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Article Type:
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SWOT Data Assimilation with Correlated Error Reduction: Fitting Model and Error Together

Sarah T. Gille Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Yu Gao Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Bruce D. Cornuelle Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Matthew R. Mazloff Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Online Publication:
11 Mar 2025
Print Publication:
01 Mar 2025
DOI:
https://doi.org/10.1175/JTECH-D-24-0062.1
Page(s):
253–268
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Abstract

The Surface Water and Ocean Topography (SWOT) satellite mission provides high-resolution two-dimensional sea surface height (SSH) data with swath coverage. However, spatially correlated errors affect these SSH measurements, particularly in the cross-track direction. The scales of errors can be similar to the scales of ocean features. Conventionally, instrumental errors and ocean signals have been solved for independently in two stages. Here, we have developed a one-stage procedure that solves for the correlated error at the same time that data are assimilated into a dynamical ocean model. This uses the ocean dynamics to distinguish ocean signals from observation errors. We test its performance relative to the two-stage method using simplified dynamics and a dataset consisting of westward-propagating Rossby waves, along with correlated instrumental errors of varying magnitudes. In a series of tests, we found that the one-stage approach consistently outperforms the two-stage approach when estimating SSH signal and correlated errors. The one-stage approach can recover over 95% of the SSH signal, while skill for the two-stage approach drops significantly as error increases. Our findings suggest that solving for the correlated errors within the assimilation framework can provide an effective analysis approach, reducing the risks of confounding signal and instrument noise.

Significance Statement

The Surface Water and Ocean Topography (SWOT) satellite measures sea surface height (SSH) with unprecedented spatial resolution. However, the measured SSH can include slowly varying large-scale errors, for example, associated with spatial shifts in the orientation of the satellite antenna. This study introduces a methodology for correcting the large-scale errors in data assimilation problems. By fitting errors and ocean dynamical signals at the same time, we reduce uncertainties both in the signal and in the large-scale error.

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sarah Gille, sgille@ucsd.edu

Abstract

The Surface Water and Ocean Topography (SWOT) satellite mission provides high-resolution two-dimensional sea surface height (SSH) data with swath coverage. However, spatially correlated errors affect these SSH measurements, particularly in the cross-track direction. The scales of errors can be similar to the scales of ocean features. Conventionally, instrumental errors and ocean signals have been solved for independently in two stages. Here, we have developed a one-stage procedure that solves for the correlated error at the same time that data are assimilated into a dynamical ocean model. This uses the ocean dynamics to distinguish ocean signals from observation errors. We test its performance relative to the two-stage method using simplified dynamics and a dataset consisting of westward-propagating Rossby waves, along with correlated instrumental errors of varying magnitudes. In a series of tests, we found that the one-stage approach consistently outperforms the two-stage approach when estimating SSH signal and correlated errors. The one-stage approach can recover over 95% of the SSH signal, while skill for the two-stage approach drops significantly as error increases. Our findings suggest that solving for the correlated errors within the assimilation framework can provide an effective analysis approach, reducing the risks of confounding signal and instrument noise.

Significance Statement

The Surface Water and Ocean Topography (SWOT) satellite measures sea surface height (SSH) with unprecedented spatial resolution. However, the measured SSH can include slowly varying large-scale errors, for example, associated with spatial shifts in the orientation of the satellite antenna. This study introduces a methodology for correcting the large-scale errors in data assimilation problems. By fitting errors and ocean dynamical signals at the same time, we reduce uncertainties both in the signal and in the large-scale error.

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sarah Gille, sgille@ucsd.edu
Keywords: Rossby waves; Altimetry; Data quality control; Bayesian methods; Regression analysis; Ocean models

1. Introduction

The Surface Water and Ocean Topography (SWOT) satellite, launched in December 2022, has ushered in a new era of high-resolution sea surface height (SSH) measurements, offering unparalleled coverage of the global ocean surface (Fu et al. 2012). The SWOT satellite uses the Ka-band Radar Interferometer (KaRIn) to measure ocean and surface water levels over a 120-km-wide swath with a near-nadir gap of approximately 20-km width (Fu et al. 2012; Fu and Ubelmann 2014; Esteban-Fernandez 2017; Morrow et al. 2019). With its high-resolution swath measurements, the SWOT satellite can measure the two-dimensional structure of small-scale features, facilitating the study of small-scale currents, tides, and oceanic circulation. However, these new measurement capabilities come with significant challenges, including both measurement noise and spatially correlated errors. Spatially correlated errors impact SWOT data, predominantly in the cross-track direction (Esteban-Fernandez 2017).

The SWOT error budget sets error requirements as a function of wavelength, making SWOT the first altimetric mission to do so (Esteban-Fernandez 2017). While SWOT’s initial performance is reported to be excellent, the possibility of spatially correlated error and the large volume of data produced by the mission necessitate innovative methods (Gaultier et al. 2016; Dibarboure et al. 2022).

In advance of the satellite launch, Metref et al. (2019) proposed a strategy to reduce the spatially structured errors of the SWOT satellite mission’s SSH data through a two-stage approach. The first stage involves detrending the SSH data by projecting them onto a subspace spanned by the SWOT spatially structured errors. The second stage uses the detrended measurements as inputs to a data assimilation scheme. Metref et al. (2020) found that the assimilation of SWOT data reduces the root-mean-square error (RMSE) of the reconstructed SSH, relative vorticity, and surface currents and also improves the noise-to-signal ratio and spectral coherence of the SSH signal at the mesoscale. A limitation of the two-stage approach is that error and signal are not necessarily orthogonal. Any overlap between the assumed error structure and the actual signal can result in “leakage,” leading to ocean signals being misrepresented as measurement errors (Dibarboure and Ubelmann 2014; Dibarboure et al. 2022). Despite this limitation, two-stage approaches represent traditional practice in satellite products and in assimilation, which treat error removal as a distinct step prior to data assimilation.

In this study, we introduce a streamlined one-stage approach, extending the framework of Metref et al. (2020) while aiming to mitigate the ambiguity between correlated errors and ocean signals. Our one-stage approach directly incorporates the reduction of correlated SWOT errors into the SSH estimation process, merging the two stages of Metref et al. (2019, 2020). In addition to minimizing the risk of confounding genuine ocean signals with correlated errors, our method allows us to consider other types of slowly varying spatially correlated errors, which could include, for example, environmental corrections such as the wet troposphere path delay or geoid/mean sea surface biases.

To develop our one-stage approach, we work with an idealized testbed scenario, centered in the California Current region (Fig. 1). The California Current System is a complex region, where sea level variability is influenced by both local winds and remote forcing from equatorial winds (Verdy et al. 2014; Zaba et al. 2018). In this case, our input data are SSH anomalies (SSHAs), simplified to contain only wave-like disturbances, roughly consistent with westward-propagating Rossby waves (Watanabe et al. 2016). These input data capture the dominant SSHA variability in midlatitude regions such as the California Current System (e.g., Chelton and Schlax 1996; Ivanov et al. 2010; Todd et al. 2011; Farrar et al. 2021; Gómez-Valdivia et al. 2017), although formally observed SSHA features are more consistent with westward-propagating nonlinear vortices rather than Rossby waves (Chelton et al. 2007, 2011). The westward propagation speed for SSHA features depends on latitude and stratification and is typically around 1° longitude in 50 days.

Fig. 1.
Fig. 1.

(a) SSH anomalies in the California Current region on 1 Jan 2016. Black lines indicate the 10° latitude × 9° longitude region of interest for this study, and gray lines bound the SWOT measurement region along the California Coast.

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

To assess the one-stage approach, we employ a quasigeostrophic (QG) Rossby wave model, rather than a full general circulation model. The reduced Rossby wave model is able to capture leading-order ocean variability (Wakata and Kitaya 2002); in this case, SSHA signals are attributable to westward-propagating Rossby waves. In our simplified scenario, our reduced model is able to provide a perfect fit to our idealized data. Within this idealized framework, our goal is to solve for correlated errors as well as dynamical variations in sea surface height, using either a one-stage or a two-stage approach, and to test the extent to which a one-stage method can ameliorate the “leakage” due to the ambiguity between ocean signal and correlated errors.

