Operator-Aware Posterior Sampling for Multivariate Weather Downscaling

Carlo Saccardi; Rudolf Saathof; Geethu Joseph; Justin Dauwels

cs.CVOct 29, 2026

Operator-Aware Posterior Sampling for Multivariate Weather Downscaling

Authors: Carlo Saccardi, Rudolf Saathof, Geethu Joseph, Justin Dauwels

Organizations: Delft University of Technology, The Netherlands

Abstract

Statistical downscaling is the task of reconstructing high-resolution meteorological fields from coarse inputs. A principled approach formulates downscaling as a Bayesian inverse problem, using a diffusion prior over high-resolution fields and incorporating the coarse field only at inference through a likelihood. This decouples the prior from the coarse source, allowing one pretrained model to adapt to different sources. However, two challenges limit this flexibility: (1) when coarse and target fields come from different reanalysis systems, no fixed coarsening operator can capture their relationship; and (2) guidance gradients can differ by orders of magnitude across variables, making a single guidance strength ineffective. This paper addresses both challenges within the guidance loop, leaving the prior unchanged. We learn the forward operator from paired data and normalize the gradient per channel. We show that this normalization corresponds to steepest descent under a block-max norm, allowing a single guidance strength across variables. Our code is publicly available at https://github.com/CarloSaccardi/operator-aware-weather-downscaling.

Figures & tables

Panel of Figure 1 — Figure 1: Comparison of surface sensible heat flux (sshf) and friction velocity (zust) between the coarse ERA5 and the high-resolution CERRA. Beyond the resolution gap, the fields differ in structure and large-scale pattern, implying an unknown and system-dependent coarse-to-fine map.

Region	Operator	$u_{10}$		$v_{10}$		$t_{2m}$		sshf		zust		sp
Region	Operator	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS
C. Europe	Bicubic $\mathbf{A}$	1.199	0.739	1.174	0.718	1.338	0.806	33.83	15.48	0.199	0.119	131.2	75.46
C. Europe	Ours	0.986	0.553	0.982	0.547	1.021	0.571	32.70	13.33	0.090	0.047	92.05	55.96
Iberia	Bicubic $\mathbf{A}$	1.362	0.827	1.372	0.828	1.809	1.152	41.18	20.43	0.174	0.095	156.1	88.86
Iberia	Ours	1.197	0.678	1.217	0.684	1.377	0.824	40.23	18.28	0.096	0.052	106.7	63.28

Table 1: Operator ablation. RMSE and CRPS per variable, both regions. Both rows use per-channel normalization and the same

\zeta

; only the likelihood operator differs. The learned operator improves every variable. Best per column in bold .

Region	Guidance	$u_{10}$		$v_{10}$		$t_{2m}$		sshf		zust		sp
Region	Guidance	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS
C. Europe	DPS ( Chung et al., 2023 )	0.969	0.539	0.966	0.535	1.092	0.608	31.27	12.79	0.096	0.049	153.7	93.17
C. Europe	Ours	0.986	0.553	0.982	0.547	1.021	0.571	32.70	13.33	0.090	0.047	92.05	55.96
Iberia	DPS ( Chung et al., 2023 )	1.172	0.661	1.186	0.662	1.413	0.842	38.22	17.28	0.096	0.051	161.6	91.88
Iberia	Ours	1.197	0.678	1.217	0.684	1.377	0.824	40.23	18.28	0.096	0.052	106.7	63.28

Table 2: Normalization ablation. RMSE and CRPS per variable, both regions. Both rows use the learned operator; only the guidance scaling differs, each arm at its own tuned

\zeta

. A single per-sample scale (DPS) leaves the high-magnitude variable sp badly under-served (up to

+67\%

RMSE), whereas ours keeps every variable controlled. Best per column in bold .

