Bridging High-Resolution Environmental Sensor Observations and Process-State Prediction: A Distribution-Shift-Robust Time–Frequency Transformer (FT-Crossformer)

Yiran Guan

doi:10.3390/s26165123

Bridging High-Resolution Environmental Sensor Observations and Process-State Prediction: A Distribution-Shift-Robust Time–Frequency Transformer (FT-Crossformer)

2026-08-13

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Yiran Guan

Z

Zhaoxu Yu

H

Hui Guo *

DOI：10.3390/s26165123

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High-resolution online sensors are now common in environmental process systems, yet turning their non-stationary, heterogeneous observation streams into reliable predictions of the underlying process state remains difficult. The statistical distribution of a sensor stream changes over time, the measured variables do not coincide with the state variables of interest, and repeatedly running a mechanistic process model for forward prediction is computationally costly. We present FT-Crossformer, a time–frequency Transformer that acts as a data-driven surrogate between multi-sensor observations and multivariate process-state prediction. To handle distribution shift in the sensor streams, a time-domain distribution-transformation module, together with an inverse-mapping module, performs an affine bias correction that removes per-window non-stationary statistics at the input and restores them at the output, so the gap between training and test distributions is reduced without discarding non-stationary information. We show that this affine correction, including its learnable per-variable scale and shift, acts in the frequency domain on every non-zero frequency component as one common scaling factor that does not depend on the frequency index, so it cannot change the relative magnitudes among the components. A frequency-stability measurement module and a frequency-weighting module therefore re-weight the spectral components of the observation signal so that the stable, task-relevant ones contribute more to the reconstructed signal. The cross-dimension attention of the Crossformer backbone serves as a multi-sensor fusion mechanism that models the dependencies among the measured variables. We validate the method on public benchmark datasets from different domains as a check of generality and, most relevantly, for environmental modeling on two real cases: a wastewater nitrogen-and-phosphorus-removal process and chlorophyll forecasting from an in situ estuary sensor mooring in San Francisco Bay. On the estuary chlorophyll data, which carries a strong train-to-test distribution shift, the full FT-Crossformer demonstrates superior accuracy among the evaluated models at the next-day nowcasting horizon, and an ablation shows that both the time-domain trans- formation and the frequency-domain weighting contribute to this accuracy. FT-Crossformer produces forward predictions from distribution-shifted sensor data with a single fixed-cost forward pass in place of a repeated mechanistic solve, which makes it a practical building block for sensor-data integration and assimilation in environmental process modeling.

Keywords:

environmental sensor data

data assimilation

non-stationary sensor streams

bias correction

observation operator

multi-sensor fusion

multivariate time-series forecasting

Crossformer

distribution shift

frequency-domain analysis

wastewater treatment process

chlorophyll forecasting

estuarine monitoring

Bridging High-Resolution Environmental Sensor Observations and Process-State Prediction: A Distribution-Shift-Robust Time–Frequency Transformer (FT-Crossformer)

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