This paper is organized as follows: section 2a reviews the Bayesian estimation method that we employ, and section 2b describes our simplified ocean dataset based on the idealized Rossby wave model. In section 3, we present our implementation of the correlated error model. Section 4 describes the one-stage versus two-stage approach in an idealized multiday simulation for a single start date and then reviews statistics for a suite of different case studies. Section 5 provides a discussion and conclusions.

2. Experimental setup: Estimation and synthetic data

a. Bayesian estimation method

We employ a Bayesian estimation method, consistent with the approach outlined by Wunsch (1996). Here, we provide a brief overview of this method, as formulated for SWOT correlated error reduction, adopting the notation used by Ide et al. (1997) and Kachelein et al. (2022) with some modifications. A central concept in Bayesian estimation is that all available information is used, and thus the estimation process solves only for an update to the prior information.

A zero-mean SSHA sample is represented as a column vector h of length N, which satisfies the equation
h=Ha+r.
In this formulation, h is treated as a correction to the background state, meaning that we assume that all prior information has already been taking into account so that our fitting procedure is providing an update and not redetermining known values. The matrix H represents model basis functions and hypothetically could be any linear model. In this study, for illustrative purposes, H will provide a stripped-down representation of ocean dynamics by defining the spatial structure of a set of Rossby waves of unknown amplitude. The vector a represents the model parameters and, in our case, will become amplitudes of the Rossby waves defined by H. Finally, the vector r is the residual and represents the portion of the signal h that is not represented by the fitted model Ha. Formally, if the measured SSHA h were zero—that is, if h were equivalent to the background state—then a would be zero. When h in (1) is nonzero, the estimated parameters a provide the information needed to correct our prior knowledge. Appendix A provides a glossary as a quick reference for the variables defined in this section.
In this study, the elements of H are chosen to be Rossby waves that are expressed as sines and cosines:
Hi,n=cos⁡(kn⋅xi−ωnti),
Hi,n+Nm=sin⁡(kn⋅xi−ωnti),
where xi is a vector representing the geographic position of the ith observation in Cartesian coordinates, ti is the time of the observation, kn is the vector representation of the nth wavenumber in the model basis, Nm is the number of waves, and ωn is the frequency of the nth wavenumber. Each row of (1) can be expressed as
hi=h(xi,ti)=∑n=1Nm(anHi,n+an+NmHi,n+Nm)+ri,
where hi denotes the ith SSHA observation, and we assume a total of Nd observations. In the vector form, h(x, t) comprises a linear set of Nd equations, which represent the observations by Nm Rossby waves, and the matrix H has Nd × 2Nm elements. The amplitudes of the sinusoidal waves are an, where in the vector form, a is a 2Nm-element vector.

The dispersion relation for Rossby waves defines the relationship between their frequency and wavenumber. For these waves on a β plane, it is ωn=−βkn/(kn2+ln2+Ld−2), where ωn represents the frequency of the nth wave, kn and ln are the zonal and meridional wavenumbers, respectively, and Ld is the Rossby radius of deformation. The meridional derivative of the Coriolis parameter f is β = df/dy, which is taken to be constant. The phase speed of the waves (ωn/kn) is westward due to the negative sign and increases with increasing wavelength until the long wave limit where kn2+ln2≪Ld−2.

The depth–mean buoyancy frequency B influences the first baroclinic mode Rossby radius of deformation, described as
Ld∼BDπf,
where D is the total depth of the water column and B is stratification. The Rossby radius of deformation is larger in regions where the ocean is deep (larger D) or has strong stratification (larger B). To determine the buoyancy frequency, we used a representative stratification taken from a numerical simulation of the California Current region (Mazloff et al. 2020). The simulation provides detailed vertical profiles, capturing variations in temperature and salinity across the water column. From these simulated density profiles, the buoyancy frequency was calculated using the Brunt–Väisälä formula.
The unknown model parameters a are estimated as
a^=(HTR−1H+P−1)−1HTR−1h.
This solution minimizes both the model misfit and the magnitudes of the model parameters. The posterior covariance matrix of the difference between the estimated and true model parameters is
⟨(a−a^)(a−a^)T⟩=(HTR−1H+P−1)−1,
where R is an Nd × Nd matrix representing the error covariance of the observations and P is a 2Nm × 2Nm matrix representing the a priori error covariance of the model elements a.

Although the inversion of the matrix (HTR−1H + P−1) takes place in the model space (dimensioned 2Nm × 2Nm), the R matrix is Nd × Nd, which makes it expensive to invert unless it is diagonal. Here, we simplify further by defining R to be a multiple of the identity matrix.

If R=σd2I, where σd is the standard deviation of the measurement (data) noise, then (6) simplifies to
a^=(HTH+σd2P−1)−1HTh,
which is all in the model space. The diagonal of the matrix P is defined as (k2+l2)−2, where k and l are, respectively, the zonal and meridional components of wavenumber k. The diagonal of σd2P−1 is the noise-to-signal ratio. Larger values imply more noise relative to information and lead to solutions that are closer to the prior guess of zero.
The posterior uncertainty covariance (7) can be transformed to physical space by pre- and postmultiplying by a matrix Hmap that converts from a to either the swath or the mapping grid:
⟨(hmap−h^map)(hmap−h^map)T⟩=Hmap(HTR−1H+P−1)−1HmapT.
The mapping matrix Hmap can be defined for any set of space–time points. For example, the time of the map can be set to the beginning, middle, or end of the assimilation time range, or any time in the past or future. Although (9) produces a full uncertainty covariance matrix, the general practice is to report only the diagonal elements or the largest eigenvalues and eigenvectors of this matrix, which represent the largest orthogonal modes of posterior uncertainty.

b. Input data for the simplified Rossby wave model

To develop an approach for analyzing propagating waves as well as correlated satellite error, we start with a simplified dataset that contains realistic sea surface height variability and for which the model parameters are fully known. The data are generated by fitting a Rossby wave model to 0.25° × 0.25° gridded altimeter fields within the black box in Fig. 1, as discussed in appendix B. We use twelve 80-day test datasets, starting on the first day of each month of 2016. Each test dataset includes a 40-day period for which we have fitted model parameters to the gridded altimetry followed by a 40-day prediction period. For each test case, the initial altimeter dataset, here identified as horig, is organized on a grid that contains 40 points in longitude and 36 points in latitude, so in the initial analysis, the number of gridded data points Ng will be 1440 data points per day, making the total number of observations over 40 days, Nd = Ng × 40.

As explained in appendix B, the idealized data used for this study are obtained by projecting the 40-day record of Copernicus gridded altimeter data onto a basis set of 190 waves that have properties typical of Rossby waves. We use 190 cosine components and 190 sine components, meaning that in this implementation Nm = 190. Details about the skill of this projection in representing the original Copernicus data are peripheral to the main objectives of this study but are included in appendix B. From the projection process in appendix B, we obtain a clean and noise-free simplified sea surface height field h, for which we have perfect knowledge of the model parameters a. For the remainder of this paper, we will use these estimated fields as the model truth. Using only input data from locations within the SWOT satellite swath, our objective will be to evaluate how well we can find appropriate wave coefficients a and replicate the full SSHA field h, both during the 40-day fitting period and projecting 40 days forward in time.

Within the 120-km-wide SWOT swath, we subsample the input data h to produce a manageable input field in which adjacent data points are reasonably independent. Because the Rossby waves have a large spatial structure, our subsampling uses every eighth point in SWOT’s cross-track direction and every 16th row in SWOT’s along-track direction. For the SWOT 1-day repeat period, carried out from April to July 2023, each day yields approximately 225 points from ascending satellite passes and 225 points from descending satellite passes, mimicking the SWOT satellite sampling. To these in-swath SSHA points, we add simulated satellite sampling errors, as discussed in the next section.

