Low-rank tensor structure of precipitation and its application to satellite-reference merging

Ryan Solgi; Rohan Shankar; Hugo A. Loaiciga

cs.LGOct 7, 2026

Low-rank tensor structure of precipitation and its application to satellite-reference merging

Authors: Ryan Solgi, Rohan Shankar, Hugo A. Loaiciga

Organizations: Department of Geography, University of California-Santa Barbara, CA, USA · Department of Electrical and Computer Engineering, University of California-Santa Barbara, CA, USA · Department of Computer Science, Cornell University, NY, USA

Abstract

The intermittent and variable nature of precipitation makes its accurate estimation over extended domains difficult, yet its spatiotemporal structure suggests that a low-rank representation may be possible. This work represents daily precipitation over the contiguous United States (CONUS) as spatiotemporal tensors and applies CANDECOMP/PARAFAC factorization, showing that preserving the native spatial and temporal modes yields more accurate reconstruction than factorizing independent daily fields or unfolded space--time matrices. Building on this finding, this work presents TMerge, a tensor-based framework that integrates satellite precipitation with sparse reference observations through shared low-rank spatial and temporal factors. TMerge was applied to correct the IMERG Final Run product with climate prediction center reference observations over CONUS. During 2019-2022, TMerge increased correlation from 0.53 to 0.85 and reduced root-mean-square error and mean absolute error by 48.2% and 29.3%, respectively. TMerge consistently outperformed linear bias correction, quantile mapping, and neural networks across seasons, precipitation-intensity regimes, and regions. Improvements were spatially coherent and largest in coastal regions where IMERG errors were greatest. These results demonstrate that low-rank tensor structure parsimoniously approximates the dominant spatiotemporal variability of precipitation and provides a practical mechanism for improving satellite estimates under limited reference observations over extended domains.

Figures & tables

Parameter Budget	Method	Correlation Coefficient ( $r$ )	RMSE
0.5%	STT	0.791 $\pm$ 0.001	3.926 $\pm$ 0.009
	SM	0.734 $\pm$ 0.045	4.413 $\pm$ 0.399
	UM	0.343 $\pm$ 0.001	6.053 $\pm$ 0.032
1.0%	STT	0.845 $\pm$ 0.001	3.435 $\pm$ 0.013
	SM	0.668 $\pm$ 0.223	5.266 $\pm$ 3.028
	UM	0.343 $\pm$ 0.002	6.054 $\pm$ 0.012

Table 1: Performance comparison of STT versus SM and UM baselines for daily precipitation reconstruction across CONUS during 2018–2022 under a 10% withholding ratio. Values are mean

\pm

maximum deviation across 5 random runs with different withholding sets. Parameter budget denotes the percentage of model parameters relative to the total number of data entries. The best method per metric is shown in bold.

Figure 1: (a) Correlation coefficient ( $r$ ) and (b) RMSE of STT, SM, and UM against the CPC reference over CONUS across 20 non-overlapping three-month periods during 2018–2022, using a 10% withholding ratio and a matched parameter budget of about 5.0%. Bars show averages over five runs.

Figure 2: Comparison of low-rank models at a 5% parameter budget under different withholding ratios.

Figure 3: Spatial distribution of RMSE for IMERG and TMerge daily precipitation estimates, together with the RMSE reduction of TMerge relative to IMERG, over CONUS during 2018–2022. White cells inside CONUS denote fitting locations excluded from evaluation, while surrounding white areas outside the CONUS region are not included in either model fitting or evaluation.

Method	Correlation Coefficient ( $r$ )	RMSE	MAE
TMerge	0.819 $\pm$ 0.001	3.716 $\pm$ 0.012	1.903 $\pm$ 0.006
MLP	0.577 $\pm$ 0.002	5.205 $\pm$ 0.009	2.278 $\pm$ 0.014
Linear	0.542 $\pm$ 0.001	6.085 $\pm$ 0.008	2.251 $\pm$ 0.004
QM	0.564 $\pm$ 0.001	5.879 $\pm$ 0.008	2.202 $\pm$ 0.004
IMERG	0.542 $\pm$ 0.001	6.296 $\pm$ 0.007	2.329 $\pm$ 0.004

Table 2: Performance comparison of TMerge, MLP, Linear, QM, and IMERG for daily precipitation estimation across CONUS during 2018–2022. Values are reported as mean

\pm

standard deviation across five random splits of CPC reference and evaluation locations. The best method per metric is shown in bold.

