Estimating Nearshore Morphological Change through Ensemble Optimal Interpolation with Altimetric Data

Abstract

Nearshore bathymetry changes on scales of hours to months in ways that strongly impact coastal processes. However, even at the best-monitored sites, surveys are typically not conducted with sufficient frequency to capture important changes such as sandbar migration. As a result, nearshore models often rely on outdated bathymetric boundary conditions, which may introduce significant errors. In this study, we investigate ensemble optimal interpolation (EnOI) as a method to update survey-derived bathymetry with altimetric measurements that are spatially sparse but have high temporal availability. We present the results of two synthetic examples and two field data experiments that demonstrate the ability of the method to accurately track morphological change between surveys. The method reduces the RMSE relative to a static bathymetry (corresponding to the day before the first assimilation step) by 23% to 68%. When compared with an estimate linearly interpolated between survey-derived bathymetries, the EnOI analysis reduces the RMSE by 19% to 47% in three out of the four experiments.

1. Introduction

On open-coast intermediate sandy beaches, surf-zone bathymetry is frequently dominated by the presence of sandbars, sub-aqueous sandy features that evolve in response to changing wave conditions. Sandbars migrate onshore and offshore, exhibit a range of morphologies (e.g., shore-parallel or complex alongshore variable and crescentic patterns), and modify surf-zone hydrodynamic processes and coastal response to storms. The continuous evolution of surf-zone sandbars means that surf-zone bathymetry is rarely constant in time, with changes occurring on time scales of hours to months to years to decades depending on the forcing conditions. The ability to predict the evolution of surf-zone sandbars continues to be a large focus area of nearshore researchers, as we still have poor predictive skill at integrating sediment transport over daily to annual time scales [1], despite the relevance and ubiquity of these morphological features.

Bathymetry is commonly recognized as a critical boundary condition for almost all aspects of nearshore modeling. Plant et al. [2] demonstrated the sensitivity of wave and flow model results to bathymetric resolution and suggested that this sensitivity varies both spatially and temporally. Ruggiero et al. [3] provided evidence from both modeled results and field data that alongshore variability in seasonal morphological change arises from variability in the initial bathymetry. The importance of a detailed bathymetry for predicting runup, setup, and swash was emphasized in [4], while Wilson et al. [5] asserted that errors in the input bathymetry may affect model accuracy as much as process errors. In short, the role of bathymetry in modeling nearshore processes cannot be overstated.

Despite the sensitivity of nearshore variables of interest to the bottom boundary condition, modelers continue to rely on a static and/or outdated bathymetry due to a lack of available data. In their evaluation of a nearshore wave, tide, and current prediction system, Allard et al. [6] attributed a lack of model skill to an out-of-date bathymetry and a failure to account for sandbar migration. More recently, Bak et al. [7] found that modeled significant wave height results were more accurate when using an updated, remotely sensed bathymetry (which has some error) as opposed to a static bathymetry that is out of date. Clearly, methods to provide accurate and timely estimates of surf-zone bathymetry on sandy coastlines are needed both to understand the processes driving morphological evolution and to provide accurate boundary conditions for nearshore modeling.

Measurements of nearshore bathymetry can be divided into two approaches: in situ observations, which typically use acoustic instrumentation, and remotely sensed observations, which can be direct (e.g., airborne bathymetric lidar) or inferred (e.g., derived from observations of other variables). Vessel-based acoustic survey approaches are typically used to quantify spatial variability in surf-zone bathymetry at a particular instant in time, and repeat surveys can be analyzed to understand temporal evolution. However, due to the expense associated with survey collection, timely bathymetry is often unavailable for even the best-monitored sites, which may have daily bathymetry during focused experiments but typically have measurements at monthly or quarterly intervals. This lack of availability is exacerbated during storm events when conditions change most significantly and simultaneously restrict access for survey vehicles.