3. Correlated error model

Following Metref et al. (2020) and Esteban-Fernandez (2017), our correlated error reduction procedure considers four error terms, defined by seven (unknown) coefficients αi. The timing error α0 is treated as a constant and is attributed to the instrument timing drift. Roll error, expressed as α1xc, results from the satellite’s roll angle and is assumed to increase linearly in the cross-track coordinate relative to nadir, xc. The baseline dilation error α2xc2 originates from variations in the satellite mast length. Last, the phase error results from relative variations in phase between the satellite’s left and right antennas, leading to distinct cross-track linear errors in each half swath. This is represented using Heaviside functions H(x), which equal 1 when x ≥ 0 and 0 otherwise:
(α3+α4xc)H(−xc)+(α5+α6xc)H(xc)
.
Together, the total error is modeled as
etotal=α0+α1xc+α2xc2+(α3+α4xc)H(−xc)+(α5+α6xc)H(xc).
Following the general approach of Metref et al. (2019), within our relatively small domain, we solve for coefficients under the assumption that they are effectively constant for each pass along the track. In other words, we assume that the spatial decorrelation scale of the along-track error exceeds our domain size. Mathematically, the terms represented by α0 and α1 are redundant with the inclusion of Heaviside functions for the left and right swaths, which will lead to a rank deficient matrix. In this case, since we use a regularized inverse, we retain all of the terms proposed by Metref et al. (2019). The prior covariance matrix P sets the expected relative sizes of the full swath adjustments represented by α0 and α1 and the left and right adjustments represented by α3, α4, α5, and α6. The term α2 is not repeated separately for the left and right sides, but its covariance is of course still represented by P.
In matrix form, for satellite pass m, the error model can be written as
etotalm=Herrmaerrm,
where vector etotalm has Ns elements corresponding to each observation within the swath and Herrm is an Ns × 7 matrix:
Herrm=[1xc1xc12H(−xc1)xc1H(−xc1)H(xc1 )xc 1H(xc1) ⋮⋮⋮⋮⋮⋮⋮ 1x cixci2 H(−x ci) xci H(−x ci) H(xci)xciH(xci)⋮⋮⋮⋮⋮⋮⋮1 xcNsxcNs2 H(−xcNs )x cNs H(−x cNs )H(xc Ns )xcNsH(x cNs)],
where xci refers to the ith element of a vector of cross-track positions xc. The corresponding fitted parameters are
aerrm=[α0mα1mα2mα3mα4mα5mα6m]T.
Since the error evolves in time, the error vectors are concatenated, and the error matrix is augmented to represent each ascending or descending satellite pass, which are assumed to have the same sampling on every pass, for a total of M passes; this results in a block diagonal matrix consisting of one Ns × 7 matrix per satellite pass:
Herr=[Herr10⋯00Herr2⋯0⋮⋮⋱⋮00⋯HerrM],
aerr=[aerr1aerr2⋮aerrM].
The total number of data points Nd (and rows in H) is thus Ns × M.

Further errors in SWOT data could stem from a range of environmental corrections, including both uncorrelated noise and large-scale correlated signals. Among the possible correlated errors is the mean dynamic topography (MDT), which is correlated in time rather than space. Total SSH required for data assimilation makes use of SSHA computed relative to the time-averaged measured mean sea surface. SSHA values are added to the estimated MDT to infer total dynamic topography. While the accuracy of the MDT is an ongoing challenge (e.g., Mazloff et al. 2014), for this proof-of-concept study, we have bypassed the issue by considering only SSHA.

a. Two-stage approach

The first stage of the two-stage approach is error removal. Following Metref et al. (2019), we consider only data collected within satellite swaths, here identified as hswath. To remove the correlated errors from the signal, in the two-stage approach, we calculate the projection of hswath onto the subspace spanned by the modeled errors in (10), minimizing the cost function:
J1(aerr)=(hswath−Herraerr)TR−1(hswath−Herraerr)+aerrTPerr−1aerr,
where J1 is a scalar representing the squared misfit between the data and the fitted error, summed over all Ns data points within the swath and over all M satellite passes included in the analysis, weighted by the data covariance R, with aerr representing the modeled best estimate of the error parameters. The term aerrTPerr−1aerr imposes an additional constraint to prevent aerr from becoming large relative to the prior model covariance for the error terms Perr. We then define a proxy version of the SSHA data h˜swath as the difference between the SWOT signal hswath and the projection onto Herr in the vector form:
h˜swath=hswath−Herra^err.
The second stage of the two-stage approach is solving for sea surface height. We fit the difference h˜swath to the subspace spanned by the Rossby wave model given by (2) and (3). This is applicable to all Ns data points within each swath and for all M time samples used in the fitting period:
J2(aw)=(h˜swath−Hswath aw)TR−1(h˜swath−Hswath aw)+awTP−1aw.
The Hswath matrix is the representation of (2) and (3) for data points within the swath only:
Hswath=[H1,1H1,2⋯H1,NmH1,Nm+1⋯H1,2Nm⋮⋮⋱⋮⋮⋱⋮HMNs,1HMNs,2⋯HMNs,NmHMNs,Nm+1⋯HMNs,2Nm].
The true wave parameters are
aw=[a1a2⋯aNmaNm+1⋯a2Nm]T,
the estimated wave parameters are a^w, and P represents the covariance matrix corresponding to the wave solutions. Using these estimates, we can find estimated values of SSHA both in the swath (h^swath) and throughout the full domain (h^).

b. One-stage approach

The one-stage approach combines the error projection in (16) and SSHA model fit in (18) and solves for errors and SSHA signals simultaneously. We achieve this using an augmented matrix Htotal that combines Hw and Herr together with an augmented parameter vector atotal:
Htotal=[Hswath|Herr],
atotal=[awaerr].
The analysis minimizes the cost function:
J(atotal)=(hswath−Htotal atotal)TR−1(hswath−Htotal atotal)+atotalTPtotal−1atotal,
again where atotalTPtotal−1atotal is the regularization term that prevents the model parameters from becoming large relative to their prior estimates. The portion of atotal representing the waves is aw, with a^w representing the best estimate. This determines the time-evolving wave-related SSHA within the swath h^swath=Hswatha^w or over the full domain: h^=Ha^w.

c. Experiment setup

In the experiments that follow, we apply both the one-stage approach and the two-stage approach to the set of 12 sea surface height datasets. To these sea surface height estimates, we add error in one of two ways. For our baseline experiments, we create correlated error using the same (10) (Metref et al. 2019) that we use to solve for the error. We refer to this as “synthetic error.” For each ascending or descending satellite pass, cross-track distances are scaled by 105 to represent 100 km, and the seven separate SWOT correlated error parameters are drawn at random from a Gaussian distribution. This approach guarantees that the correlated error will be consistent with the basis functions used to fit the error, allowing us to verify that the fitting procedure works but not providing a means to test sensitivity to error that differs from (10). In a second set of tests, we used the SWOT simulator (Gaultier et al. 2016) to create correlated error, which we refer to as “simulator error.” The simulator error incorporates errors due to baseline dilation, roll error, phase error, and timing error, as well as the “KaRIn” error, which is structured as random noise. We omit the SWOT simulator’s large orbital error, on the assumption that the orbital error should be corrected separately prior to starting analysis, and we also omit the wet troposphere correction, which is not the focus of this study and which relies on an interpolation scheme that is no longer supported by Python. Note that the simulator error has a larger spatiotemporal variance than the synthetic error: in these tests, the smaller synthetic error is envisioned to represent the residual correlated error that might remain in the data after first-pass data processing.

4. Results

a. Case study: One-stage versus two-stage approach

We first consider a case study, starting 1 January 2016, using synthetic error. We apply the one-stage and two-stage correlated error reduction procedures to 40 days of synthetic satellite data and then use the estimated model parameters to make 40-day forecasts. We consider the SWOT satellite calibration/validation orbit, which uses a 1-day repeat, meaning that 40 days of observations correspond to 80 separate satellite passes: 40 ascending passes and 40 descending passes. The misfit between the fitted solution and the data (the model–data misfit) is measured by the normalized mean-squared error (NMSE):
NMSE=mean[(hswath−h^swath)2]mean(hswath2),
where hswath is the truth (i.e., the synthetic satellite data) and h^swath is our estimate. NMSE quantifies how much the estimated values deviate from the true values, relative to the variance of those values. The percentage of variance explained,
%variance explained=100×(1−NMSE),
also referred to as “skill,” is a useful metric for assessing the performance of a model and reflects the percentage of the total variance in the observed data (i.e., the mean of hswath2) that the model successfully captures.

