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

arXiv:2610.11000 · cs.LG, physics.ao-ph · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Low-rank tensor structure of precipitation and its application to satellite-reference merging".

Tom: The gist:

Jane: First, who's behind it and why it matters.

Title and authors: Tom: Now, what’s the big takeaway from this paper? It seems like they are proposing a way to merge different types of precipitation data—like dense satellite estimates and sparse reference points—by exploiting this shared low-rank structure.

Jane: That’s right. They show that TMerge outperforms standard methods like linear bias correction, quantile mapping, and even neural networks across all the conditions they tested, which is pretty strong.

Lu: The paper highlights that their method improves correlation from zero point five three to zero point eight five over the period of two thousand nineteen to two thousand twenty-two when applied to correcting the IMERG Final Run product with climate prediction center reference observations across the CONUS, which is a huge jump.

Meng: And they didn't just get better correlation; they reduced root-mean-square error by forty-eight point two percent and mean absolute error by twenty-nine point three percent, which is a significant reduction in average error metrics for precipitation estimates.

Lalam: The core idea is that by imposing a multilinear structure across space, time, and source, TMerge estimates the precipitation value at an unseen location at time t as Xˆ i,j,t equals Yˆ(i, j, t ref), which is how they fuse the different data sources.

Tom: So what does this mean for us? It suggests that preserving the native spatial resolution is important if you want to capture the true physical variability of precipitation instead of just relying on simpler factorizations.

Jane: It means that for complex, intermittent phenomena like rain, treating it as a tensor where space and time are inherently linked is a much more efficient way to model it than treating each day or location independently.

Lu: This approach moves beyond just daily fields; it factors in the shared evolution of precipitation across all locations simultaneously.

Meng: From an engineering standpoint, this means the model is capturing these shared modes efficiently, which implies a compact representation compared to trying to model every single point explicitly.

The paper's summary: Tom: Let’s talk more about how they frame this problem. They start by pointing out that precipitation has asymmetric and intermittent spatiotemporal patterns, making accurate estimation difficult over extended domains, which is the main challenge in the field.

Jane: They argue that because of these patterns, a low-rank representation might be possible where preserving those native spatial and temporal modes yields better reconstruction than factorizing independent daily fields or unfolded space–time matrices.

Lu: They represent daily precipitation over the CONUS as spatiotemporal tensors and apply CANDECOMP/PARAFAC factorization to demonstrate this finding, which is a key technique for low-rank modeling in multiway data.

Meng: This formulation Xˆ(i, j, t) equals X R r=one lambda r A(i, r)B(j, r)C(t, r), where A and B describe spatial factors and C describes temporal evolution through shared latent factors <ref:2610.11000#pg2>.

Lalam: And then they use this to build TMerge by introducing an additional source mode L equal to two for the satellite and reference sources into a joint tensor Y in R I×J×T × L, using the same low-rank model structure across space, time, and source.

Tom: So they take that initial finding about daily precipitation tensors and extend it to a practical framework called TMerge that explicitly handles merging those satellite estimates with sparse reference observations through shared factors.

Jane: It’s a way to combine information from different sources by forcing them to share the same underlying low-rank spatial and temporal structure, which is quite clever.

Lu: This is fundamentally about leveraging the inherent structure of the data rather than just applying a correction layer on top of a pre-existing product.

Meng: The paper shows that this shared factorization is effective because it couples all days through common spatial modes and all locations through shared temporal evolution, which ties everything together nicely.

The paper's improvements: Tom: The results show some very tangible improvements, like the correlation increasing from zero point five three up to zero point eight five when correcting the IMERG Final Run product with climate prediction center reference observations between two thousand nineteen and two thousand twenty-two.

Jane: And they also report a reduction in error metrics: root-mean-square error dropped by forty-eight point two percent and mean absolute error dropped by twenty-nine point three percent, which is substantial improvement over the previous methods they compared it against.

Lu: They noted that TMerge consistently outperformed linear bias correction, quantile mapping, and neural networks across different seasons, precipitation-intensity regimes, and regions where they tested it.

Meng: And importantly for practical application, these improvements were spatially coherent and largest in coastal regions where the IMERG errors were already greatest because the method effectively handles those sharp land–ocean emissivity gradients.

Lalam: In a specific experiment using CPC resolution, TMerge improved the correlation coefficient to zero point eight five four while reducing RMSE by forty-eight point two percent and MAE by twenty-nine point three percent, reaching an RMSE of three point two seven two mm day−one and an MAE of one point six three eight mm day−one.

Tom: They also looked at how sensitive the method is to having sparse reference data, showing it remains effective even when coverage is limited, though it does improve with denser observations.

Jane: And they found that the best performance was achieved using a three-month window, which they used for their main results, and that TMerge achieved its strongest coastal performance by increasing correlation from zero point four eight nine to zero point eight three one in those areas alone.

Lu: They also explored how rank affects things; they saw rapid gains at small ranks, moving from r = zero point six nine three at rank twenty-five to r = zero point eight one three at rank one hundred followed by a clear plateau where the rank stayed between r = zero point eight three one and r = zero point eight three six for ranks of one hundred fifty to two hundred fifty.