To measure bathymetric evolution during storms, field experiments have used in situ approaches, including frequent vessel-based surveys as well as sonic altimeters installed near the seabed, which measure seafloor evolution at high temporal resolution (seconds to minutes) but at sparse spatial locations (typically on the order of 10 locations per experiment) [8,9]. Combining surveys and altimetric observations together can then help provide high-resolution observations in space and time. For example, Moulton et al. [10] updated survey bathymetry with altimeter-based estimates of accretion and erosion to obtain bathymetry estimates that were more accurate than those obtained with temporal interpolation of survey data to learn about the morphological evolution of a rip channel. The recent emergence of autonomous systems with acoustic sensors that can follow survey lines [11,12] or flying/diving/swarming systems that can provide point observations [13,14] suggests that in situ observations of bathymetry may become more universally available, and robust methods to combine data with variable sampling schemes in space and time will be in higher demand.

Given the need for timely bathymetry and the obstacles to obtaining data of sufficiently high resolution (both spatially and temporally) in using in situ approaches, investigators have also explored a variety of methods to infer bathymetry from other more easily observed properties. Drawing on earlier work in estimating wave properties from optical signals [15,16,17], Holman et al. [18] introduced cBathy, a method for estimating bathymetry by directly inverting the dispersion relation. Van Dongeren et al. [19] also used a direct inversion method that leveraged video-derived roller dissipation and intertidal bathymetry in addition to wave celerity. Over the past decade, several model-based inversion frameworks have also appeared. For example, Wilson et al. [20] introduced a model-based inversion framework that applied the ensemble Kalman filter (EnKF) to update the bathymetry by using observations of wave celerity, currents, and shoreline position derived from optical, infrared, and radar signals. After updating the climatological background bathymetry, they were able to detect an observed rip current that was undetected by using the background bathymetry. Alternatives to classical ensemble techniques include [21], which presented synthetic test cases that demonstrate the advantages of the compressed-state Kalman filter (CSKF) for bathymetry inversion using wave height and celerity, while Salim and Wilson [22] employed a variational data assimilation scheme to estimate bathymetry from observed and modeled wave heights and currents. More recently, Collins et al. [23] applied a fully convolutional neural network to infer bathymetry directly from sea-surface imagery.

The challenges faced in the implementation of these methods are as diverse as the approaches themselves. Inversion methods require robust models and observations of sufficient quality and quantity. Brodie et al. [24] concluded that cBathy should not be used to quantify morphological evolution including sandbar migration during storm events and attributed large errors to discrepancies between observed wave speeds and those predicted by linear wave theory and optical image artifacts. Salim and Wilson [22] reported a less accurate estimate when assimilating both currents and wave heights than when assimilating wave heights alone, which indicates that data assimilation does not always lead to improved outcomes. They provided evidence that the lower accuracy was due to a greater degree of nonlinearity in the relationship between longshore current and depth and the linearization technique employed in the inversion method. Wilson et al. [20] cited errors in wave boundary conditions as a potential barrier to a data assimilation system based on purely remotely sensed observations. Due to these types of challenges, there are few frameworks for timely bathymetry estimation that are in operational use.

The current work focuses on the assimilation of altimetric observations to update survey-based bathymetry by using ensemble optimal interpolation (EnOI). Conceptually, this work is similar to [10] in that we use altimetric observations to track morphological change given temporally sparse yet spatially dense survey data. However, the ensemble-based approach used in this work has more in common with [20]. As an ensemble-based method, it is unnecessary to explicitly prescribe covariances as in traditional optimal interpolation. Additionally, the implementation we employ includes a method for constructing the ensemble by using historical data, which reduces the need for assumptions that are not based on observed data. Though the focus of this work is on combining in situ observations, it could also be extended to include remotely sensed input data in the future.

In this paper, we provide an overview of the site layout and data collection (Section 2.1) and briefly review ensemble-based data assimilation before discussing specific details of the implementation used in this work (Section 2.2). We next describe four numerical experiments using first synthetic data in one and two dimensions and then real data in one and two dimensions. These experiments are used to evaluate the ability of the EnOI approach to track morphological change (Section 2.4), and Section 3 presents the experimental results. Section 4 discusses the strengths and limitations of the approach. We summarize our findings in Section 5.

2. Materials and Methods

2.1. Observational Data

Survey and altimetric measurements are the two sources of observational data considered in this study. This section provides a general description of the data. We explain details that are specific to individual experiments in Section 2.4.

