Neural Stochastic Processes for Satellite Precipitation Refinement

Shunya Nagashima; Takumi Bannai; Shuitsu Koyama; Tomoya Mitsui; Shuntaro Suzuki

cs.CVApr 12, 2026

Neural Stochastic Processes for Satellite Precipitation Refinement

Authors: Shunya Nagashima, Takumi Bannai, Shuitsu Koyama, Tomoya Mitsui, Shuntaro Suzuki

Organizations: Neurogica Inc. · LTS, Inc. · ME-Lab Japan, Inc.

Abstract

Accurate precipitation estimation is critical for flood forecasting, water resource management, and disaster preparedness. Satellite products provide global hourly coverage but contain systematic biases; ground-based gauges are accurate at point locations but too sparse for direct gridded correction. Existing methods fuse these sources by interpolating gauge observations onto the satellite grid, but treat each time step independently and therefore discard temporal structure in precipitation fields. We propose Neural Stochastic Process (NSP), a model that pairs a Neural Process encoder conditioning on arbitrary sets of gauge observations with a latent Neural SDE on a 2D spatial representation. NSP is trained under a single variational objective with simulation-free cost. We also introduce QPEBench, a benchmark of 43{,}756 hourly samples over the Contiguous United States (2021--2025) with four aligned data sources and six evaluation metrics. On QPEBench, NSP outperforms 13 baselines across all six metrics and surpasses JAXA's operational gauge-calibrated product. An additional experiment on Kyushu, Japan confirms generalization to a different region with independent data sources.

Figures & tables

Figure 1: Motivation for the proposed Neural Stochastic Process (NSP). Satellite precipitation estimates provide gridded coverage but carry systematic biases, while gauges are sparse but locally accurate. NSP encodes satellite estimates, elevation, and gauges with the encoder $E_{\varphi}$ , evolves a latent state through stochastic dynamics, and maps it to a calibrated precipitation estimate with the decoder $D_{\theta}$ ; “Eval only” indicates that the radar reference is used only for comparison, not as a model input.

Figure 2: Architecture of Neural Stochastic Process (NSP). (a) During training, gauge observations are split into context and target sets: the black markers in the input maps denote context gauges encoded through the mask $\mathbf{m}_{t}$ , while the right panel shows target gauges $\mathbf{y}_{t}$ used by $\mathcal{L}_{\mathrm{rec}}$ . The shared encoder maps consecutive inputs to $\mathbf{z}_{t}$ and $\mathbf{z}_{t+1}$ , and $\mathcal{L}_{\mathrm{trans}}$ aligns the Neural SDE prior $p_{\theta}(\mathbf{z}_{t+1}\mid\mathbf{z}_{t})$ with the encoder posterior $q_{\varphi}(\mathbf{z}_{t+1}\mid\mathbf{x}_{t+1})$ . (b) At inference, all available gauges are treated as context, and a single encoder–decoder pass predicts $\hat{\mathbf{y}}_{t}$ without SDE integration.

Method	$\text{RMSE}_{r}\downarrow$	$\text{MAE}_{r}\downarrow$	$\text{RMSE}_{g}\downarrow$	$\text{MAE}_{g}\downarrow$	$r_{r,\text{coll}}\uparrow$	$\text{FSS}_{R}\uparrow$
Quantile mapping Wood et al. (2004)	4.073 $\pm$ 0.184	2.012 $\pm$ 0.063	0.885 $\pm$ 0.027	0.173 $\pm$ 0.006	0.288 $\pm$ 0.014	0.479 $\pm$ 0.010
EMOS Gneiting et al. (2005)	3.885 $\pm$ 0.127	1.844 $\pm$ 0.078	1.115 $\pm$ 0.096	0.179 $\pm$ 0.019	0.305 $\pm$ 0.013	0.478 $\pm$ 0.029
XGBoost Chen and Guestrin (2016)	3.741 $\pm$ 0.129	1.827 $\pm$ 0.077	1.062 $\pm$ 0.088	0.179 $\pm$ 0.020	0.284 $\pm$ 0.016	0.487 $\pm$ 0.008
GWR Brunsdon et al. (1996)	3.739 $\pm$ 0.230	1.627 $\pm$ 0.081	0.662 $\pm$ 0.021	0.155 $\pm$ 0.004	0.376 $\pm$ 0.016	0.229 $\pm$ 0.017
Cokriging Cressie (1993)	3.706 $\pm$ 0.121	1.789 $\pm$ 0.069	1.065 $\pm$ 0.081	0.220 $\pm$ 0.015	0.279 $\pm$ 0.016	0.461 $\pm$ 0.009
GSMaP Kubota et al. (2020)	3.638 $\pm$ 0.126	1.826 $\pm$ 0.076	1.017 $\pm$ 0.086	0.179 $\pm$ 0.020	0.288 $\pm$ 0.014	0.483 $\pm$ 0.014

Table 1: Quantitative comparison of precipitation refinement methods. Values are mean

\pm

standard deviation over three-fold time-series cross-validation. The best score in each column is in bold .