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Model	$u_{10}$		$v_{10}$		$t_{2m}$		sshf		zust		sp
Model	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS
CondDiff	0.904	0.489	0.903	0.486	0.843	0.452	27.08	10.22	0.075	0.039	59.98	35.56
CorrDiff [ Mardani et al., 2025 ]	0.884	0.477	0.877	0.471	0.831	0.449	25.76	9.782	0.072	0.038	72.01	41.82
U-Net †	0.856	0.616	0.852	0.610	0.831	0.641	25.42	12.76	0.071	0.049	116.0	87.82
Bicubic $\mathbf{A}$	1.199	0.739	1.174	0.718	1.338	0.806	33.83	15.48	0.199	0.119	131.2	75.46
DPS [ Chung et al., 2023 ]	0.969	0.539	0.966	0.535	1.092	0.608	31.27	12.79	0.096	0.049	153.7	93.17

Appendix

Table 3: Central Europe: full comparison. Conditional baselines (top) condition on the coarse field at training time; the inference-time methods (bottom) use it through the likelihood, with Ours our full method. Best per column in bold (lower is better). † The U-Net is deterministic, so its CRPS is a single-prediction MAE, not comparable to the

8

-member ensemble CRPS; it is excluded from the CRPS comparison.

Model	$u_{10}$		$v_{10}$		$t_{2m}$		sshf		zust		sp
Model	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS
CondDiff	0.020	0.010	0.020	0.010	0.029	0.015	3.29	0.841	0.0018	0.0010	3.76	2.41
CorrDiff [ Mardani et al., 2025 ]	0.020	0.010	0.020	0.009	0.028	0.015	3.21	0.867	0.0016	0.0009	2.38	1.34
U-Net †	0.019	0.012	0.019	0.012	0.026	0.018	3.11	1.06	0.0015	0.0011	3.09	2.40
Bicubic $\mathbf{A}$	0.035	0.023	0.031	0.019	0.045	0.031	3.32	1.06	0.0084	0.0054	9.20	5.15
DPS [ Chung et al., 2023 ]	0.021	0.010	0.021	0.009	0.035	0.021	3.39	1.00	0.0036	0.0018	13.85	8.96

Appendix

Table 4: Central Europe: bootstrap intervals. Each entry is the half-width (

\pm

) of the

95\%

interval for the matching entry in Table 3 ; the full interval is that table’s value

\pm

the half-width here. From

2000

resamples of the test fields in blocks of

24

. The interval covers the test draw only; with a single training seed it says nothing about seed-to-seed variability. † U-Net CRPS is a single-member MAE (see Table 3 ).

Model	$u_{10}$		$v_{10}$		$t_{2m}$		sshf		zust		sp
Model	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS
CondDiff	1.039	0.552	1.070	0.571	0.916	0.500	26.47	9.923	0.077	0.040	74.09	45.00
CorrDiff [ Mardani et al., 2025 ]	1.020	0.541	1.036	0.550	0.917	0.507	25.28	9.548	0.074	0.039	79.20	46.70
U-Net †	0.998	0.702	1.015	0.714	0.916	0.733	25.08	12.30	0.073	0.050	127.4	98.46
Bicubic $\mathbf{A}$	1.362	0.827	1.372	0.828	1.809	1.152	41.18	20.43	0.174	0.095	156.1	88.86
DPS [ Chung et al., 2023 ]	1.172	0.661	1.186	0.662	1.413	0.842	38.22	17.28	0.096	0.051	161.6	91.88

Appendix

Table 5: Iberia: full comparison. Layout and conventions as in Table 3 . † The U-Net CRPS is a single-member MAE and is excluded from the CRPS comparison.

Model	$u_{10}$		$v_{10}$		$t_{2m}$		sshf		zust		sp
Model	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS
CondDiff	0.025	0.012	0.027	0.014	0.022	0.011	1.99	0.592	0.0017	0.0008	6.34	4.34
CorrDiff [ Mardani et al., 2025 ]	0.027	0.013	0.027	0.014	0.023	0.013	2.10	0.629	0.0017	0.0009	2.95	1.85
U-Net †	0.025	0.016	0.027	0.017	0.023	0.016	2.00	0.767	0.0017	0.0011	4.96	4.08
Bicubic $\mathbf{A}$	0.031	0.020	0.028	0.017	0.039	0.029	2.54	1.11	0.0078	0.0034	10.99	5.70
DPS [ Chung et al., 2023 ]	0.024	0.013	0.026	0.013	0.043	0.029	2.60	1.08	0.0024	0.0012	11.05	5.99

Appendix

Table 6: Iberia: bootstrap intervals. Each entry is the half-width (

\pm

) of the

95\%

interval for the matching entry in Table 5 ; the full interval is that table’s value

\pm

the half-width here. Resampling as in Table 4 .