Bathymetric surfaces constructed from survey data establish the initial state (background) for assimilation, inform ensemble properties (as described in Section 2.3.1), and provide ground truth to evaluate the approach. We use altimetric data to perform daily updates of the background state. Both types of observations were collected at the Field Research Facility (FRF) in Duck, North Carolina, at the location displayed in Figure 1.

The FRF generally conducts monthly surveys, but special field experiments feature surveys with up to daily frequency. Survey transects have a nominal alongshore spacing of 45 m, and measurements have a resolution of approximately 5 m in the cross-shore direction. Bak et al. [12] reported a Total Vertical Uncertainty (TVU) for survey measurements of 0.04–0.08 m at the 95% confidence level. We interpolated survey data to a 5 m grid by using cubic spline interpolation for the field data experiments described in Section 2.4.

Echologger ECS400 acoustic altimeters are arranged at the FRF in three transects of five altimeters with an alongshore spacing of approximately 90 m and a cross-shore spacing of 50 m. However, field challenges have limited the availability of these observations, leading to the irregular spacing shown in Figure 1. The altimeters provide bottom position information at variable temporal frequency that depends on the signal-to-noise ratio [24]. A Kalman filter is applied to mitigate the effects of spurious altimetric readings that may arise under adverse conditions. For this study, measurements were averaged over daily intervals.

The FRF features a sandy, microtidal, intermediate beach that is often characterized by a single or double bar. Sand grain size varies from approximately 1 mm to 0.1 mm, and sand is finer and better sorted in the offshore direction [25]. Northeasters commonly occur in winter, and tropical hurricanes occur less frequently [25]. Bathymetric change is highly dependent on storm characteristics and varies seasonally according to the frequency, duration, and intensity of storm events.

It is well known that during storms, the seafloor may be obscured to an acoustic altimeter by suspended sediment. However, due to near-continuous backscatter sampling, acoustic altimeters are typically able to provide accurate measurements even during highly energetic events. Brodie et al. [24] found that the altimeters considered in the present study failed to provide accurate measurements only for significant wave heights greater than 4 m when the seafloor was obscured for up to 45 h. The range of significant wave heights during the period in which altimetric measurements were collected for the current study is 0.26 to 4.36 m. However, significant wave heights greater than or equal to 4 m were only observed for several hours during one of the experiments (FIELD2), and daily average observations were available for the duration of the experiments described in Section 2.4.

2.2. Data Assimilation

2.2.1. Data Assimilation Background

We approach data assimilation conceptually as an optimization problem that provides an updated estimate of a system state given prior approximation and observations. We represent the system with the state vector $\mathbf{x}$. Let ${\mathbf{x}}^b$ denote our best prior knowledge of the state, which we refer to as the “background”, and let $\mathbf{y}$ denote available observations. Specifically, we seek an analysis ${\mathbf{x}}^a$ that solves minimization problem (1) [26].

(1)$\begin{array}{c}{\mathbf{x}}^a=\underset{\mathbf{x}}{\mathrm{argmin}}J\hfill \\ J\left(\mathbf{x}\right)={(\mathbf{x}-{\mathbf{x}}^b)}^T{\left[{\mathsf{P}}^b\right]}^{-1}(\mathbf{x}-{\mathbf{x}}^b)+{(\mathbf{y}-\mathsf{H}\mathbf{x})}^T{\mathsf{R}}^{-1}(\mathbf{y}-\mathsf{H}\mathbf{x})\hfill \end{array}$

where ${\mathsf{P}}^b$ is the background error covariance matrix and $\mathsf{R}$ is the observation error covariance matrix. $\mathsf{H}$ is the observation operator which maps the state vector to the observation space. We consider the case where $\mathsf{H}$ is a linear operator, although it may be nonlinear in general. Intuitively, (1) penalizes departures from both the background and observations according to the uniqueness and uncertainty of the information contained in each. Where error covariances for the background or observations are large, departures are less penalized.

Given the assumptions of (1) unbiased observations, (2) unbiased background state, and (3) mutually independent observation and background errors, Bouttier and Courtier [26] show that the solution to (1) is given by (2).

