State-Space Unlearning for Non-Stationary Bias in Land Surface Forecasting

Anidipta Pal

cs.LGSep 30, 2026

State-Space Unlearning for Non-Stationary Bias in Land Surface Forecasting

Authors: Anidipta Pal

Organizations: Heritage Institute of Technology, Kolkata, India

Abstract

Operational land surface forecasting systems built on Mamba-family Structured State Space Models absorb non-stationary confounding events (unrecorded irrigation booms, dam-operation shifts, sensor recalibrations) into their state-transition matrices, silently biasing NDVI, LST, and crop phenology predictions long after the physical cause ends. This paper introduces SSU-LSF (State-Space Unlearning for Land Surface Forecasting), the first machine-unlearning framework purpose-built for geoscientific Mamba-based SSMs. We develop EKFac influence functions specialized to the Mamba state matrices via a closed-form matrix-exponential gradient, use spectral-radius-weighted elbow thresholding to localize a temporal confounding footprint $Φ$ , and apply Hessian-free projected gradient ascent within a KL-divergence trust region augmented by spatial total-variation (TV) regularization. Proposition 1 establishes that residual confounding is bounded by $\mathcal{O}\big((1-ρ(\bar{A})^{T_c})/((1-ρ(\bar{A}))μ)\big)$ , which grows with the window length $T_c$ . Across three heterogeneous benchmarks and eleven baselines, SSU-LSF achieves confounding reduction rates of $0.773$ (CropHarvest), $0.821$ (NDVI-LST), and $0.859$ (ERA5), with worst-case clean-domain RMSE degradation of $4.2\%$ on ERA5, converging in 3--5 epochs at $8.4\times$ lower GPU-cost per unlearning request than full retraining. Code: https://github.com/Anidipta/SSU-LSF

Figures & tables

Figure 1: SSU-LSF pipeline. (a) A confounding event biases $(\bar{A},\bar{B},C)$ and forward-propagates spurious NDVI elevation beyond $t_{e}$ . (b) EKFac influence scores $\varphi_{t}$ ; elbow threshold $\gamma$ isolates footprint $\Phi$ (red). (c) RMSE convergence: SSU-LSF (blue) reaches oracle-level RMSE in five epochs while CM (red) stays elevated.

	NDVI-LST Fusion			ERA5 Patchified			CropHarvest
Method	RMSE $\downarrow$	CRR $\uparrow$	TCI $\uparrow$	SKT-R $\downarrow$	MIA $\downarrow$	ACC $\uparrow$	Sp-R $\downarrow$	SCS $\downarrow$	F1 $\uparrow$	GPU-h/req $\downarrow$
CM	0.0681	0.000	0.601	2.97	0.84	0.612	0.0543	0.083	0.697	–
OR †	0.0398	1.000	0.891	1.19	0.51	0.903	0.0371	0.079	0.871	47.3
SISA [ 13 ]	0.0418	0.612	0.833	1.26	0.52	0.871	0.0395	0.088	0.834	58.4
GR [ 11 ]	0.0461	0.741	0.774	1.47	0.57	0.843	0.0428	0.127	0.799	2.3
SR [ 11 ]	0.0439	0.778	0.798	1.38	0.55	0.858	0.0411	0.102	0.817	7.9

Table 1: Per-benchmark results.

\dagger

: upper bound. CRR for ERA5 and CropHarvest and

\Delta

RMSE for all benchmarks are in Table 5 . GPU-h is the ERA5 cost per unlearning request including EKFac re-estimation (per-benchmark costs in Table 9 ).

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Module	Configuration	Key parameters	Params (M)
Patch embedding	Conv2d(5,256,4,stride 4)	$4\times 4$ , mean pool	1.0
Mamba encoder	$6\times$ block	$d{=}256$ , $E{=}2$ , $N{=}16$ , drop 0.1	39.8
Output norm	LayerNorm(256)	extract $h[:,-1,:]$	0.0
Forecast head	MLP 256-512-512-60, GELU	$\tau{=}6$ , $d_{\mathrm{out}}{=}5$	6.5
Total			47.3

Appendix

Table 2: MambaForecaster module specifications.