Within the SWOT swath, we compare the performance of the one-stage and two-stage approaches, evaluating results in a 1-day snapshot midway through the training period. For this case study, the error components (i.e., the true aerr) that together determine the correlated error were drawn from a random distribution, scaled such that when the terms are summed, the standard deviation of the correlated error is 34% of the SSHA standard deviation. This magnitude was also reflected in the σd and P error covariance prescribed in the model. In this example for the 40-day fitting period, the one-stage data assimilation approach effectively recovers over 99% of the SSHA variance on the satellite swath (Fig. 2c), while the two-stage approach removes correlated errors less effectively, recovering only 66% of the SSHA signal (Fig. 2e). For the correlated error, the one-stage approach successfully recovers 93.5% of the true error (Fig. 2h), while the two-stage approach demonstrates no skill in estimating errors (Fig. 2j). Differences are less pronounced when we consider the skill at reconstructing total SSHA (h + e): both approaches recover more than 99% of h + e (Figs. 2m,o), implying that the shortcomings in the two-stage approach represent a failure to distinguish correlated errors from the SSHA signal. The superior performance of the one-stage approach suggests that solving for correlated errors as part of the assimilation is more effective than implementing separate procedures to solve for correlated errors and propagating dynamical SSHA signals, since the slow propagation of the Rossby waves provides key information that allows them to be separated from correlated error.

Fig. 2.
Fig. 2.

SSHA h and uncertainties e (both in m) along SWOT satellite swath for 21 Jan 2016, showing (a) the true h used for this study, derived by projecting gridded altimetry onto a set of Rossby waves (along-swath values are not shown but match the gridded values); (b) h^1−stage from the one-stage approach; (c) h−h^1−stage; (d) h^2−stage from the two-stage approach; (e) h−h^2−stage; (f) correlated error etotal used in this simulation; (g) estimated correlated error e^1−stage from the one-stage approach; (h) difference, Δe=etotal−e^1−stage; (i) estimated error e^2−stage from the two-stage approach; (j) difference, Δe=etotal−e^2−stage; (k) total SSHA, h + etotal, used as input; (l) total SSHA h^+e^1−stage from one-stage approach; (m) difference between (k) and (l); (n) total SSHA h^+e^2−stage from two-stage approach; and (o) difference between (k) and (n). In this case, R=σd2I=0.01I, and the coefficients αi have standard deviation 0.0125, with a resulting noise-to-signal ratio (root-mean-squared error divided by root-mean-squared signal) of 0.34.

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

Outside the satellite swath, the Rossby wave model provides a dynamical framework, allowing information measured by the satellite to propagate beyond the swath boundaries, filling in the full study domain. In the case study considered here, the one-stage approach shows better skill outside of the swath, with 62% of variance explained over the entire domain for the 40-day training period (Fig. 3b) compared with 29% in the two-stage approach (Fig. 3e). The two-stage approach also shows significant errors at the swath edges due to misinterpretation of error signals (see Figs. 3e,f). To quantify the results more completely, in the next section, we consider statistics based on a set of 12 different model start dates.

Fig. 3.
Fig. 3.

(a) The true SSHA h on 21 Jan 2016 (see appendix B for method to determine h); (b) SSHA estimate of a 190-wave Rossby wave model in the one-stage approach; (c) difference between true SSHA and estimated SSHA of one-stage approach, with black lines showing the center points of the outermost input data; (d) the gridded Copernicus SSHA on 21 Jan 2016; (e) SSHA estimate of a 190-wave Rossby wave model in the two-stage approach; and (f) the difference between the true SSHA and estimated SSHA of a two-stage approach. Noise-to-signal ratio is the same as in Fig. 2.

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

b. Multidataset analysis

We now extend the case study to consider the set of 12 start dates and a range of synthetic error fields created with 30 different noise levels. For simplicity, for each noise level, we use a single value to set the standard deviation of each of the seven coefficients αi that define the correlated error in (10), even though the coefficients have different units. We use a total of 30 different noise values, evenly spaced between 5 × 10−4 and 2.95 × 10−2. These values lead to total correlated error perturbations of 3 cm or less. Results show that on average the one-stage approach outperforms the two-stage approach (Fig. 4a). When the overall error is small relative to the signal (i.e., the ratio of root-mean-squared error to root-mean-squared SSHA, RMSE/RMS_SSHA≲0.05 ), there is little difference between the one-stage (orange stars) and two-stage (blue × marks) skills, but the two-stage skill drops as the error increases. Moreover, in the two-stage approach, for large RMSE/RMS_SSHA, there is a large spread between the results from the different cases, indicating that the two-stage Rossby wave parameter estimation is less robust. For both the one-stage and two-stage approaches, correlated errors (Fig. 4b) are more accurately estimated when the error is large relative to the signal (larger RMSE/RMS_SSHA). Since we use the percentage variance explained as a skill metric, for small errors, the two-stage approach can sometimes estimate errors that are unrelated to the true error, resulting in negative skill, while the one-stage approach consistently provides skillful error estimates.

Fig. 4.
Fig. 4.

(a) Percentage variance explained in the SSHA estimate for all experiments, plotted as a function of the error input to the problem (root-mean-squared error) normalized by root-mean-squared SSHA for each date. In total, there are 360 experiments, encompassing 30 noise values for each of 12 start dates. (b) Percentage variance explained in error estimate and (c) percentage variance explained of the total input signal h + e, for the one-stage and two-stage approaches as a function of the ratio of root-mean-squared error to root-mean-squared SSHA, RMSE/RMS_SSHA ratio. Blue “×” markers represent results from the two-stage approach, and orange “*” markers represent the one-stage approach. Since the input data are the same for the one-stage and two-stage approaches, each blue marker can be matched to an orange marker with the same RMSE/RMS_SSHA value. Since SSH data vary in time, and uncertainty has a random component, the RMSE/RMS_SSHA ratios differ for each of the 360 datasets.

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

Finally, since no random white noise is added to the observations, we expect that the fit should be able to fully reconstruct the combined SSHA and correlated error. As shown in Fig. 4c, this is indeed the case: the total “observed” signal, hswath = h + etotal, is well represented in both the one-stage and two-stage, with over 96% of the variance explained in all cases. The one-stage approach explains a slightly higher percentage of the overall variance; this is because this estimate has better prior information and fits all of the parameters simultaneously, which allows it to explain more of the normalized data variance for the same normalized model parameter cost.

The challenge in these experiments stems from the fact that the Rossby wave signal projects onto the correlated error, and the correlated error signal projects onto the Rossby wave. In Fig. 5, we explore this effect by simplifying the simulated dataset to consist of only one contribution: either the Rossby wave signal or the correlated error. Figure 5a shows that the models are skillfully able to reconstruct the signal, except in cases when the correlated error is assumed to be very noisy (blue “×” marks with small RMSE/RMS_SSHA ratios), when the prior covariance matrix assumes that the correlated error is so small that it cannot be skillfully reconstructed. In Fig. 5b, we reverse the process, fitting the correlated error to the Rossby wave model (blue “×” marks) and fitting the Rossby wave signal to the correlated error model (orange stars). While the Rossby waves and correlated error are largely distinct, these results show that the signals project onto each other, so that an average of 3%–4% of the variance (and as much as 17%) can be explained by the “wrong” model. The fact that the signals project onto each other is the reason why the two-stage approach shows lower overall skill than the one-stage approach. As a final check, in Fig. 5c, we replace the signals with random white noise while retaining the same fitting process and same covariance matrices used in the other cases. White noise does not project onto the fitted models, and less than 1% of the white noise variance can be explained by either model (red “+” signs and cyan dots).

Fig. 5.
Fig. 5.

(a) Percentage variance explained in experiments in which the signal is simplified, so that only the Rossby wave component of the signal is fitted by the Rossby wave model (orange) and the correlated error is fitted only with the correlated error model (blue “×” marks). Cases shown here are the same as in Fig. 4. (b) Percentage variance explained in the inverse case, in which the Rossby wave component of the signal is fitted by the correlated error model (orange stars), and the correlated error component of the signal is fitted with the Rossby wave model (blue “×” marks). This provides a measure of the projection of each part of the signal onto the model for the other part of the signal. (c) Percentage variance explained when a signal consisting only of uncorrelated white noise is fitted with the Rossby wave model (red “+” signs) or the correlated error model (cyan dots).

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

The average over 12 test cases of the percentage of variance explained illustrates the performance of both approaches (Fig. 6). In all parts of the domain and for all times, the one-stage approach performance is 20–30 percentage points better than the two-stage approach. Within the swath, the one-stage approach is able to explain almost all of the variance during the 40-day fitting period (blue line) and more than 90% of the variance in the 40 days after fitting. The two-stage approach explains more than 50% of the within-swath SSHA variance (red line). Outside the swath, skill increases noticeably over time as the Rossby waves propagate westward to fill in the full domain. In the one-stage approach, the percentage of variance is consistently positive (gold line), while the two-stage approach initially shows negative skill outside the swath (purple line), consistent with no correlation between the fitted estimates and the true SSH. As we would expect, the percentage of variance explained for the full domain (green line for one-stage; dark blue for two-stage) is intermediate to the in-swath and out-of-swath results.

