{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T08:31:12Z","timestamp":1786609872863,"version":"build-2736575974"},"reference-count":39,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T00:00:00Z","timestamp":1786579200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004772","name":"Ningxia Natural Science Foundation","doi-asserted-by":"publisher","award":["2026AAC030163"],"award-info":[{"award-number":["2026AAC030163"]}],"id":[{"id":"10.13039\/501100004772","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>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\u2013frequency 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.<\/jats:p>","DOI":"10.3390\/s26165123","type":"journal-article","created":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T08:06:16Z","timestamp":1786608376000},"page":"5123","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bridging High-Resolution Environmental Sensor Observations and Process-State Prediction: A Distribution-Shift-Robust Time\u2013Frequency Transformer (FT-Crossformer)"],"prefix":"10.3390","volume":"26","author":[{"given":"Yiran","family":"Guan","sequence":"first","affiliation":[{"name":"School of Information Engineering, Ningxia University, Yinchuan 750021, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhaoxu","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Information Engineering, Ningxia University, Yinchuan 750021, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-7809-1908","authenticated-orcid":false,"given":"Hui","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Information Engineering, Ningxia University, Yinchuan 750021, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,13]]},"reference":[{"key":"ref_1","first-page":"118","article-title":"Activated sludge models ASM1, ASM2, ASM2d and ASM3","volume":"9","author":"Henze","year":"2000","journal-title":"IWA Sci. 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