Figure 4: Performance comparison of TMerge, MLP, QM, Linear correction, and the IMERG Final product against the CPC reference at CPC resolution over CONUS during 2018–2022. (a) Correlation coefficient ( $r$ ) across three-month periods. (b) RMSE across three-month periods. (c) RMSE across CPC reference precipitation bins.

Figure 5: Sensitivity analysis of TMerge performance over CONUS in 2018 compared to baseline methods. Correlation coefficient ( $r$ ) is shown as a function of (a) reference observation fraction, (b) temporal window length, (c) number of spatial divisions, and (d) CP rank and corresponding parameter ratio.

Method	Correlation Coefficient ( $r$ )	RMSE	MAE
TMerge	0.854 $\pm$ 0.002	3.272 $\pm$ 0.026	1.638 $\pm$ 0.018
MLP	0.771 $\pm$ 0.008	4.012 $\pm$ 0.095	1.698 $\pm$ 0.050
Linear	0.534 $\pm$ 0.002	6.093 $\pm$ 0.058	2.235 $\pm$ 0.033
QM	0.558 $\pm$ 0.002	5.848 $\pm$ 0.055	2.191 $\pm$ 0.034
IMERG	0.534 $\pm$ 0.002	6.320 $\pm$ 0.061	2.317 $\pm$ 0.035

Table 3: Performance comparison of TMerge, MLP, Linear, QM, and IMERG for daily precipitation estimation at IMERG native resolution across CONUS during 2019–2022. Values are reported as mean

\pm

standard deviation across runs. The best method per metric is shown in bold.

Figure 6: Sensitivity of TMerge performance to rank, MLP performance to architecture size, and QM performance to the number of IDW neighbors and the IDW power, evaluated using the 2018 period.

Figure 7: Predicted versus reference (CPC) daily precipitation at withheld CONUS test cells (2019–2022) for TMerge, MLP, QM, Linear, and IMERG: (a) density scatter colored by point count with KGE and NSE annotated and (b) bivariate density colored by modal month, both on log axes with the dashed 1:1 line.

Figure 8: Performance comparison of TMerge, MLP, QM, Linear correction, and the IMERG Final product against the CPC reference at IMERG resolution over CONUS during 2019–2022. (a) Correlation coefficient ( $r$ ) across three-month periods. (b) RMSE across three-month periods. (c) RMSE across CPC reference precipitation bins.

Figure 9: Temporally averaged precipitation and pixel-wise RMSE at the withheld CONUS cells during 2019–2022. For visibility, each test cell is shown as an enlarged marker covering approximately $115\times$ its true 0.1 ∘ grid-cell area. All remaining CPC locations within CONUS were used for model fitting, while areas outside the CONUS domain were excluded from both fitting and evaluation.

Figure 10: Performance comparison of TMerge, MLP, QM, Linear correction, and the IMERG Final product against the CPC reference at native IMERG spatial resolution over the CONUS coastal zone during 2019–2022: (a) correlation coefficient ( $r$ ), RMSE, and MAE, (b) density scatter colored by point count with KGE and NSE annotated, (c) bivariate density colored by modal month, and (d) bivariate density colored by mean distance to the coastline.

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Remote Sensing Generative Modeling Multimodal Sensor Fusion

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Reliable precipitation monitoring is essential for disaster risk reduction, water resources management, and agricultural decision-making. Multi-source satellite observations, particularly the combination of geostationary infrared and passive microwave measurements, have become a primary means of precipitation detection. Traditional multi-source satellite precipitation estimation methods remain computationally inefficient, and many deep learning methods lack the flexibility to incorporate new sensors without retraining the full model. Here we introduce PRISMA (Precipitation Inference from Satellite Modalities via generAtive modeling), a plug-and-play latent generative framework for multi-sensor precipitation estimation. PRISMA learns an unconditional precipitation prior from IMERG Final fields and constrains it through independently trained, sensor-specific conditional branches, allowing new observation sources to be incorporated without retraining the generative backbone. Applied to FY-4B AGRI infrared and GPM GMI microwave observations, PRISMA improves Critical Success Index by up to 40.3% and reduces root-mean-square error by 22.6% relative to infrared-only estimation within microwave swaths, while also improving probabilistic skill and maintaining an average inference time of about 37 s. Independent rain-gauge validation across China confirms consistent gains, and typhoon case studies show that microwave conditioning restores eyewall and spiral rainband structures, reducing storm-core mean absolute error by up to 42.3%. PRISMA thus provides an extensible and efficient framework for multi-sensor precipitation estimation.

Abstract

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