Figure 3: Qualitative result at 03:00 UTC on March 16, 2025. NSP better matches the radar reference despite the satellite overestimation.

Figure 4: Regional zoom over the southeastern United States at 16:00 UTC on March 16, 2025.

Method	$\text{RMSE}_{r}\!\downarrow$	$\text{RMSE}_{g}\!\downarrow$	$r_{r,\text{coll}}\!\uparrow$
GWR Brunsdon et al. (1996)	1.788	0.655	0.389
CNP Garnelo et al. (2018a)	1.779	1.953	0.437
EMOS Gneiting et al. (2005)	1.738	1.082	0.169
Cokriging Cressie (1993)	1.715	1.077	0.139
Linear regression	1.688	0.768	0.356
Kriging Cressie (1993)	1.640	0.973	0.114

Table 2: Kyushu (Japan) regional results.

Architecture ablations
	Model	$\text{RMSE}_{r}\downarrow$	$\text{MAE}_{r}\downarrow$	$\text{RMSE}_{g}\downarrow$	$\text{MAE}_{g}\downarrow$	$r_{r,\text{coll}}\uparrow$	$\text{FSS}_{R}\uparrow$
(1-i)	w/o Neural SDE	2.888 $\pm$ 0.039	1.487 $\pm$ 0.053	0.338 $\pm$ 0.066	0.062 $\pm$ 0.018	0.479 $\pm$ 0.020	0.509 $\pm$ 0.022
(1-ii)	Deterministic	2.885 $\pm$ 0.080	1.492 $\pm$ 0.044	0.355 $\pm$ 0.147	0.060 $\pm$ 0.028	0.487 $\pm$ 0.033	0.495 $\pm$ 0.012
(1-iii)	w/o Satellite residual	3.141 $\pm$ 0.089	1.613 $\pm$ 0.038	0.252 $\pm$ 0.024	0.041 $\pm$ 0.005	0.475 $\pm$ 0.018	0.153 $\pm$ 0.047
(1-iv)	Homoscedastic	2.932 $\pm$ 0.062	1.492 $\pm$ 0.020	0.260 $\pm$ 0.016	0.052 $\pm$ 0.003	0.471 $\pm$ 0.023	0.445 $\pm$ 0.012
(1-v)	Log-space MSE	3.554 $\pm$ 0.104	1.726 $\pm$ 0.048	0.405 $\pm$ 0.031	0.043 $\pm$ 0.006	0.475 $\pm$ 0.019	0.525 $\pm$ 0.022

Table 3: Ablation study on CONUS. Mean

\pm

std over three folds; best in bold .

(a) RMSE by intensity
Intensity	GSMaP	GC	NSP	Rel. improv.
(mm/h)				vs. GSMaP	vs. GC
0	0.592	0.377	0.308	$+$ 48.0%	$+$ 18.3%
0–1	0.693	0.436	0.356	$+$ 48.6%	$+$ 18.3%
1–5	4.011	2.528	2.055	$+$ 48.8%	$+$ 18.7%
5–10	7.725	5.597	5.190	$+$ 32.8%	$+$ 7.3%

Table 4: Error analysis. (a) Pixel-level RMSE by radar precipitation intensity (fold 3, 8,759 samples). (b) Median spatial displacement (km) between predicted and radar exceedance masks, measured by the shift maximizing 2-D cross-correlation; lower is better.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Property	Value
Temporal coverage	January 2021 – December 2025 (5 years)
Total hourly steps	43,824 (43,756 after quality filtering)
Grid resolution	$0.1^{\circ}\times 0.1^{\circ}$ ( ${\sim}$ 11 km)
Grid size	$260\times 590$ (153,400 pixels)
Spatial extent	$24.05^{\circ}$ – $49.95^{\circ}$ N, $124.95^{\circ}$ – $66.05^{\circ}$ W
Mean rainy fraction	4.6% $\pm$ 2.0% of radar pixels per sample

Appendix

Table 5: QPEBench dataset overview.

