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.
(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.
The dispersion relation for Rossby waves defines the relationship between their frequency and wavenumber. For these waves on a β plane, it is
Although the inversion of the matrix (
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
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
b. One-stage approach
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
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
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)
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.
(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,
(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).
(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.
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
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
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
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 |
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 |
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 |
Best estimate of sea surface height measurements represented as a column vector | |
In two-stage method, sea surface height with estimated correlated error removed | |
Model basis functions | |
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 |
| |
The portion of | |
The covariance matrix representing prior uncertainty in all model parameters | |
r |
Residual |
| |
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
Using the set of 190 wavenumbers, we estimate model coefficients
(a) SSHA (horig) from 21 Jan 2016, the midpoint of the fitting period for the analysis beginning on 1 Jan 2016. (b) SSHA (
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).
(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.
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
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0062.1
REFERENCES
Chelton, D. B., and M. G. Schlax, 1996: Global observations of oceanic Rossby waves. Science, 272, 234–238, https://doi.org/10.1126/science.272.5259.234.
Chelton, D. B., M. G. Schlax, R. M. Samelson, and R. A. de Szoeke, 2007: Global observations of large oceanic eddies. Geophys. Res. Lett., 34, L15606, https://doi.org/10.1029/2007GL030812.
Chelton, D. B., M. G. Schlax, and R. M. Samelson, 2011: Global observations of nonlinear mesoscale eddies. Prog. Oceanogr., 91, 167–216, https://doi.org/10.1016/j.pocean.2011.01.002.
Copernicus Marine Service Information, 2023: Global ocean gridded L 4 sea surface heights and derived variables reprocessed 1993 ongoing. Marine Data Store (MDS), accessed 2 August 2023, https://doi.org/10.48670/moi-00148.
Dibarboure, G., and C. Ubelmann, 2014: Investigating the performance of four empirical cross-calibration methods for the proposed SWOT mission. Remote Sens., 6, 4831–4869, https://doi.org/10.3390/rs6064831.
Dibarboure, G., and Coauthors, 2022: Data-driven calibration algorithm and pre-launch performance simulations for the SWOT mission. Remote Sens., 14, 6070, https://doi.org/10.3390/rs14236070.
Esteban-Fernandez, D., 2017: SWOT project mission performance and error budget, revision A. Jet Propulsion Laboratory Tech. Rep. JPL D-79084, 117 pp., https://swot.jpl.nasa.gov/system/documents/files/2178_2178_SWOT_D-79084_v10Y_FINAL_REVA__06082017.pdf.
Farrar, J. T., T. Durland, S. R. Jayne, and J. F. Price, 2021: Long-distance radiation of Rossby waves from the equatorial current system. J. Phys. Oceanogr., 51, 1947–1966, https://doi.org/10.1175/JPO-D-20-0048.1.
Fu, L.-L., and C. Ubelmann, 2014: On the transition from profile altimeter to swath altimeter for observing global ocean surface topography. J. Atmos. Oceanic Technol., 31, 560–568, https://doi.org/10.1175/JTECH-D-13-00109.1.
Fu, L.-L., D. Alsdorf, R. Morrow, E. Rodriguez, and N. Mognard, 2012: SWOT: The Surface Water and Ocean Topography mission: Wide-swath altimetric measurement of water elevation on Earth. JPL Tech. Rep. 12-05, 228 pp., https://dataverse.jpl.nasa.gov/file.xhtml?fileId=45898&version=2.0.
Gaultier, L., C. Ubelmann, and L.-L. Fu, 2016: The challenge of using future SWOT data for oceanic field reconstruction. J. Atmos. Oceanic Technol., 33, 119–126, https://doi.org/10.1175/JTECH-D-15-0160.1.
Gómez-Valdivia, F., A. Parés-Sierra, and A. L. Flores-Morales, 2017: Semiannual variability of the California undercurrent along the Southern California Current System: A tropical generated phenomenon. J. Geophys. Res. Oceans, 122, 1574–1589, https://doi.org/10.1002/2016JC012350.
Ide, K., P. Courtier, M. Ghil, and A. C. Lorenc, 1997: Unified notation for data assimilation: Operational, sequential and variational. J. Meteor. Soc. Japan, 75, 181–189, https://doi.org/10.2151/jmsj1965.75.1B_181.
Ivanov, L. M., C. A. Collins, T. M. Margolina, and V. N. Eremeev, 2010: Nonlinear Rossby waves off California. Geophys. Res. Lett., 37, L13602, https://doi.org/10.1029/2010GL043708.
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.
Mazloff, M. R., S. T. Gille, and B. Cornuelle, 2014: Improving the geoid: Combining altimetry and mean dynamic topography in the California coastal ocean. Geophys. Res. Lett., 41, 8944–8952, https://doi.org/10.1002/2014GL062402.
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.
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.
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.
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.
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.
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.
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.
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.
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.