# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## 5 Conclusions  
This work produces a first-principles estimate of correlated instrument error with results that approximate observed statistical behavior. We believe that this model presents a significant advancement in the estimation of the spatial and temporal correlation structure of instrument error for remote sensing systems and, in particular, for the unique considerations of GNSS-R measurements by CYGNSS. In essence, this work answers a theoretical exercise for enumerating the plausible engineering reasons why two data points from the same observing constellation can share correlated sources of error. We evaluate the correlated error structure for CYGNSS by examining the potential plausible sources of correlated structure from individual components of the instrumentation, combining these sources from first principles as a tuned engineering model, and evaluating the efficacy of the model via a robust validation during a period when two satellites with nearly identical observing geometry captured near-simultaneous and near-collocated samples. The instrument-originating sources of correlated error is likely to be a small component of the overall correlated structures of observation error. For instance, a likely significant source of correlated error structure is the Geophysical Model Function retrieval that converts observed NBRCS to surface wind speed. These errors can be multifaceted, both encompassing representation error, as the ground truth for training this retrieval is reanalysis data, which may not capture the spatial or temporal dynamics of wind speed [27]. A companion work will explore how the retrieval that maps from NBRCS to wind speed produces correlated error structures in space and time. Further, the fact that the single-differenced observations generally drive the double-differenced autocorrelation behavior suggests that [PERSON] et al. 's metric of simultaneity and collocation [42] generally applies for CYGNSS when satellites are in a near overlap condition. This interesting result suggests that double-differencing may not be required for statistical estimations of GNSS-R errors given a sufficiently large dataset of observations that nearly overlap. This implication may relax further validation requirements of \(R_{mod}\) and may enable near-real-time calibration strategies when samples are sufficiently close without the need to generate reanalysis-driven forward model NBRCS, which introduces significant latencies. Finally, while the correlated error model \(R_{mod}\) was designed to compute the estimated correlated error from arbitrary CYGNSS samples, this may be unnecessary in practice as the tuning makes the model quite stable between samples and tracks. Further, given sufficient assumptions about the stationarity of the instrument-correlated error and the overall magnitude of the instrument-originating sources of correlated error, this information could be conveyed as a static look-up table. Conceptualization, C.E.P. and C.S.R. ; methodology, C.E.P. ; software, C.E.P., D.S.M., T.W. and A.R. ; validation, C.E.P. ; formal analysis, C.E.P. ; investigation, C.E.P. ; resources, C.S.R. ; data curation, C.E.P. and A.R. ; writing--original draft preparation, C.E.P. ; writing--review and editing, all authors; visualization, C.E.P. ; supervision, C.S.R. ; project administration, C.E.P. and C.S.R. ; funding acquisition, C.S.R. All authors have read and agreed to the published version of the manuscript. This work was supported in part by the National Aeronautics and Space Administration Science Mission Directorate under Contract NNL13 AQ00C with the University of Michigan and Contract 80 NM0018D0004 with the California Institute of Technology Jet Propulsion Laboratory. Publicly available datasets were analyzed in this study. CYGNSS Level 1 data can be found at NASA's Physical Oceanography Distributed Active Archive Center (PO.DAC) [43]. ERA5 reanalysis can be found at [44]. We acknowledge the insightful feedback of two anonymous reviewers who improved the quality of this manuscript.