Fig. 6.
Fig. 6.

Percentage of SSHA variance explained in one-stage and two-stage approaches, as a function of time, for in-swath only, out-of-swath only, and total domain, for correlated error with standard deviation 0.0125, as in Fig. 2. In this figure, model truth is determined from the gridded input fields, for example, in Fig. 3a. The gray line indicates the skill that would be achieved by assuming persistence of conditions from 21 day, here shown only for the full domain. All solid lines show statistics computed for the set of results based on the 12 analysis start dates, beginning from the first of each month in 2016. Error bars indicate 1 standard error of the mean. The gray dashed line indicates the 40-day mark when the forecast starts.

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

A standard benchmark is to compare forecast model performance against a baseline assumption of persistence—an assumption that conditions do not change relative to reference data. The gray line in Fig. 6 shows the skill achieved by persistence relative to the day-21 SSHA field (i.e., in this case, h gridded across the domain). The Rossby waves propagate slowly through the domain, so for short time separations, persistence is relatively skillful. Over longer time periods, the percent variance explained by persistence degrades quickly. By the end of the prediction period at day 80, both the one-stage and two-stage forecasts based on the Rossby wave model show greater skill than persistence. Keep in mind that these persistence results are expected to show greater skill than if we had data only on the swath, with a guess of zero SSHA off swath.

c. SWOT simulator error impacts

In a final set of tests, we use the Gaultier et al. (2016) simulator error in place of the idealized synthetic error. Using the SWOT simulator, we generate one realization of the error per satellite pass (in contrast with the synthetic error, for which we generated 30 different error fields of varying amplitudes).

The change in the noise necessitates reevaluation of the covariance matrix P. The covariance matrix defines the prior knowledge about the amplitude of the error coefficients, effectively constraining the size of the adjustments that the fitting procedure is able to make for each coefficient. In analogy with Fig. 6, here Fig. 7 shows the percentage variance explained by the one-stage and two-stage approaches, illustrated for two different covariance matrices P. We find limited skill if we adopt the same covariance matrix that we used in Fig. 6, assuming all error parameters to have a standard deviation of 0.0125; then, a maximum of about 80% of variance is explained using the one-stage approach (light blue line) and a maximum of less than 50% using the two-stage approach (light pink line). As an alternative, we set the diagonal of P to be consistent with the known variability, as determined from preliminary one-stage and two-stage fits to the data. The assumed standard deviations are 0.015 m for α0, α3, and α5; 0.15 × 10−10 m−1 for α2; and 0.3 × 10−5 (unitless) for α1, α4, and α6. In this case, the one-stage approach explains almost 100% of the variance (dark blue line), and the two-stage approach (red line) evolves in time similarly to what we found for the two-stage approach in Fig. 6. Overall, the percentage of SSH variance explained is slightly lower for the case using simulator error rather than synthetic error, as might be expected given the presence of additional terms (e.g., KaRIn noise and slowly varying correlated error) in the simulator fields.

Fig. 7.
Fig. 7.

Percentage of SSHA variance explained when error derives from the SWOT simulator (Gaultier et al. 2016) rather than the simplified synthetic model. Results presented here are analogous to those in Fig. 6, but only the in-swath variance is shown. Two sets of covariance matrices are considered. Dark blue and red lines show results using a covariance matrix P that has been adjusted to reflect the covariance of the simulator error, as determined based on the preliminary one-stage and two-stage fits to the data. Light blue and pink lines show results that employ the same covariance matrix used for the results in Fig. 6, with the same covariance assumed for all coefficients αi from (10).

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

As Fig. 7 illustrates, the one-stage approach consistently outperforms the two-stage approach during the 40-day fitting period, even when the covariance matrix P has not been adjusted to represent the known covariance of the error field. In forecast mode, after the first 40 days, the steep decrease of the light blue line in Fig. 7 shows that when the wrong covariance matrix P is used, the fit can diverge rapidly from the true SSH. However, with an appropriate covariance matrix, the one-stage approach provides a 40-day forecast that in these cases can explain more than 60% of the SSHA variance (blue line).

5. Conclusions

This study has aimed to provide a comprehensive analysis of the impact of roll error and other correlated errors on SWOT sea surface height (SSH) data assimilation. We have introduced a novel one-stage data assimilation approach that incorporates the process of correlated error reduction directly into the assimilation framework, contrasting with a two-stage methodology in which errors and ocean signals are analyzed separately. Our findings suggest that the one-stage approach enhances the robustness and accuracy of SSH estimates, especially in the presence of increasing correlated errors.

For demonstration purposes, we have used a simplified Rossby wave model to construct a clean dataset capturing a leading-order pattern of SSHA variability in the California Current region. We then added spatially correlated noise to the filtered data and analyzed it using the same form of the simplified Rossby wave model. This has allowed us to focus on the performance of the fit, without having to consider whether misfits were due to model shortcomings rather than noise in the SSHA.

Through a series of analyses, we varied the amplitude of the correlated errors and assessed their impact on SSHA estimation. We also used the SWOT simulator (Gaultier et al. 2016) to generate correlated errors consistent with SWOT errors as anticipated prior to the satellite’s launch. The one-stage approach outperformed the two-stage approach, particularly under conditions of high correlated error. While the two-stage approach showed diminished skill in estimating SSHA with increased levels of correlated error, the one-stage approach provided robust and skillful results, consistently explaining close to 100% of SSHA variance within the SWOT satellite swath, regardless of the error magnitude. Outside the satellite swath, the one-stage approach provides more skillful estimates of SSHA both during and after the fitting/assimilation time window. The skill in estimating correlated error terms is evaluated based on the percentage of variance explained. Large errors are more easily estimated than small errors since they represent a larger percentage of the total signal. Small errors can be difficult to estimate and therefore have large fractional uncertainties, but fortunately, since the errors are small, they have minimal impact on SSH.

Our results underscore the importance of addressing correlated errors as part of the data assimilation process. By doing so, we reduce the likelihood of misinterpreting instrument errors as ocean signals. SWOT level 2 (L2) data have been released with guidelines for reducing or removing roll error, and an initial estimate of roll error has been removed from the level 3 (L3) product produced by AVISO. Nonetheless, L3 products have the potential to contain remnants of the correlated error that could be reduced using a one-stage assimilation approach.

Some caveats should be considered in evaluating these results. This study has been carried out using a toy problem to test the core concepts underlying the correlated error correction within assimilation systems. We have considered a relatively small domain, with an idealized truth h that features only larger-scale Rossby wave-like propagating signals, and with a perfect model. Both the error model used as input to this problem and the fitting procedure have assumed an idealized error model that is constant (or nearly constant) over the study domain. We have specifically considered a case in which the correlated error and the propagating Rossby wave signal project onto each other and are difficult to separate; in cases with distinct models, correlated error and ocean signal might be more easily separated, simply by using a well-chosen prior. New challenges will likely arise in efforts to implement this procedure with realistic data containing small-scale noise and forms of signal correlated error that might not be anticipated by the fitting or assimilation procedures.

In summary, our research demonstrates that for data assimilation of the type done with SWOT measurements, an integrated one-stage approach that concurrently addresses correlated errors and ocean signal estimation has the potential to provide a reliable and robust representation of ocean dynamics. While the approach documented in this paper is idealized, it establishes a methodology that could be implemented more generally in SWOT data assimilation and that could be extended to other types of correlated errors, for example, due to satellite orbit errors, geoid uncertainties, tides, or near-inertial oscillations. This methodology offers the possibility of refining our representation of oceanic processes in assimilating models.

Acknowledgments.

This study has been supported by the National Aeronautics and Space Administration (NASA) Surface Water and Ocean Topography (SWOT) Science Team, Awards 80NSSC20K1136 and 80NSSC24K1657.

Data availability statement.

The code for this project is available at https://github.com/sgille/swot_correlated_error or https://doi.org/10.5281/zenodo.13841195. Data can be accessed via https://doi.org/10.5281/zenodo.10963448.