(2)${\mathbf{x}}^a={\mathbf{x}}^b+{\mathsf{P}}^b{\mathsf{H}}^T{(\mathsf{H}{\mathsf{P}}^b{\mathsf{H}}^T+\mathsf{R})}^{-1}(\mathbf{y}-\mathsf{H}{\mathbf{x}}^b)$

From the perspective of Bayesian inference and with the further assumption of Gaussian error distributions for both the background and observations, (2) provides the maximum likelihood estimator of the true state [26]. In other words, it is the estimate of the true state for which observations $\mathbf{y}$ are most probable.

2.2.2. EnKF and EnOI

While (2) provides the solution to (1), this expression includes the background error covariance matrix, which may be difficult, if not impossible, to define explicitly. Furthermore, the size of this matrix is $n^2$, where n is the size of the state vector. This renders direct manipulation and storage of the matrix potentially impractical from a computational standpoint.

The ensemble Kalman filter (EnKF) addresses the problems of both explicitly defining and manipulating the background error covariance matrix by employing an ensemble representation of the background [27]. The ensemble implicitly stores a low-rank approximation of the background error covariance matrix, and operations involving this matrix, such as in (2), are executed through equivalent operations on the ensemble itself. In the remainder of this section, we will use the term “background covariance” to refer to the low-rank ensemble representation. With the ensemble representation, we may rewrite (2) as [28]

(3)${\mathbf{x}}_i^a={\mathbf{x}}_i^b+{\mathsf{P}}_e^b{\mathsf{H}}^T{(\mathsf{H}{\mathsf{P}}_e^b{\mathsf{H}}^T+\mathsf{R})}^{-1}({\mathbf{y}}_i-\mathsf{H}{\mathbf{x}}_i^b)$

where ${\mathbf{x}}_i$, $i=1,\dots ,N$ is the ith member of an ensemble representing the distribution of possible state configurations. As Burgers et al. [29] explain, observations are also treated in an ensemble manner, so that ${\mathbf{y}}_i=\mathbf{y}+{\mathsf{ϵ}}_i$, where ${\mathsf{ϵ}}_i$ is drawn from a zero-mean Gaussian distribution. The subscript e emphasizes that the background covariance is an ensemble approximation.

The analysis equations for EnOI are identical to those of the EnKF. The difference between the two approaches is that the statistical properties of the ensemble are determined by prior knowledge (e.g., historical data) in EnOI, whereas statistical properties are updated at each assimilation step in the EnKF [30]. While EnOI is the formal approach adopted in this study, we reference the EnKF because most of the techniques we employ in EnOI were developed within the EnKF framework.

To evolve bathymetry from one daily assimilation step to next, we use the identity as the state transition model and perturb the ensemble mean with realizations sampled from a covariance model constructed by using approximately 40 years of survey data. There are three primary reasons for choosing the identity as the state transition model. First, given the known challenges of capturing morphological evolution, we wanted to explore a parsimonious approach that might serve as a benchmark for future advances in coastal data assimilation. Second, coastal morphological modeling is highly parameter-dependent. Generally, morphological models must be specifically calibrated for particular events of interest. This detracts from their usefulness in the framework presented, where the emphasis is on a general high-temporal-resolution update that might be leveraged for near-real-time estimates. Finally, we observe that using the identity as the forward model provides a reasonably accurate approximation of the bathymetry over the daily time step of the update procedure. The bathymetric evolution procedure is similar to the approach by Li et al. [31], who implemented a random walk model in applying the ${\mathcal{H}}^2$ Kalman filter (HiKF). However, Li et al. [31] sampled from a Gaussian process, whereas we sample from a covariance model generated from historical survey data.

To reduce spurious correlations at larger distances in the background covariance [27] and increase the effective rank of the background covariance [32], we employ covariance localization as described by Houtekamer and Mitchell [33]. Specifically, the localization takes the form of the Schur product $\rho \circ {\mathsf{P}}_e^b$, where $\rho$ is a fifth-order polynomial correlation function [34]. While we do not provide a detailed comparison, we have observed in practice that applying a localized algorithm with a 200-member ensemble for the applications below produces a more accurate analysis than EnOI with a 1000-member ensemble and no localization.