Conclusion: Tom: So, to wrap up what we’ve heard about this paper, the main point is that daily precipitation has an exploitable low-rank structure that is much better represented when its native spatial and temporal dimensions are kept in tensor form.

Jane: TMerge successfully integrates dense satellite estimates with sparse reference observations by factoring them through shared low-rank spatial and temporal factors, substantially improving IMERG precipitation estimates while retaining their native spatial resolution.

Lu: This establishes a low-rank tensor structure as an organizing property of precipitation variability, which has potential benefits for several hydrologic and Earth system applications that need high-resolution inputs.

Meng: From an engineering perspective, this means we have a framework that can provide high-resolution precipitation estimates over extended regions even with limited reference observation coverage.

Lalam: The paper demonstrates that low-rank tensor structure is an organizing property of precipitation variability and shows its effectiveness in improving high-resolution precipitation estimates over extended regions with limited reference-observation coverage, which is what this study was about.

Tom: That’s it for today, folks. We’ve been talking about the "Low-rank tensor structure of precipitation and its application to satellite–reference merging," but we'll be back next time after a break.

Ryan Solgi, Rohan Shankar, Hugo A. Loáiciga

Department of Geography, University of California-Santa Barbara · Department of Electrical and Computer Engineering, University of California-Santa Barbara

cs.LG, physics.ao-ph

Submitted: 2026-10-07

Updated: 2026-10-07

Code: https://github.com/rmsolgi/TMerge

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 92/100

The gist: The gist: Low-rank tensor structure of precipitation and its application to satellite–reference merging demonstrates that preserving native spatial and temporal modes in tensor form allows for more

Key concepts

Low-Rank Tensor Structure: This concept suggests that complex data like precipitation can be accurately described using only a small number of underlying, shared patterns (factors). Instead of modeling every single daily value independently, the paper shows that the spatial and temporal variations are driven by a few common 'modes' or latent factors that link different locations and times together.
STT Formulation: The STT formulation represents daily precipitation as a third-order tensor (X in R I×J×T), where I is space, J is location, and T is time. This mathematical structure captures how precipitation varies across latitude, longitude, and time simultaneously. It simplifies the variability by showing that these complex interactions can be broken down into shared spatial factors (A) and temporal factors (C).
Tensor-Based Merging (TMerge): TMerge is a framework that combines dense satellite precipitation data with sparse reference observations. It achieves this by creating a joint tensor incorporating both sources. By factorizing this combined tensor, TMerge learns shared low-rank spatial and temporal factors that link the full satellite grid to the limited reference points, allowing for accurate estimation at any location.
Coastal Region Performance: TMerge showed its strongest improvements in coastal areas where existing satellite products (IMERG) had the largest errors. By exploiting the shared low-rank structure, TMerge reduced errors significantly in these regions—improving correlation and lowering RMSE—demonstrating that this tensor approach is particularly effective where traditional methods struggle most.

Summary

The gist: Low-rank tensor structure of precipitation and its application to satellite–reference merging demonstrates that preserving native spatial and temporal modes in tensor form allows for more accurate reconstruction than factorizing independent daily fields or unfolded space–time matrices

Low-Rank Tensor Structure of Precipitation

The intermittent and variable nature of precipitation suggests that a low-rank representation may be possible. This work represents daily precipitation over the contiguous United States (CONUS) as spatiotemporal tensors and shows that preserving the native spatial and temporal modes yields more accurate reconstruction than factorizing independent daily fields or unfolded space–time matrices. Tensor factorization extends low-rank modeling to multiway data, allowing spatial, temporal, and data-source dimensions to be represented jointly. The formulation STT represents daily precipitation fields within a spatial grid and time as a third-order tensor X ∈ R I×J×T. This model reduces to Xˆ(i, j, t) = X R r=1 λrA(i, r)B(j, r)C(t, r), where A ∈ R I×R and B ∈ R J×R describe spatial factors and C ∈ R T ×R. This formulation represents precipitation variability through shared latent spatial and temporal factors, coupling all days through common spatial modes and all locations through shared temporal evolution.

Tensor-Based Merging Framework (TMerge)

The work presents TMerge, a tensor-based framework that integrates satellite precipitation with sparse reference observations through shared low-rank spatial and temporal factors. TMerge exploits the latent structure by factorizing a joint satellite–reference precipitation tensor into shared low-rank spatial and temporal factors. This joint tensor is constructed by introducing an additional source mode, yielding Y ∈ R I×J×T ×L, where L = 2 corresponds to the satellite (sat) and reference (ref) sources. The low-rank model is given by Yˆ(i, j, t, l) = X R r=1 λrA(i, r)B(j, r)C(t, r)D(l, r), where A ∈ R I×R and B ∈ R J×R represent spatial factors. The satellite source is observed over the full spatial grid whereas the reference source is observed only at the reference locations. By imposing a multilinear structure across space, time, and source, TMerge estimates the precipitation value at an unseen location (i, j) at time t as Xˆ i,j,t = Yˆ(i, j, t,ref).