APPENDIX A

Glossary

a

True model parameters

a^

Best estimate of model parameters

aw

Rossby wave model parameters

aerr

Model parameters for correlated error terms

B

Buoyancy frequency

c

Phase speed of the first baroclinic Rossby wave

D

Depth of the water column

e

True correlated error, represented as a column vector

e^

Best estimate of the correlated error represented as a column vector

f

Coriolis parameter

h

True or measured sea surface height, represented as a column vector

h^

Best estimate of sea surface height measurements represented as a column vector

h˜

In two-stage method, sea surface height with estimated correlated error removed

H

Model basis functions

Hi,n, Hi,n+Nm

Elements of Rossby waves expressed as cosines and sines, respectively.

i

Index for an observed SSHA measurement, at position xi and time ti

k, l

Zonal and meridional wavenumbers

k

Vector wavenumber, with components defined by k and l

Ld

First baroclinic Rossby wave deformation radius

M

Number of satellite swaths included; for daily data two passes per day

Ng

The number of regularly gridded mapped SSHA values in the study domain; in this case, 40 points in longitude by 36 points in latitude

Nd

The number of SSHA observations input to the fitting procedure (defined by Ng to develop the simplified SSHA field and by Ns to test the one-stage and two-stage approaches)

Nm

The number of waves included in the model

Ns

The number of observations contained within the swath

P

σw2I, the portion of Ptotal representing the Rossby wave model parameters

Perr

The portion of Ptotal representing correlated error parameters

Ptotal

The covariance matrix representing prior uncertainty in all model parameters

r

Residual

R

σd2I, matrix represents the measurement (data) noise

ti

Time of ith observation

xi

Geographic position in Cartesian coordinates

α0

Timing error parameter

α1

Roll error parameter

α2

Baseline dilation error parameter

α3, α4, α5, α6

Phase error parameters

β

Meridional derivative of the Coriolis parameter (df/dy)

σd

Standard deviation of the measurement (data) noise

σw

Standard deviation of the signal

ωn

Frequency of Rossby waves

APPENDIX B

Synthetic Data from Sea Surface Height

The SSH data used as input for this analysis are generated starting from daily level 4 (L4, multisatellite) altimeter fields within the region shown in Figure 1 from the Copernicus Global Ocean Gridded L4 sea surface heights and derived variables reprocessed dataset (Copernicus Marine Service Information 2023), sometimes referred to as CMEMS. In this appendix, we summarize the method used to generate these input files. We use L4 SSHA fields from 1 January 2016 through early 2017, which are mapped to a 0.25° grid. As discussed in section 2b, we set up a total of twelve 80-day test datasets, starting on the first day of each month in 2016.

For each start date, we project a 40-day sequence of daily fields onto a basis set of 190 waves with properties typical of Rossby waves—190 cosine components and 190 sine components—meaning that in this implementation Nm = 190. The wavelengths include 10 zonal modes (0–5.1 cycles per degree in space) and 19 meridional modes (from −5.24 to +5.04 cycles per degree latitude), in both cases evenly incremented at intervals of 0.571 radians per degree (1 cycle in 11°). In contrast with classic Fourier transforms, here the modes are chosen to include wavelengths slightly larger than the domain size to avoid periodicity within the space and time domain of the simulation. The meridional wavenumbers are asymmetric in the positive and negative directions to avoid having a large number of modes with wavenumber 0.

By construction, this basis set is not orthogonal over our test region. The use of a nonorthogonal basis set is intentional since it allows us to capture low-wavenumber structures that are larger than our study domain. In a classic ordinary least squares problem, the nonorthogonal basis set would be rank deficient with no robust solution. As discussed in section 2, we avoid these problems through the use of the regularized inverse with P covariance matrices.

Using the set of 190 wavenumbers, we estimate model coefficients a^orig, where the hat indicates that these are estimated coefficients, and the subscript “orig” signals that they are derived from fitting the original gridded altimeter data, which contains patterns of variability that extend beyond the 190 fitted wave signals. The fitted coefficients a^orig are able to represent about 95% of the SSHA variance in the domain during the 40-day fitting period starting 1 January 2016 (Fig. B1). We then use a^orig in the wave model to project SSHA forward in time, computing an estimated SSHA h^orig, as a function of time both for the fitting period and for 40 days afterward. The core findings of this study do not depend on the ability of the Rossby wave model to replicate the gridded Copernicus SSHA fields, but for completeness in this appendix, we summarize our assessment of the skill of the Rossby wave fitting procedure. At the midpoint of the 40-day fitting period, Fig. B1 shows the original SSHA on 21 January 2016 (Fig. B1a), the fitted SSHA (Fig. B1b), and the residual difference (h^orig−horig) (Fig. B1c).

Fig. B1.
Fig. B1.

(a) SSHA (horig) from 21 Jan 2016, the midpoint of the fitting period for the analysis beginning on 1 Jan 2016. (b) SSHA (h^orig=h) represented by 190 Rossby waves and (c) residual difference, corresponding to about 3.1% of the variance on this date.

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

The success of the fitting is largely due to the fact that the wave coefficients allow SSHA to propagate westward, as illustrated in the Hovmöller diagram in Fig. B2, which shows the original SSHA data (Fig. B2a), the Rossby wave fit (Fig. B2b), and the difference between the SSHA data and Rossby wave fit (Fig. B2c) both for the fitting period (days 1–40) and for the prediction (days 40–80).

Fig. B2.
Fig. B2.

(a) SSHAs from Copernicus gridded fields (sometimes referred to as AVISO) at 34.625°N latitude, (b) smoothed version of SSHA created by projecting a gridded Copernicus SSHA values onto a set of 190 wave modes consistent with large-scale Rossby waves, and (c) difference between original and smoothed SSHA data. In all three panels, the horizontal blue line at 40 days indicates the transition from from the fitting period to prediction.

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

To obtain 12 test datasets, we repeated this fitting procedure 12 times, starting at the first day of each month of 2016. On average, the propagating Rossby wave model represents 70%–90% of the SSHA variance over the 40-day fitting period, as illustrated in Fig. B3 for the 12 test cases. The fitting period is the first 40 days to the left of the vertical dashed line. Gray lines show individual cases, and the red line is the mean over the 12 cases. As a baseline measure, we compare the Rossby wave model with a null hypothesis prediction that the SSHA is constant, pegged at conditions on the 21st day (blue line). Over the 40-day fitting period, on average the Rossby wave model explains a higher percentage of variance (also known as the “skill”) than does persistence (blue line), except within ±5 days on either side of day 21.

Fig. B3.
Fig. B3.

Percentage of gridded Copernicus SSHA variance explained by the Rossby wave model as a function of day for 12 start dates, beginning from the first of each month in 2016. Gray lines indicate individual realizations, the red line shows the mean, and red shading indicates twice the standard error of the mean 2σ/12, where σ is the standard deviation of the 12 monthly realizations. Data from the first 40 days are used in the least squares fitting procedure (to the left of the vertical dashed gray line). After 40 days, SSHA is predicted based only on information from the first 40 days. The blue line indicates persistence from the midpoint of the fitting period (day 21), with shading indicating twice the standard error of the mean.

Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1

After the 40-day fitting period (right of the vertical dashed line in Fig. B3), the skill of the Rossby wave model varies considerably, as indicated by the spread of the gray lines: in some cases, the Rossby wave model continues to explain a large percentage of the gridded SSHA data, and in other cases, the model diverges significantly. Differences could arise for a number of reasons: the Rossby wave model could omit frequency–wavenumber combinations that are important at some times; the system could experience occasional external forcing (e.g., from wind) that excites new propagating waves, the waves could propagate at speeds that differ from the linear Rossby wave phase speed (Chelton et al. 2007, 2011), or the model could be incomplete for other reasons. The skill decreases slightly less steeply for the Rossby wave model (red) than for persistence (blue), indicating that the Rossby wave model carries useful information about the evolution of the SSHA field. The objective of this paper is focused on demonstrating the feasibility of including correlated error corrections within a model, and we leave for other studies the possibility of carrying out more detailed exploration of Rossby wave or QG representations of altimeter data. Within the main body of the text, we define our true sea surface height and true model coefficients based on the fitted fields:
h=h^orig
a=a^orig.

