar 2026).

DeepEarth positions Earth4D as the “universal” spatio-temporal anchor of the model. Event-centered multi-modal data are sampled, each modality is encoded independently, Earth4D encodes the event coordinates, all embeddings are fused into “universal tokens,” and the system is trained with masked reconstruction (Legel et al., 7 Mar 2026). This suggests that Earth4D is not a standalone Earth model; it is the address system that lets a broader world model bind observations to continuous planetary space-time.

2. Representation and mathematical structure

Earth4D extends multi-resolution hash encoding to four dimensions through a fully decomposable construction. The paper states that, following Grid4D, it extends NVIDIA’s multi-resolution hash encoding “by concatenating features from one spatial ( $xyz$ ) and three spatio-temporal ( $xyt, yzt, xzt$ ) grids” (Legel et al., 7 Mar 2026). The representation implied by the text and appendix is

$\phi_{\text{Earth4D}}(x,y,z,t)=\operatorname{concat}\Big( \phi_{xyz}(x,y,z),\; \phi_{xyt}(x,y,t),\; \phi_{yzt}(y,z,t),\; \phi_{xzt}(x,z,t) \Big).$

Each of the four component encoders is itself a multi-resolution hash-encoded 3D field. If each grid has $L$ levels and each level stores a feature vector of dimension $F$ , then the output dimensionality is

$D = 4 \times L \times F.$

The appendix gives the default configuration as $L=24$ levels per grid and $F=2$ feature dimensions per level, yielding

$D = 4 \times 24 \times 2 = 192.$

The paper states this directly: Earth4D “Outputs 192D per $xyz$ 0 coordinate from 4 grids $xyz$ 1 24 levels $xyz$ 2 2D feature per level,” and the forecasting section likewise says that “Earth4D encodes $xyz$ 3 into a 192D vector” (Legel et al., 7 Mar 2026).

The input description uses two equivalent views of the coordinates. Conceptually, Earth4D consumes latitude, longitude, elevation, and time. In implementation-oriented sections, the experiments are described using $xyz$ 4. The intended pathway is therefore geodetic Earth coordinates plus time, converted into an internal Cartesian representation and then queried against the four decomposed 3D grids (Legel et al., 7 Mar 2026). The paper, however, does not provide the explicit latitude/longitude/elevation-to- $xyz$ 5 transformation, temporal normalization constants, the hash function, or the interpolation equation. A faithful summary must therefore note that these details are not specified in the provided text (Legel et al., 7 Mar 2026).

3. Integration within DeepEarth

Within DeepEarth, Earth4D is fused with modality-specific encoders in a self-supervised world-model pipeline. The operational sequence described in the paper is: sample multi-modal data around an event; encode each modality with its own encoder; encode the event coordinates with Earth4D; fuse all embeddings into universal tokens; jointly encode these tokens; and decode masked observations or simulate missing data (Legel et al., 7 Mar 2026).

The architecture is inspired by PerceiverIO, V-JEPA 2, Galileo, and AlphaEarth, but Earth4D is the new positional component that provides a continuous Earth-centered 4D address system (Legel et al., 7 Mar 2026). This makes Earth4D modality-agnostic in principle: it is not tied to imagery, language, or sensor data individually, but to the shared spatial and temporal reference frame in which those modalities are observed.

A second use case in the paper isolates Earth4D as a feature extractor for ecological forecasting. On Globe-LFMC 2.0, the forecasting architecture is deliberately simple: Earth4D encodes $xyz$ 6 into a 192D vector, this vector is concatenated with a learnable species embedding initialized randomly, and an MLP predicts LFMC percentage (Legel et al., 7 Mar 2026). The significance of that setup is methodological: it tests how much predictive structure is present in the Earth4D encoding itself, without adding satellite imagery, weather variables, or topography.

The training objective of the broader DeepEarth model is described only at a high level as masked reconstruction. The provided text does not include a formal loss equation for the self-supervised world-model stage, nor does it specify the exact fusion operator between Earth4D embeddings and modality embeddings (Legel et al., 7 Mar 2026). This omission is material because it means Earth4D’s embedding design is documented much more concretely than the full DeepEarth optimization stack.