Component	NDVI-LST	ERA5	CropHarvest	Method
Forecast head only $L_{\mathrm{head}}$	4.1	4.1	4.1	Spectral norm product
Full encoder $L_{\mathrm{enc}}$	17.8	18.3	16.9	Randomized smoothing

Appendix

Table 3: Empirical Lipschitz constants (99th percentile over test set) via randomized smoothing (

\sigma_{\mathrm{smooth}}{=}0.01

Benchmark	MMD (ref. model)	$\hat{\mu}_{\mathrm{ref}}$	$p$ -value
NDVI-LST	$0.161\pm 0.013$	$0.160$	$<0.001$
ERA5	$0.214\pm 0.016$	$0.212$	$<0.001$
CropHarvest	$0.109\pm 0.010$	$0.108$	$<0.001$

Appendix

Table 4: MMD and RKHS separation

\hat{\mu}

estimated on clean reference model (not

\boldsymbol{\theta}^{*}

). All three benchmarks show significant separation (

p{<}0.001

, permutation test with 10,000 permutations).

	NDVI-LST	ERA5	CropHarvest
CRR $\uparrow$	$0.821{\pm}0.010$	$0.859{\pm}0.009$	$0.773{\pm}0.012$
$\Delta$ RMSE% $\downarrow$	$3.5{\pm}0.2$	$4.2{\pm}0.3$	$2.1{\pm}0.2$

Appendix

Table 5: Per-benchmark SSU-LSF CRR and clean-domain

\Delta

RMSE% (vs. OR), mean

\pm

std over 3 seeds.

Method	SCS $\downarrow$	FSS ( $n{=}8$ ) $\uparrow$
EWC-UL	0.134	0.571
GR	0.127	0.588
ARCANE	0.109	0.619
FAST	0.099	0.638
SR	0.102	0.633
ForgetFS	0.096	0.649

Appendix

Table 6: SCS vs. FSS (

n{=}8

neighbourhood) across all methods on CropHarvest. Pearson

r{=}-0.91

, Spearman

\rho_{s}{=}-0.93

(

p{<}0.001

). The FSS improvement from GR to SSU-LSF is

0.115

	NDVI-LST			CropHarvest
Method	CRR $\uparrow$	$\Delta$ RMSE% $\downarrow$	SCS $\downarrow$	CRR $\uparrow$	$\Delta$ RMSE% $\downarrow$	SCS $\downarrow$
ForgetFS	0.814	4.3	0.096	0.772	3.8	0.093
ForgetFS+TV	0.815	4.4	0.084	0.773	3.9	0.083
MULL	0.817	3.8	0.089	0.779	3.0	0.086
MULL+TV	0.818	3.9	0.082	0.780	3.1	0.082
SSU-LSF	0.821	3.5	0.081	0.773	2.1	0.081

Appendix

Table 7: Adding SSU-LSF’s TV term to ForgetFS and MULL (suffix +TV). This isolates the contribution of the TV term from the influence footprint. SSU-LSF outperforms both +TV variants on CRR on NDVI-LST and on

\Delta

RMSE on both benchmarks; +TV closes most of the SCS gap, confirming TV is the decisive factor for SCS.

\Delta

RMSE% is vs. OR.

	$T_{c}{=}6$ mo	$T_{c}{=}12$ mo	$T_{c}{=}24$ mo	$T_{c}{=}36$ mo
SSU-LSF CRR	$0.961{\pm}0.008$	$0.903{\pm}0.011$	$0.821{\pm}0.010$	$0.712{\pm}0.015$
Fitted $1{-}kT_{c}$	$0.953$	$0.906$	$0.812$	$0.718$
Oracle Retrain CRR	$0.993{\pm}0.003$	$0.991{\pm}0.003$	$0.987{\pm}0.004$	$0.985{\pm}0.004$
GR CRR	$0.891{\pm}0.018$	$0.831{\pm}0.021$	$0.741{\pm}0.018$	$0.628{\pm}0.024$
Window-estimate error $\pm 3$ mo	$0.944{\pm}0.012$	$0.887{\pm}0.014$	$0.803{\pm}0.013$	$0.691{\pm}0.018$

Appendix

Table 8: CRR vs. confounding window length

T_{c}

on NDVI-LST (mean

\pm

std, 3 seeds). Fitted curve:

1{-}kT_{c}

(

k{=}0.0078

, least squares), the

\rho_{\mathrm{eff}}\to 1

limit of Proposition 1’s first term.

R^{2}{=}0.995

with one fitted constant over four points (3 residual d.o.f.); this supports the monotone trend, not the exact functional form.