Percentile	Value (mm/h)
P50	0.82
P90	3.55
P95	5.67
P99	14.45
Max	192.20

Appendix

Table 6: Distribution of non-zero radar precipitation across the full dataset.

Channel	Transform
Satellite (GSMaP)	$\log(1+x)$
Gauge observations	$\log(1+x)$
Elevation (DEM)	$x\,/\,2000$
Context mask	Binary $\{0,1\}$
Radar (target)	None (raw mm/h)

Appendix

Table 7: Input normalization.

Hyperparameter	Value
Optimizer	AdamW ( $\beta_{1}{=}0.9$ , $\beta_{2}{=}0.999$ )
Learning rate	$3\times 10^{-3}$ (OneCycleLR)
Weight decay	$1.35\times 10^{-3}$
Epochs	4
Batch size	4 per GPU ( $\times 8=32$ effective)
Mixed precision	FP16

Appendix

Table 8: Hyperparameters of NSP.

Fold	Train	Validation	Test
1	2021	2022	2023
2	2021–2022	2023	2024
3	2021–2023	2024	2025

Appendix

Table 9: Expanding-window time-series cross-validation splits over the 2021–2025 period.

Method	$\text{RMSE}_{r}\downarrow$	$\text{MAE}_{r}\downarrow$	$\text{RMSE}_{g}\downarrow$	$\text{MAE}_{g}\downarrow$	$r_{r,\text{coll}}\uparrow$
ViT	2.979	2.010	1.993	1.275	0.483
Quantile mapping	1.876	1.073	1.377	0.623	0.216
GWR	1.788	0.812	0.655	0.252	0.389
CNP	1.779	1.559	1.953	1.815	0.437
EMOS	1.738	0.954	1.082	0.326	0.169
Cokriging	1.715	0.951	1.077	0.384	0.139

Appendix

Table 10: Full quantitative comparison on the Kyushu (Japan) regional dataset. Single train/validation/test split (2020/2021/2022); best in bold .

Threshold (mm/h)	GSMaP GC	NSP	Note
0.1	33 km (3 px)	33 km (3 px)	Equal
1.0	33 km (3 px)	55 km (5 px)	GC better
5.0	440 km (40 px)	220 km (20 px)	NSP $2\times$ better
10.0	not reached	not reached	Both below 0.5

Appendix

Table 11: Minimum useful scale: smallest neighborhood where

\text{FSS}\geq 0.5

[ 52 ] . Lower indicates spatially sharper predictions.

Threshold	GC	NSP	Improv.	NSP bias	GC bias
(mm/h)	cRMSE	cRMSE
0.1	3.66	3.54	$+$ 3.3%	$-$ 0.52	$-$ 0.34
1.0	5.41	4.87	$+$ 10.0%	$-$ 0.92	$-$ 0.41
5.0	9.72	9.24	$+$ 4.9%	$-$ 3.54	$-$ 2.00
10.0	13.38	11.88	$+$ 11.2%	$-$ 4.20	$-$ 2.49
25.0	21.32	14.70	$+$ 31.1%	$-$ 0.23	$+$ 3.40

Appendix

Table 12: Conditional intensity RMSE (cRMSE) and bias, computed only on pixels where both prediction and radar exceed the threshold.

Category	Definition	MSE share
Hit bias (intensity)	pred $\geq 0.1$ and radar $\geq 0.1$	67.0%
Miss (position)	pred $<0.1$ and radar $\geq 0.1$	14.4%
False alarm (position)	pred $\geq 0.1$ and radar $<0.1$	18.6%
Intensity-related		67.0%
Position-related		33.0%

Appendix

Table 13: MSE source decomposition into mutually exclusive pixel categories (

\tau{=}0.1

mm/h).

GC accuracy	$N$	GC RMSE	NSP RMSE	$\Delta$	NSP wins
RMSE $<1$ (accurate)	7,236	0.592	0.575	$-$ 0.017	52.7%
RMSE 1–2	1,415	1.193	1.014	$-$ 0.179	81.0%
RMSE 2–4	34	2.365	0.836	$-$ 1.529	100.0%
Overall	8,686				57.5%

Appendix

Table 14: Per-sample comparison conditioned on GSMaP GC accuracy (fold 3). 73 samples with missing GC data are excluded; 1 sample with GC RMSE

{}\geq 4

is included in Overall but omitted from bins.