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  • Todd, R. E., D. L. Rudnick, M. R. Mazloff, R. E. Davis, and B. D. Cornuelle, 2011: Poleward flows in the southern California Current System: Glider observations and numerical simulation. J. Geophys. Res., 116, C02026, https://doi.org/10.1029/2010JC006536.

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  • Verdy, A., M. R. Mazloff, B. D. Cornuelle, and S. Y. Kim, 2014: Wind-driven sea level variability on the California coast: An adjoint sensitivity analysis. J. Phys. Oceanogr., 44, 297–318, https://doi.org/10.1175/JPO-D-13-018.1.

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  • Wakata, Y., and S. Kitaya, 2002: Annual variability of sea surface height and upper layer thickness in the Pacific Ocean. J. Oceanogr., 58, 439–450, https://doi.org/10.1023/A:1021205129971.

    • Search Google Scholar
    • Export Citation

  • Watanabe, W. B., P. S. Polito, and I. C. da Silveira, 2016: Can a minimalist model of wind forced baroclinic Rossby waves produce reasonable results? Ocean Dyn., 66, 539–548, https://doi.org/10.1007/s10236-016-0935-1.

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  • Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge University Press, 442 pp.

  • Zaba, K. D., D. L. Rudnick, B. D. Cornuelle, G. Gopalakrishnan, and M. R. Mazloff, 2018: Annual and interannual variability in the California Current System: Comparison of an ocean state estimate with a network of underwater gliders. J. Phys. Oceanogr., 48, 2965–2988, https://doi.org/10.1175/JPO-D-18-0037.1.

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  • Kachelein, L., B. D. Cornuelle, S. T. Gille, and M. R. Mazloff, 2022: Harmonic analysis of non-phase-locked tides with red noise using the red_tide package. J. Atmos. Oceanic Technol., 39, 1031–1051, https://doi.org/10.1175/JTECH-D-21-0034.1.

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  • Mazloff, M. R., B. Cornuelle, S. T. Gille, and J. Wang, 2020: The importance of remote forcing for regional modeling of internal waves. J. Geophys. Res. Oceans, 125, e2019JC015623, https://doi.org/10.1029/2019JC015623.

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  • Metref, S., E. Cosme, J. Le Sommer, N. Poel, J.-M. Brankart, J. Verron, and L. Gómez Navarro, 2019: Reduction of spatially structured errors in wide-swath altimetric satellite data using data assimilation. Remote Sens., 11, 1336, https://doi.org/10.3390/rs11111336.

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  • Metref, S., E. Cosme, F. Le Guillou, J. Le Sommer, J.-M. Brankart, and J. Verron, 2020: Wide-swath altimetric satellite data assimilation with correlated-error reduction. Front. Mar. Sci., 6, 822, https://doi.org/10.3389/fmars.2019.00822.

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  • Morrow, R., and Coauthors, 2019: Global observations of fine-scale ocean surface topography with the Surface Water and Ocean Topography (SWOT) mission. Front. Mar. Sci., 6, 232, https://doi.org/10.3389/fmars.2019.00232.

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  • Todd, R. E., D. L. Rudnick, M. R. Mazloff, R. E. Davis, and B. D. Cornuelle, 2011: Poleward flows in the southern California Current System: Glider observations and numerical simulation. J. Geophys. Res., 116, C02026, https://doi.org/10.1029/2010JC006536.

    • Search Google Scholar
    • Export Citation

  • Verdy, A., M. R. Mazloff, B. D. Cornuelle, and S. Y. Kim, 2014: Wind-driven sea level variability on the California coast: An adjoint sensitivity analysis. J. Phys. Oceanogr., 44, 297–318, https://doi.org/10.1175/JPO-D-13-018.1.

    • Search Google Scholar
    • Export Citation

  • Wakata, Y., and S. Kitaya, 2002: Annual variability of sea surface height and upper layer thickness in the Pacific Ocean. J. Oceanogr., 58, 439–450, https://doi.org/10.1023/A:1021205129971.

    • Search Google Scholar
    • Export Citation

  • Watanabe, W. B., P. S. Polito, and I. C. da Silveira, 2016: Can a minimalist model of wind forced baroclinic Rossby waves produce reasonable results? Ocean Dyn., 66, 539–548, https://doi.org/10.1007/s10236-016-0935-1.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge University Press, 442 pp.

  • Zaba, K. D., D. L. Rudnick, B. D. Cornuelle, G. Gopalakrishnan, and M. R. Mazloff, 2018: Annual and interannual variability in the California Current System: Comparison of an ocean state estimate with a network of underwater gliders. J. Phys. Oceanogr., 48, 2965–2988, https://doi.org/10.1175/JPO-D-18-0037.1.

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    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (a) SSH anomalies in the California Current region on 1 Jan 2016. Black lines indicate the 10° latitude × 9° longitude region of interest for this study, and gray lines bound the SWOT measurement region along the California Coast.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    SSHA h and uncertainties e (both in m) along SWOT satellite swath for 21 Jan 2016, showing (a) the true h used for this study, derived by projecting gridded altimetry onto a set of Rossby waves (along-swath values are not shown but match the gridded values); (b) h^1−stage from the one-stage approach; (c) h−h^1−stage; (d) h^2−stage from the two-stage approach; (e) h−h^2−stage; (f) correlated error etotal used in this simulation; (g) estimated correlated error e^1−stage from the one-stage approach; (h) difference, Δe=etotal−e^1−stage; (i) estimated error e^2−stage from the two-stage approach; (j) difference, Δe=etotal−e^2−stage; (k) total SSHA, h + etotal, used as input; (l) total SSHA h^+e^1−stage from one-stage approach; (m) difference between (k) and (l); (n) total SSHA h^+e^2−stage from two-stage approach; and (o) difference between (k) and (n). In this case, R=σd2I=0.01I, and the coefficients αi have standard deviation 0.0125, with a resulting noise-to-signal ratio (root-mean-squared error divided by root-mean-squared signal) of 0.34.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a) The true SSHA h on 21 Jan 2016 (see appendix B for method to determine h); (b) SSHA estimate of a 190-wave Rossby wave model in the one-stage approach; (c) difference between true SSHA and estimated SSHA of one-stage approach, with black lines showing the center points of the outermost input data; (d) the gridded Copernicus SSHA on 21 Jan 2016; (e) SSHA estimate of a 190-wave Rossby wave model in the two-stage approach; and (f) the difference between the true SSHA and estimated SSHA of a two-stage approach. Noise-to-signal ratio is the same as in Fig. 2.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) Percentage variance explained in the SSHA estimate for all experiments, plotted as a function of the error input to the problem (root-mean-squared error) normalized by root-mean-squared SSHA for each date. In total, there are 360 experiments, encompassing 30 noise values for each of 12 start dates. (b) Percentage variance explained in error estimate and (c) percentage variance explained of the total input signal h + e, for the one-stage and two-stage approaches as a function of the ratio of root-mean-squared error to root-mean-squared SSHA, RMSE/RMS_SSHA ratio. Blue “×” markers represent results from the two-stage approach, and orange “*” markers represent the one-stage approach. Since the input data are the same for the one-stage and two-stage approaches, each blue marker can be matched to an orange marker with the same RMSE/RMS_SSHA value. Since SSH data vary in time, and uncertainty has a random component, the RMSE/RMS_SSHA ratios differ for each of the 360 datasets.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    (a) Percentage variance explained in experiments in which the signal is simplified, so that only the Rossby wave component of the signal is fitted by the Rossby wave model (orange) and the correlated error is fitted only with the correlated error model (blue “×” marks). Cases shown here are the same as in Fig. 4. (b) Percentage variance explained in the inverse case, in which the Rossby wave component of the signal is fitted by the correlated error model (orange stars), and the correlated error component of the signal is fitted with the Rossby wave model (blue “×” marks). This provides a measure of the projection of each part of the signal onto the model for the other part of the signal. (c) Percentage variance explained when a signal consisting only of uncorrelated white noise is fitted with the Rossby wave model (red “+” signs) or the correlated error model (cyan dots).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Percentage of SSHA variance explained in one-stage and two-stage approaches, as a function of time, for in-swath only, out-of-swath only, and total domain, for correlated error with standard deviation 0.0125, as in Fig. 2. In this figure, model truth is determined from the gridded input fields, for example, in Fig. 3a. The gray line indicates the skill that would be achieved by assuming persistence of conditions from 21 day, here shown only for the full domain. All solid lines show statistics computed for the set of results based on the 12 analysis start dates, beginning from the first of each month in 2016. Error bars indicate 1 standard error of the mean. The gray dashed line indicates the 40-day mark when the forecast starts.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Percentage of SSHA variance explained when error derives from the SWOT simulator (Gaultier et al. 2016) rather than the simplified synthetic model. Results presented here are analogous to those in Fig. 6, but only the in-swath variance is shown. Two sets of covariance matrices are considered. Dark blue and red lines show results using a covariance matrix P that has been adjusted to reflect the covariance of the simulator error, as determined based on the preliminary one-stage and two-stage fits to the data. Light blue and pink lines show results that employ the same covariance matrix used for the results in Fig. 6, with the same covariance assumed for all coefficients αi from (10).