2.3. Assimilation Method

To perform the assimilation, we exploit knowledge of local conditions and use the procedure described in Section 2.3.1 to construct the background ensemble. At assimilation times, each member ${\mathbf{x}}_i^b$ of the background ensemble is updated according to Equation (3), and we take the mean of the resulting ensemble to be the analysis for that assimilation step. At each subsequent step, we construct the new background ensemble by using the previous analysis as the background. In this work, we use a step size of 1 day. The Parallel Data Assimilation Framework (PDAF) [35] provides the key routines for the assimilation.

2.3.1. Ensemble Construction

Of vital importance to the EnOI approach is the ability to accurately approximate the true background covariances [32]. Equivalently, the background ensemble should reflect the error statistics associated with approximating the true state with the ensemble mean [28]. In the original formulation of the EnKF, Evensen proposed constructing the ensemble by perturbing a “best guess” background with zero-mean pseudorandom fields that reflected the true scales of the system [36].

While the EnOI analysis demonstrates skill with the pseudorandom perturbation approach, we have observed artifacts present in the analysis that appear to be inherited from the ensemble. With the goal of constructing an ensemble that more accurately represents plausible physical states, we explore an alternative method based on finite Karhunen–Loève expansions of a covariance matrix derived from the anomalies between each background state and a historical data matrix.

To generate the covariance matrix, we construct a data matrix using the FRF survey digital elevation model (DEM) product. The DEM is constructed by kriging survey data with a linear variogram model. Each column of the matrix is a survey DEM interpolated to the analysis domain. The survey dates associated with the DEMs range from 1989 to within a year prior to the first assimilation step. More recent surveys are omitted to prevent the inclusion of information from states that we attempt to predict. We perform the singular value decomposition (SVD) of the historical anomaly matrix to extract the predominant modes.

We then proceed to generate the sample by using a finite Karhunen–Loève expansion, which is a well-known tool for modeling stochastic processes [37]. Each expansion $\mathbf{u}$ is given by (4)

(4)$\mathbf{u}=\overline{\mathbf{u}}+\sum _{j=1}^J\sqrt{{\lambda }_j}{\varphi }_j{\alpha }_j$

where ${\mathbf{\varphi }}_j$ and ${\lambda }_j$ are the eigenvectors and eigenvalues of the covariance matrix, $\overline{\mathbf{u}}$ is the mean of the distribution for $\mathbf{u}$, and ${\alpha }_j$ is a normally distributed random variable. J is the number of modes selected for the expansion. We make the assumption under this formulation that the process itself is Gaussian. By applying (4) to the historical survey data matrix, we are able to generate an arbitrary number of realizations that reflect the underlying covariances of the anomaly matrix. Figure 2 displays a scree plot and the first three modes of the SVD of the historical anomaly matrix for survey-derived bathymetry from 17 October 2018. Based on the leftmost panel of Figure 2, we chose $J=10$ for the number of modes for the expansions.

2.4. Numerical Experiments

In this section, we present an overview of four experiments that explore the effectiveness of using the EnOI approach with point observations to estimate bathymetry. Two experiments use synthetic data for observations, reference bathymetry, and background, and two use observations from FRF altimeters and background and reference bathymetry derived from survey data. Each experiment updates a 200-member ensemble, derived from FRF historical data as described in Section 2.3.1, on a daily time step. The only data provided to the EnOI routine are the background (prior) state, the associated ensemble, and simulated or actual altimetric observations available at each assimilation step. For each experiment, we apply the EnOI algorithm with covariance localization as explained in Section 2.2.2. We specify the observation error covariance matrix as a scalar matrix whose entries correspond to the observation error variance, which we assume to be 0.01 m² for all observations.