Dominant error in 3,692 failure cases
	Count	Proportion
NSP better than GC	4,994	57.5%
NSP worse than GC	3,692	42.5%
Intensity error (hit bias)	2,681	72.6%
False precipitation	571	15.5%
Missed precipitation	440	11.9%

Appendix

Table 15: Per-sample comparison with GSMaP GC and failure mode decomposition (

\tau{=}0.1

mm/h).

Mode	Count	Proportion
Intensity error (hit bias)	7,125	81.3%
False precipitation	930	10.6%
Missed precipitation	704	8.0%

Appendix

Table 16: Dominant error mode for all 8,759 test samples (

\tau{=}0.1

mm/h).

Variant	$\text{RMSE}_{r}\downarrow$	$\text{MAE}_{r}\downarrow$	$\text{RMSE}_{g}\downarrow$	$\text{MAE}_{g}\downarrow$	$r_{r,\text{coll}}\uparrow$	$\text{FSS}_{R}\uparrow$
Matched variance	2.883 $\pm$ 0.139	1.489 $\pm$ 0.076	0.341 $\pm$ 0.062	0.063 $\pm$ 0.015	0.478 $\pm$ 0.021	0.522 $\pm$ 0.018
Naive temporal penalty	2.853 $\pm$ 0.121	1.470 $\pm$ 0.089	0.366 $\pm$ 0.037	0.070 $\pm$ 0.011	0.478 $\pm$ 0.018	0.519 $\pm$ 0.042
Full transition KL	2.818 $\pm$ 0.062	1.444 $\pm$ 0.026	0.393 $\pm$ 0.047	0.076 $\pm$ 0.013	0.478 $\pm$ 0.021	0.527 $\pm$ 0.022

Appendix

Table 17: Temporal objective variants on CONUS. The matched-variance row uses the Girsanov special case

\sigma_{\varphi}^{2}=\sigma_{\theta}^{2}\Delta t

; the naive temporal penalty replaces the KL with mean-only temporal matching. Best values are in bold .

Panel of Figure 5 — Figure 5: Hour-by-hour event tracking using the existing radar-evaluated metrics $\mathrm{RMSE}_{r}\downarrow$ and $\mathrm{FSS}_{R}\uparrow$ on two 12-hour fold 3 events. NSP is compared against the no- $\mathcal{L}_{\mathrm{trans}}$ ablation and ConvCNP. In both events, removing $\mathcal{L}_{\mathrm{trans}}$ degrades $\mathrm{RMSE}_{r}$ and usually lowers $\mathrm{FSS}_{R}$ , with the effect especially pronounced in the late-October event. ConvCNP improves over naive gauge-only interpolation but still fails to preserve thresholded spatial structure, yielding consistently low $\mathrm{FSS}_{R}$ under dense-grid prediction.

Event	Model	$\mathrm{RMSE}_{r}\downarrow$	$\sigma(\mathrm{RMSE}_{r})\downarrow$	$\mathrm{FSS}_{R}\uparrow$
March 15, 2025	NSP	2.770	0.392	0.707
March 15, 2025	NSP w/o $\mathcal{L}_{\mathrm{trans}}$	2.828	0.403	0.671
March 15, 2025	ConvCNP	3.123	0.427	0.069
October 30–31, 2025	NSP	1.892	0.191	0.539
October 30–31, 2025	NSP w/o $\mathcal{L}_{\mathrm{trans}}$	2.295	0.323	0.472
October 30–31, 2025	ConvCNP	2.166	0.249	0.035

Appendix

Table 18: Event-wise temporal analysis on two 12-hour CONUS events in fold 3. Values are means over the 12 hourly steps;

\sigma(\mathrm{RMSE}_{r})

denotes the standard deviation of

\mathrm{RMSE}_{r}

across the event window. ConvCNP denotes a reimplementation adapted to dense-grid prediction (Appendix D.3 ).

Event	Mode	$\mathrm{RMSE}_{r}\downarrow$	$\mathrm{FSS}_{R}\uparrow$	Mean latency (ms)
March 15, 2025	NSP (standard inference)	2.776	0.713	2.60
March 15, 2025	NSP (pure SDE rollout)	3.239	0.351	2.22
March 15, 2025	NSP (hybrid, $\alpha{=}0.5$ )	3.102	0.497	2.92
October 30–31, 2025	NSP (standard inference)	1.891	0.543	2.60
October 30–31, 2025	NSP (pure SDE rollout)	2.043	0.313	2.21
October 30–31, 2025	NSP (hybrid, $\alpha{=}0.5$ )	1.851	0.503	2.93

Appendix

Table 19: SDE rollout analysis on the same two 12-hour fold 3 events. NSP (standard inference) denotes the deployed inference mode used throughout the paper, which re-encodes each hour independently. NSP (pure SDE rollout) propagates the latent state without re-encoding after initialization, and NSP (hybrid) linearly blends rolled-out and re-encoded latent states with

\alpha=0.5

. Values are means over the 12 hourly steps; latency is the mean per-hour inference time within each mode.