  • <sc>Fig</sc>. B1.
    View in gallery
    Fig. B1.

    (a) SSHA (horig) from 21 Jan 2016, the midpoint of the fitting period for the analysis beginning on 1 Jan 2016. (b) SSHA (h^orig=h) represented by 190 Rossby waves and (c) residual difference, corresponding to about 3.1% of the variance on this date.

  • <sc>Fig</sc>. B2.
    View in gallery
    Fig. B2.

    (a) SSHAs from Copernicus gridded fields (sometimes referred to as AVISO) at 34.625°N latitude, (b) smoothed version of SSHA created by projecting a gridded Copernicus SSHA values onto a set of 190 wave modes consistent with large-scale Rossby waves, and (c) difference between original and smoothed SSHA data. In all three panels, the horizontal blue line at 40 days indicates the transition from from the fitting period to prediction.

  • <sc>Fig</sc>. B3.
    View in gallery
    Fig. B3.

    Percentage of gridded Copernicus SSHA variance explained by the Rossby wave model as a function of day for 12 start dates, beginning from the first of each month in 2016. Gray lines indicate individual realizations, the red line shows the mean, and red shading indicates twice the standard error of the mean 2σ/12, where σ is the standard deviation of the 12 monthly realizations. Data from the first 40 days are used in the least squares fitting procedure (to the left of the vertical dashed gray line). After 40 days, SSHA is predicted based only on information from the first 40 days. The blue line indicates persistence from the midpoint of the fitting period (day 21), with shading indicating twice the standard error of the mean.

Fig. 1.

(a) SSH anomalies in the California Current region on 1 Jan 2016. Black lines indicate the 10° latitude × 9° longitude region of interest for this study, and gray lines bound the SWOT measurement region along the California Coast.
Fig. 2.

SSHA h and uncertainties e (both in m) along SWOT satellite swath for 21 Jan 2016, showing (a) the true h used for this study, derived by projecting gridded altimetry onto a set of Rossby waves (along-swath values are not shown but match the gridded values); (b) h^1−stage from the one-stage approach; (c) h−h^1−stage; (d) h^2−stage from the two-stage approach; (e) h−h^2−stage; (f) correlated error etotal used in this simulation; (g) estimated correlated error e^1−stage from the one-stage approach; (h) difference, Δe=etotal−e^1−stage; (i) estimated error e^2−stage from the two-stage approach; (j) difference, Δe=etotal−e^2−stage; (k) total SSHA, h + etotal, used as input; (l) total SSHA h^+e^1−stage from one-stage approach; (m) difference between (k) and (l); (n) total SSHA h^+e^2−stage from two-stage approach; and (o) difference between (k) and (n). In this case, R=σd2I=0.01I, and the coefficients αi have standard deviation 0.0125, with a resulting noise-to-signal ratio (root-mean-squared error divided by root-mean-squared signal) of 0.34.
Fig. 3.

(a) The true SSHA h on 21 Jan 2016 (see appendix B for method to determine h); (b) SSHA estimate of a 190-wave Rossby wave model in the one-stage approach; (c) difference between true SSHA and estimated SSHA of one-stage approach, with black lines showing the center points of the outermost input data; (d) the gridded Copernicus SSHA on 21 Jan 2016; (e) SSHA estimate of a 190-wave Rossby wave model in the two-stage approach; and (f) the difference between the true SSHA and estimated SSHA of a two-stage approach. Noise-to-signal ratio is the same as in Fig. 2.
Fig. 4.

(a) Percentage variance explained in the SSHA estimate for all experiments, plotted as a function of the error input to the problem (root-mean-squared error) normalized by root-mean-squared SSHA for each date. In total, there are 360 experiments, encompassing 30 noise values for each of 12 start dates. (b) Percentage variance explained in error estimate and (c) percentage variance explained of the total input signal h + e, for the one-stage and two-stage approaches as a function of the ratio of root-mean-squared error to root-mean-squared SSHA, RMSE/RMS_SSHA ratio. Blue “×” markers represent results from the two-stage approach, and orange “*” markers represent the one-stage approach. Since the input data are the same for the one-stage and two-stage approaches, each blue marker can be matched to an orange marker with the same RMSE/RMS_SSHA value. Since SSH data vary in time, and uncertainty has a random component, the RMSE/RMS_SSHA ratios differ for each of the 360 datasets.
Fig. 5.

(a) Percentage variance explained in experiments in which the signal is simplified, so that only the Rossby wave component of the signal is fitted by the Rossby wave model (orange) and the correlated error is fitted only with the correlated error model (blue “×” marks). Cases shown here are the same as in Fig. 4. (b) Percentage variance explained in the inverse case, in which the Rossby wave component of the signal is fitted by the correlated error model (orange stars), and the correlated error component of the signal is fitted with the Rossby wave model (blue “×” marks). This provides a measure of the projection of each part of the signal onto the model for the other part of the signal. (c) Percentage variance explained when a signal consisting only of uncorrelated white noise is fitted with the Rossby wave model (red “+” signs) or the correlated error model (cyan dots).
Fig. 6.

Percentage of SSHA variance explained in one-stage and two-stage approaches, as a function of time, for in-swath only, out-of-swath only, and total domain, for correlated error with standard deviation 0.0125, as in Fig. 2. In this figure, model truth is determined from the gridded input fields, for example, in Fig. 3a. The gray line indicates the skill that would be achieved by assuming persistence of conditions from 21 day, here shown only for the full domain. All solid lines show statistics computed for the set of results based on the 12 analysis start dates, beginning from the first of each month in 2016. Error bars indicate 1 standard error of the mean. The gray dashed line indicates the 40-day mark when the forecast starts.
Fig. 7.

Percentage of SSHA variance explained when error derives from the SWOT simulator (Gaultier et al. 2016) rather than the simplified synthetic model. Results presented here are analogous to those in Fig. 6, but only the in-swath variance is shown. Two sets of covariance matrices are considered. Dark blue and red lines show results using a covariance matrix P that has been adjusted to reflect the covariance of the simulator error, as determined based on the preliminary one-stage and two-stage fits to the data. Light blue and pink lines show results that employ the same covariance matrix used for the results in Fig. 6, with the same covariance assumed for all coefficients αi from (10).
Fig. B1.

(a) SSHA (horig) from 21 Jan 2016, the midpoint of the fitting period for the analysis beginning on 1 Jan 2016. (b) SSHA (h^orig=h) represented by 190 Rossby waves and (c) residual difference, corresponding to about 3.1% of the variance on this date.
Fig. B2.

(a) SSHAs from Copernicus gridded fields (sometimes referred to as AVISO) at 34.625°N latitude, (b) smoothed version of SSHA created by projecting a gridded Copernicus SSHA values onto a set of 190 wave modes consistent with large-scale Rossby waves, and (c) difference between original and smoothed SSHA data. In all three panels, the horizontal blue line at 40 days indicates the transition from from the fitting period to prediction.
Fig. B3.

Percentage of gridded Copernicus SSHA variance explained by the Rossby wave model as a function of day for 12 start dates, beginning from the first of each month in 2016. Gray lines indicate individual realizations, the red line shows the mean, and red shading indicates twice the standard error of the mean 2σ/12, where σ is the standard deviation of the 12 monthly realizations. Data from the first 40 days are used in the least squares fitting procedure (to the left of the vertical dashed gray line). After 40 days, SSHA is predicted based only on information from the first 40 days. The blue line indicates persistence from the midpoint of the fitting period (day 21), with shading indicating twice the standard error of the mean.
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