Our objective is to be able to provide accurate estimates that reflect true bathymetric variability during times when survey information is unavailable. This presents a challenge in terms of evaluating performance because validation data are not available at the resolution of interest (daily). For this reason, we first implement the method by using synthetic bathymetry realizations for both reference and observation data. This allows for a comparison to a reference bathymetry for each assimilation step. Additionally, we compare the EnOI analysis to a survey-derived bathymetry held constant over the assimilation period and an estimate linearly interpolated (in time) between survey-derived bathymetry realizations bounding the assimilation period. These simpler estimates are what nearshore modelers often use in practice and serve as a reference used to assess relative improvement in our ability to estimate bathymetry.

We examined two synthetic cases: one-dimensional offshore bar migration (SYNTH1) and two-dimensional transitions between crescentic and alongshore uniform bar patterns (SYNTH2). For each case, we used bathymetry realizations generated with the parametric model for barred equilibrium beach profiles (parametric beach tool [PBT]) [38,39]. Given site-specific parameters that are based on historical characteristics of the site, the PBT takes a user-defined shoreline and bar locations to generate bathymetry estimates. We generated 14 realizations that together simulate changes in bar position over a 14-day period. We used the parameters associated with the FRF [38,39]. Each of the synthetic experiments uses a 200-member ensemble constructed as described in Section 2.3.1.

2.5. One-Dimensional Synthetic Experiment

For SYNTH1, the domain represents a 350 m transect at the FRF between $x=75$ m and $x=425$ m in the local shore-normal coordinate system with 5 m resolution. The shoreline position is fixed at $x=125$ m, and the bar position increases monotonically from $x=175$ m to $x=325$ m over the 14 realizations. We choose the locations of x = (150, 200, 250, 300, 350) m for simulated observations to correspond to the current layout of the FRF altimeter array. By taking Realization 0 to be the background on day 0, we perform the assimilation with observations taken at the specified x-coordinates from Realization i for each day i, $i=1,\dots ,13$, of the experiment. We perturb the simulated observations with random noise sampled from a Gaussian distribution with zero mean and a standard deviation of 0.1 m, which is the assumed root-mean-square observational error.

2.6. Two-Dimensional Synthetic Experiment

For SYNTH2, the domain is between $x=125$ and $x=475$ m in the cross-shore direction and between $y=695$ and $y=1055$ m in the alongshore direction with 5 m resolution in both directions. The shoreline location is again fixed at $x=125$ m, and we take the altimetric observations from the same cross-shore locations as in SYNTH1 along three transects at $y=(775,875,975)$ m. Equation (5) defines the sandbar position for the two-dimensional experiment.

(5)$x\left(t\right)=D\left(t\right)+\left|2Acos\left(\frac{\pi y}{L}\right)\right|-\frac{2A}{\pi }$

where x and y are the cross-shore and alongshore coordinates of the bar, A is one half of the cross-shore distance between consecutive horns and bays, D is the alongshore average offshore bar position, and L is the distance between consecutive horns. Over the first eight days, we increase D from 225 to 275 m while simultaneously decreasing A from 25 to 0 m. Then, over the last six days, we decrease D from 275 to 205 m and increase A from 0 to 25 m (its original value). We maintain L at 269 m for the first eight days and then adjust it to 610 m for the last six days. The overall effect is a crescentic formation that migrates offshore while evolving to an alongshore-uniform state followed by the onshore migration and formation of a crescentic pattern with a longer wavelength. Figure 3 displays three stages of this evolution.

2.6.1. One-Dimensional Field Experiment

The first field-based experiment (FIELD1) uses data from the BathyDuck experiment (October 2015), where seven surveys were conducted at the FRF in 30 days. This offers a rare opportunity to observe bathymetric change on a submonthly scale. During BathyDuck, the FRF collected altimetric data at $x=(150,200,300)$ m along a single transect at $y=940$ m in the local coordinate system. Data are available from the two altimeters at $x=150$ and $x=200$ m for the entire month of October and from the altimeter at $x=300$ for the final two weeks of the month.

To perform the experiment, we discretize the domain as a single transect at $y=940$ m and $75\le x\le 425$ m with a 5 m resolution such that altimeter locations coincide with the resulting grid. The 15 September 2015 DEM (produced from the latest survey prior to BathyDuck) informs the background state for the experiment. For each daily assimilation step, altimetric measurements are averaged over the day to mitigate the effects of noise.