Panel of Figure 6 — Figure 6: Effect of the inference-time context ratio on NSP performance (fold 3, model trained with 50% context). (a) Radar-evaluated metrics are best at 10–25% context and degrade with denser input, revealing a gauge over-interpolation effect. (b) Gauge-evaluated metrics improve monotonically with denser context, as expected.

Panel of Figure 7 — Figure 7: Effect of the training-time context ratio on NSP performance (fold 3, seed 42). Each point retrains NSP from scratch with the indicated fixed context ratio; the 100% setting diverged and is omitted. At 75%, $\text{FSS}_{R}$ collapses despite a slight $\text{RMSE}_{r}$ improvement, confirming that dense gauge supervision during training induces overfitting to point observations.

Panel of Figure 8 — Figure 8: Effect of the fixed station-network ratio on NSP performance (fold 3, fixed 50% context split within each retained station subset). Unlike the earlier context-ratio sweep, this experiment first fixes the available gauge network and then partitions only that network into context and held-out targets. (a) Radar-evaluated metrics remain stable across 10–75% retained stations. (b) Gauge-evaluated metrics are shown for completeness, but their values are not directly comparable to the earlier sweeps because the evaluated station subset changes with the retained network.

Figure 9: Gauge context ablation for the widespread convective event shown in Figure 3 (03:00 UTC, March 16, 2025). Removing gauge context degrades $\text{RMSE}_{r}$ by 16.7%; the $|\Delta\,\text{prediction}|$ panel confirms that gauge corrections concentrate in active precipitation regions.

Figure 10: Gauge context ablation for a localized winter precipitation event (13:00 UTC, December 24, 2025). Removing gauge context changes $\text{RMSE}_{r}$ by only 1.5%; the spatial structure is predominantly satellite-driven.

Method	Category	Latency (ms)	Parameters
GSMaP GC	Operational product	0.04	—
Linear regression	Statistical	1.05	—
IDW	Interpolation	1.88	—
NSP	Deep learning	2.64	4.19M

Appendix

Table 20: Inference latency and model size. Measured on a single NVIDIA H200 GPU with batch size 1, excluding data loading and host–device transfer.

Panel of Figure 11 — Figure 11: Two success cases, each showing a full CONUS view (top) and regional zoom (bottom). Case 1: A warm-front system over the southeastern US—GSMaP overestimates the spatial extent; GSMaP GC attenuates but introduces discontinuities; NSP produces a coherent field matching radar and preserves the frontal band orientation. Case 2: An intense squall line from Texas to the Great Lakes—GSMaP inflates the convective core far beyond the radar boundary; NSP recovers the sharp leading edge and trailing stratiform gradient.

Panel of Figure 12 — Figure 12: Failure case at 13:00 UTC on January 22, 2024. Top: Full CONUS—radar detects moderate precipitation over the upper Midwest, but the satellite input is near-zero; without a spatial prior, NSP’s residual correction cannot reconstruct the field. Bottom: Regional zoom—GSMaP fails to detect shallow warm-rain precipitation; GSMaP GC produces diffuse, incoherent patches; NSP, dependent on the satellite residual, cannot recover the missing structure.

Method	$\text{RMSE}_{r}\downarrow$	$\text{MAE}_{r}\downarrow$	$\text{RMSE}_{g}\downarrow$	$\text{MAE}_{g}\downarrow$	$r_{r,\text{coll}}\uparrow$	$\text{FSS}_{R}\uparrow$
DDPM	7.482 $\pm$ 2.534	6.441 $\pm$ 2.365	8.178 $\pm$ 2.814	7.605 $\pm$ 2.555	0.361 $\pm$ 0.161	0.060 $\pm$ 0.039
NSP (Ours)	2.818 $\pm$ 0.062	1.444 $\pm$ 0.026	0.393 $\pm$ 0.047	0.076 $\pm$ 0.013	0.478 $\pm$ 0.021	0.527 $\pm$ 0.022

Appendix

Table 21: Conditional DDPM baseline vs. NSP on CONUS (mean

\pm

std over three-fold cross-validation). The DDPM performs far worse than NSP on every metric, with large cross-fold variance. All metrics follow the same protocol as Table 1 .

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