Performance Comparison and Evaluation

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. The improvements were spatially coherent and largest in coastal regions where IMERG errors were greatest. TMerge achieved the highest correlation coefficient and lowest RMSE in every three-month period during 2018–2022. In the CPC resolution experiment, TMerge improved the correlation coefficient to 0.854 while reducing RMSE by 48.2% and MAE by 29.3%, reaching RMSE = 3.272 mm day−1 and MAE = 1.638 mm day−1.

Sensitivity Analysis

The sensitivity analyses further show that TMerge remains effective under sparse reference coverage while improving with denser observations. The best performance is obtained with the 3-month window, which is used in the main results discussed above. TMerge achieved its best performance when the domain was modeled as a single CONUS-wide tensor, with r = 0.835, and performance decreased modestly as the domain was divided into smaller spatial tiles, reaching r = 0.808 for 32 divisions. The rank ablation in Figure 5d shows rapid gains at small ranks, from r = 0.693 at rank 25 to r = 0.813 at rank 100, followed by a clear plateau, with r remaining between 0.831 and 0.836 for ranks 150–250.

Coastal Region Performance

TMerge achieved the strongest coastal performance, increasing the correlation from r = 0.489 for IMERG to r = 0.831, while reducing RMSE from 9.32 to 4.98 mm day−1 and MAE from 3.84 to 2.44 mm day−1. TMerge consistently improved precipitation estimation across CONUS, with the strongest error reductions in coastal regions where IMERG errors were highest. The resulting TMerge parameterization represented approximately 1.2% of the space–time–source tensor, whereas the MLP contained 3.15 million trainable parameters, equivalent to approximately 11.5% of the tensor size.

Conclusion

This study demonstrates that daily precipitation contains an exploitable low-rank structure that is represented more accurately and parsimoniously when its native spatial and temporal dimensions are preserved in tensor form. TMerge integrates dense satellite estimates with sparse reference observations through shared low-rank spatial and temporal factors, substantially improving IMERG precipitation estimates while retaining their native spatial resolution. These findings establish low-rank tensor structure as an organizing property of precipitation variability with potential benefits for several hydrologic and Earth system applications, and demonstrate its effectiveness in improving high-resolution precipitation estimates over extended regions with limited reference-observation coverage.

The paper is a scientific study investigating the use of low-rank tensor factorization to model precipitation variability and merge satellite data with sparse reference observations for improved estimation. This research matters because it establishes that preserving the native spatial and temporal organization of daily precipitation in tensor form allows for more accurate reconstruction than traditional matrix-based methods, leading to a framework called TMerge that significantly outperforms existing correction techniques across various conditions, particularly in coastal regions where satellite errors are most pronounced.

How it works

The core methodology involves representing daily precipitation fields as tensors (multidimensional arrays) to preserve their spatial, temporal, and multi-source structure. A daily gridded precipitation field can be represented as a second-order tensor (matrix), whose two modes correspond to latitude and longitude. When daily gridded precipitation fields are stacked over time, the resulting dataset forms a third-order tensor, with modes corresponding to latitude, longitude, and time. The work adopts CP factorization to examine and leverage the low-rank structure of spatiotemporal precipitation data. This formulation is named low-rank spatiotemporal tensor model (STT) and reduces to Xˆ(i, j, t) = X R r=1 λrA(i, r)B(j, r)C(t, r), where A ∈ R I×R and B ∈ R J×R describe spatial factors and C ∈ R T ×R. This formulation represents precipitation variability through shared latent spatial and temporal factors.

Tensor-Based Merging (TMerge)

TMerge extends the spatiotemporal low-rank formulation from a single precipitation tensor to a joint satellite–reference representation.

Improvements for AI systems

Bold header: Tensor-based satellite–reference merging (TMerge) for precipitation estimation. TMerge fuses satellite precipitation and sparse reference observations via shared low-rank spatial and temporal factors, allowing it to improve the IMERG Final Product over CONUS by leveraging a joint low-rank representation of both sources.
Bold header: Outperforming traditional correction baselines. TMerge consistently outperformed linear bias correction, quantile mapping, and neural networks across seasons, precipitation-intensity regimes, and regions, demonstrating that the low-rank tensor structure provides a compact multilinear representation of precipitation variability.
Bold header: Enhanced performance in high-error coastal regions. TMerge showed gains were largest in coastal regions where IMERG errors were greatest, specifically because it effectively mitigates errors stemming from sharp land–ocean emissivity gradients and coastal precipitation regimes.
Bold header: Robustness under sparse reference data constraints. The framework remains effective even with limited observations, as demonstrated by the finding that TMerge remains competitive under sparse reference coverage and improved with denser observations, unlike baselines which showed limited gains from additional reference observations along the full range of observation fractions.
Bold header: Preservation of native spatial resolution. By using TMerge in a high-resolution experiment, it achieved superior results compared to methods using coarser CPC resolution, demonstrating that the low-rank formulation is less affected by spatial resolution as long as the dominant spatiotemporal structure can be captured, thus preserving its native 0.1◦ resolution.

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Key concepts

Summary

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