4. Hash tables, learned probing, and empirical behavior

Earth4D’s scalability depends on bounded-capacity hash tables. The appendix specifies up to $xyz$ 7 entries per level, 2D feature per entry, a total of 724M trainable parameters, and roughly 11 GB GPU memory during training (Legel et al., 7 Mar 2026). The core trade-off is that hash encoding compresses spatial features into fixed memory, but collisions occur when different coordinates map to the same memory location.

The paper treats collision management as a central problem and introduces learned hash probing as the main improvement over vanilla Earth4D. It states that Earth4D with learned probing “learns to select optimal hash table indices from a candidate set” and functions as “an end-to-end differentiable system that learns optimal memory allocation patterns for the data” (Legel et al., 7 Mar 2026). The exact probing equations are not given, but the conceptual mechanism is clear: candidate hash slots are considered, a learned module selects among them, and the process is optimized jointly with the downstream task.

The reported empirical gains are substantial. The appendix attributes to learned probing a 33% collision reduction at 1M points and an 18% reduction in validation loss on RGB reconstruction (Legel et al., 7 Mar 2026). On Globe-LFMC 2.0, Earth4D with learned probing achieves

MAE = 11.7 pp
RMSE = 18.7 pp
$xyz$ 8

on a train/test split of 76,467 / 13,297 samples (Legel et al., 7 Mar 2026). The baseline Galileo model, which uses Sentinel-2 optical imagery, Sentinel-1 SAR, VIIRS night lights, ERA5 weather, TerraClimate soil/water data, SRTM topography, $xyz$ 9, and species type, achieves

MAE = 12.6 pp
RMSE = 18.9 pp
$xyt, yzt, xzt$ 0

(Legel et al., 7 Mar 2026). The paper emphasizes that Earth4D surpasses Galileo without satellite imagery, weather data, or topography, using only $xyt, yzt, xzt$ 1 and species embeddings (Legel et al., 7 Mar 2026).

The appendix also provides the key ablation against vanilla hash encoding. Standard multi-resolution hash encoding without learned probing yields

RMSE = 26.0 pp
MAE = 16.6 pp
$xyt, yzt, xzt$ 2
800M parameters
$xyt, yzt, xzt$ 3 hash capacity

whereas the learned-probing version reaches the 11.7 / 18.7 / 0.783 result above, corresponding to 29.5% MAE reduction and 35.0% $xyt, yzt, xzt$ 4 improvement (Legel et al., 7 Mar 2026). A compressed setting with 5M parameters, 99.3% parameter reduction, and $xyt, yzt, xzt$ 5 hash capacity still attains MAE = 15.0 pp and $xyt, yzt, xzt$ 6, while providing 4× training speedup and 93% memory reduction (Legel et al., 7 Mar 2026).

These findings suggest that Earth4D’s performance depends less on the abstract idea of four decomposed grids alone than on the memory-allocation mechanism used to query them. The paper is explicit that vanilla collisions materially degrade performance, and that learned probing is central rather than incidental (Legel et al., 7 Mar 2026).

5. Earth4D in adjacent 4D environment-modeling literature

In adjacent papers, “Earth4D-style” is used more broadly for systems that reconstruct, generate, or render time-varying environments with explicit geometry, weather, or atmosphere. These works do not redefine Earth4D as a positional encoder, but they illustrate how the term is being extended across world-model, digital-twin, and environment-rendering contexts.

System	Relation to Earth4D	Scope
DeepEarth / Earth4D	Planetary-scale 4D space-time positional encoder	Positional embedding (Legel et al., 7 Mar 2026)
One4D	Unified RGB+pointmap 4D generation and reconstruction	General 4D world modeling (Mi et al., 24 Nov 2025)
Weather-Magician	Weather-conditioned rendering layer for reconstructed real scenes	Local scene-scale 4D weather synthesis (Sang et al., 26 May 2025)
Cloud4D	Reconstructs 3D liquid water content over time from cameras	Localized atmospheric 4D retrieval (Lin et al., 24 Nov 2025)
EarthCrafter	Scalable large-scale 3D Earth generation from aerial priors	Static 3D foundation, not temporal (Liu et al., 22 Jul 2025)
EarthScape	Multimodal 2.5D Earth-surface dataset	Surface-focused, not true 4D (Massey et al., 19 Mar 2025)

One4D defines “4D” as synchronized RGB frames and pointmaps, unifying single-image generation, sparse-frame completion, and full-video reconstruction under a Unified Masked Conditioning mechanism. The paper positions this as a step toward “general, high-quality geometry-based 4D world modeling using video diffusion models” (Mi et al., 24 Nov 2025). This suggests a complementary direction to Earth4D: instead of encoding where-and-when explicitly, One4D focuses on joint appearance-and-geometry generation under varying observation sparsity.