Panel of Figure 2 — Figure 2: Power spectral density, Central Europe. Test split, 8-member ensembles. The inset magnifies the highest $0.15$ decades of wavenumber, where the curves separate.

Panel of Figure 3 — Figure 3: Marginal distributions, Central Europe. $\log_{10}$ of the probability density over the test split, so the tails stay visible.

Panel of Figure 4 — Figure 4: Power spectral density, Iberia. Test split, 8-member ensembles. The inset magnifies the highest $0.15$ decades of wavenumber, where the curves separate.

Panel of Figure 5 — Figure 5: Marginal distributions, Iberia. $\log_{10}$ of the probability density over the test split, so the tails stay visible.

Region	Coarse source	$u_{10}$		$v_{10}$		$t_{2m}$		sshf		zust		sp
Region	Coarse source	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS
C. Europe	ERA5 $\sim$ $25$ km	0.986	0.553	0.982	0.547	1.021	0.571	32.70	13.33	0.090	0.047	92.05	55.96
C. Europe	ERA5 $\sim$ $50$ km	0.994	0.557	0.989	0.551	1.055	0.590	36.79	14.80	0.091	0.047	95.68	58.85
Iberia	ERA5 $\sim$ $25$ km	1.197	0.678	1.217	0.684	1.377	0.824	40.23	18.28	0.096	0.052	106.7	63.28
Iberia	ERA5 $\sim$ $50$ km	1.207	0.683	1.224	0.686	1.403	0.842	42.49	19.31	0.098	0.053	107.3	63.73

Appendix

Table 7: Transfer to a

\sim

50

km coarse source. RMSE and CRPS per variable, both regions. Only

\mathbf{A}_{\phi}

is refit; the prior is unchanged. The

\sim

25

km rows repeat Ours from Tables 3 and 5 .

Region	Coarse source	$u_{10}$		$v_{10}$		$t_{2m}$		sshf		zust		sp
Region	Coarse source	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS	RMSE	CRPS
C. Europe	ERA5 $\sim$ $25$ km	0.022	0.011	0.021	0.010	0.032	0.019	3.65	1.09	0.0023	0.0011	3.62	2.46
C. Europe	ERA5 $\sim$ $50$ km	0.025	0.013	0.022	0.010	0.041	0.024	4.29	1.25	0.0027	0.0013	5.08	3.57
Iberia	ERA5 $\sim$ $25$ km	0.026	0.014	0.028	0.014	0.031	0.020	2.74	1.18	0.0023	0.0012	5.24	3.43
Iberia	ERA5 $\sim$ $50$ km	0.032	0.016	0.033	0.017	0.036	0.024	3.06	1.40	0.0025	0.0013	7.18	4.70

Appendix

Table 8: Transfer to a

\sim

50

km coarse source: bootstrap intervals. Each entry is the half-width (

\pm

) of the

95\%

interval for the matching entry in Table 7 . Resampling as in Table 4 . The intervals for the

\sim

25

km rows are the Ours rows of Tables 4 and 6 .

Panel of Figure 6 — Figure 6: sshf on a representative sample, Central Europe. (a) coarse ERA5 input, (b) CERRA target, (c) DPS with a fixed linear operator, (d) our learned operator.

Figure 7: Surface pressure on a single test timestep, Central Europe. Top: CERRA target and the $8$ -member ensemble mean from each method. Bottom: error fields on a common scale; (d) bicubic ERA5 residual, (e) ours, (f) DPS.

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