	NDVI-LST Fusion				ERA5 Patchified
Method	EKFac	FP	Ascent	Total	EKFac	FP	Ascent	Total
OR †	–	–	–	18.4	–	–	–	47.3
SISA	–	–	–	22.1	–	–	–	58.4
GR	–	–	0.9	0.9	–	–	2.3	2.3
ForgetFS	–	–	2.9	2.9	–	–	8.1	8.1
MULL	–	–	2.1	2.1	–	–	6.4	6.4

Appendix

Table 9: Per-unlearning-request GPU-hour breakdown. EKFac is re-estimated for each new confounding event identified. GPU-h/req is the operationally relevant metric: if three events per year are identified, annual cost is

3\times 5.6{=}16.8

GPU-h for SSU-LSF vs.

3\times 47.3{=}141.9

GPU-h for OR on ERA5. The OR

\dagger

/SISA/PNN-UL figures are per-retrain (no stage breakdown). SISA includes shard-retrain overhead and uses a pre-structured computational model (shards established before training) incomparable on GPU-h/req; its figure is included for completeness but excluded from the speedup comparison. EKFac: Hessian estimation. FP: footprint.Per-request speedup:

47.3/5.6{=}8.4\times

on ERA5.

Block	1	2	3	4	5	6
Mean	0.79	0.82	0.84	0.86	0.87	0.88
Std	0.02	0.02	0.03	0.03	0.03	0.03
Min	0.74	0.77	0.79	0.81	0.82	0.83
Max	0.83	0.86	0.88	0.90	0.91	0.91

Appendix

Table 10: Per-block

\rho(\bar{A})

statistics across 10 seeds on ERA5. At

\rho_{\max}{=}0.91

, the bound’s unlearning cost term

L_{\mathrm{enc}}\delta_{\mathrm{KL}}/(1{-}\rho_{\max}){=}18.3\times 0.05/0.09{\approx}10.2

K is vacuous relative to oracle RMSE

1.19

K. The bound is informative only qualitatively.

Configuration	CRR $\uparrow$	$\Delta$ RMSE% $\downarrow$	MIA $\downarrow$	SCS $\downarrow$
Full SSU-LSF	0.821(0.010)	3.5(0.2)	0.53(0.01)	0.081(0.003)
w/o $\Phi$ only	0.694(0.012)	8.2(0.4)	0.59(0.02)	0.092(0.003)
w/o KL only	0.831(0.009)	21.7(1.1)	0.52(0.01)	0.088(0.004)
w/o TV only	0.819(0.010)	3.6(0.2)	0.53(0.01)	0.131(0.006)
w/ GCS (not EKFac)	0.751(0.015)	5.1(0.3)	0.56(0.02)	0.094(0.004)
w/ Diag. Fisher	0.779(0.013)	4.6(0.3)	0.55(0.01)	0.091(0.003)

Appendix

Table 11: Full ablation (NDVI-LST). Top block: single-component removals. Middle block:

2\times 2

interaction ablations. Bottom block: sample-efficiency for clean rehearsal budget

|\mathcal{D}_{\mathrm{clean}}|

. All results: mean

\pm

std, 3 seeds.

\Delta

RMSE% is vs. OR.

$\lambda_{\mathrm{TV}}$	0	0.003	0.01	0.03	0.1
CRR $\uparrow$	0.819	0.820	0.821	0.815	0.792
$\Delta$ RMSE% $\downarrow$	3.6	3.5	3.5	4.1	6.3
SCS $\downarrow$	0.131	0.097	0.081	0.079	0.078

Appendix

Table 12:

\lambda_{\mathrm{TV}}

sensitivity (NDVI-LST, 3 seeds; default 0.01).

Figure 2: Hyperparameter sensitivity (NDVI-LST). (a) Threshold $\gamma$ . (b) KL tolerance $\delta_{\mathrm{KL}}$ . (c) Weight $\beta$ . Optimal ranges $\gamma{\in}[85,97]$ , $\delta_{\mathrm{KL}}{\in}[0.02,0.10]$ , $\beta{\in}[0.5,0.7]$ are consistent across all benchmarks.

Figure 3: LiRA MIA accuracy vs. unlearning epoch (ERA5). Dashed: oracle (0.51). Shaded: near-chance ( $\leq 0.53$ ). SSU-LSF converges near-monotonically; GR oscillates $\pm 0.04$ .

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