Weather-Magician addresses a different layer of the stack. It reconstructs a clear-weather real scene with Gaussian splatting and then renders synthesized 4D weather effects such as fog, haze, smog, rainfall, snowfall, and snow accumulation in real time (Sang et al., 26 May 2025). The paper describes this as directly relevant to Earth4D-like systems because geometry is reconstructed once and environmental state is changed by manipulating weather modules rather than retraining the whole scene (Sang et al., 26 May 2025). Its strength is an interactive weather-conditioned rendering layer rather than planetary-scale encoding.

Cloud4D reconstructs a physically meaningful 4D cloud state from six synchronized ground-based cameras, inferring 3D liquid water content at 25 m spatial and 5 s temporal resolution over a $xyt, yzt, xzt$ 7 volume (Lin et al., 24 Nov 2025). It is therefore Earth4D-like in the sense of high-resolution atmospheric state retrieval, but unlike Earth4D it is a localized retrieval framework rather than a general positional encoder (Lin et al., 24 Nov 2025).

EarthCrafter and EarthScape define the lower-dimensional edge cases. EarthCrafter is explicitly a large-scale static 3D Earth generation framework built on aerial priors and sparse latent generation (Liu et al., 22 Jul 2025). EarthScape is described as “a strong 2.5D multimodal foundation rather than a true 4D dataset,” integrating aerial imagery, DEM, terrain derivatives, hydrology, and infrastructure for surficial geologic mapping (Massey et al., 19 Mar 2025). Together, these papers clarify that Earth4D proper is not simply any Earth-related 3D or 4D system; it is a specific positional representation whose broader “Earth4D-style” interpretation is still heterogeneous.

6. Limitations, scope, and unresolved details

The paper’s strongest limitations are explicit. First, collisions are a real issue in vanilla Earth4D and materially degrade performance. Second, the strongest results rely on learned hash probing. Third, performance is better in well-sampled regions, as indicated by the geographic error maps in the LFMC study. Fourth, the default full model remains large, with 724M trainable parameters in the default configuration and 800M parameters in the vanilla ablation (Legel et al., 7 Mar 2026).

A second category of limitation concerns specification. The provided text does not include the exact geodetic-to-Cartesian conversion, temporal normalization, interpolation formula, hash function, fusion operator with modality embeddings, or the masked-reconstruction loss equation of the full DeepEarth model (Legel et al., 7 Mar 2026). These omissions matter because they limit Earth4D’s immediate reproducibility as an encoder independent of DeepEarth’s codebase.

A third limitation is conceptual scope. Earth4D is a positional encoder, not a complete Earth-system simulator. It provides a continuous, learnable representation of where and when, but the broader tasks of rendering weather, generating geometry, reconstructing dynamic scenes, or estimating atmospheric state are handled by other systems with different assumptions and scales. This suggests that Earth4D is most naturally interpreted as infrastructure for Earth-centered world models rather than as a substitute for them.

A plausible implication is that future Earth4D-style systems will combine several layers that are currently separate in the literature: a planetary-scale space-time anchor such as Earth4D (Legel et al., 7 Mar 2026), sparsity-aware joint RGB/geometry generation such as One4D (Mi et al., 24 Nov 2025), local weather-conditioned rendering such as Weather-Magician (Sang et al., 26 May 2025), and physically meaningful atmospheric retrieval such as Cloud4D (Lin et al., 24 Nov 2025). The present papers do not claim that this integration has been achieved. What they do show is that “Earth4D” has begun to designate both a specific 4D positional encoding and a wider research direction centered on explicit space-time structure for Earth observations, dynamic environment modeling, and interactive world representation.