2.6.2. Two-Dimensional Field Experiment

To explore a 2D domain, we use the period from 1 October to 30 November 2018, for which there is high data availability for the FRF altimeter array. Five altimeters are located along $y=769$ at $x=(150,200,250,300,350)$ m; two are located along $y=861$ m at $x=(200,300)$ m; and an additional two are located along $y=940$ m at $x=(200,250)$ m. In this window, all altimeters report with lapses of no more than several hours, which allows for a daily-averaged value. The period also has typical monthly CRAB/LARC survey data occurring on 17 October 2018 and 20 November 2018.

We construct the domain as a 5 m resolution rectangular grid in the FRF local coordinate system and interpolate the survey data from 17 October 2018 to the grid for use as background. We perform assimilation at a daily time step by using daily-averaged altimetric observations at each step. Due to the absence of ground-truth data between the two endpoint surveys, it is not possible to assess the performance of the analysis until the final assimilation step. Since only the background and altimetric observations are available to the EnOI filter, however, the analysis from the final assimilation step should provide an indication of the overall performance of the method, given material change between the initial and final surveys.

2.7. Evaluation

To evaluate the accuracy of the EnOI analysis, we report the root-mean-square error (RMSE) and bias. These metrics are well known and easily interpretable and give a reasonable indication of the overall error in the estimate. To better convey the spatial distribution of errors, we report RMSE and bias by cross-shore distance in bins with a width of 50 m and an equal number of analysis points in each bin. If an altimeter location falls within a bin, the bin is centered on that location. This allows the effects of an individual observation on the analysis to be more easily deduced from the error metrics.

2.8. Uncertainty Quantification

For the synthetic experiments, we consider the bathymetry realizations (reference bathymetry) to perfectly represent a true state. Additionally, we may generate an arbitrary number of representations of the true state, which allows us to evaluate performance at any desired temporal resolution. For experiments using field data, however, there is error associated with both observations and reference bathymetry, and our ability to validate the method is constrained by spatial and temporal data availability, measurement error, and representation error. Accordingly, there is a higher level of uncertainty in the reference bathymetry for the field cases, and the error metrics in FIELD1 and FIELD2 reflect disagreement between EnOI and the survey DEMs at corresponding times.

We quantify the uncertainty in the reference bathymetry as a composite of instrument error ${\sigma }_{instr}$, spatial representation error ${\sigma }_{repr}$, and temporal representation error ${\sigma }_{tavg}$. We assign constant values of ${\sigma }_{instr}=0.1$ m, based on altimeter specifications and knowledge of their performance over a period of seven years. For the spatial representation error [40], we use a value of ${\sigma }_{repr}=0.17$ m. This was derived by using the RMSE for kriging estimates during the BathyDuck experiment. Specifically, ${\sigma }_{repr}$ was determined by using BathyDuck surveys with a survey transect spacing approximately half the typical spacing (23 m versus 46 m). The kriging weights were obtained by using transects with the customary 46 m spacing, and the corresponding estimate was validated against the intervening transects. We take ${\sigma }_{tavg}$ to be the combined daily standard deviation of available altimetric observations about the daily mean on the day in which the survey data were collected. Equation (6) gives an estimate for the standard uncertainty u given the individual components discussed above.

(6)$u=\sqrt{{\sigma }_{instr}^2+{\sigma }_{repr}^2+{\sigma }_{tavg}^2}$

We use the standard deviation of the analysis ensemble to estimate analysis uncertainty.

3. Results

3.1. One-Dimensional Synthetic Experiment

Figure 4a displays the SYNTH1 PBT realizations as they evolve in time (reference bathymetry), while the EnOI analysis for each assimilation step is presented in Figure 4b, and temporal linear evolution is shown in Figure 4c. The temporal linear evolution reflects a common practice in evolving bathymetry between two states (e.g., initial and final with respect to a storm). For the SYNTH1 simulation, the EnOI analysis RMSE and bias were 0.19 m and −0.03 m, respectively. By comparison, the temporal linear interpolation (“interpolated”) RMSE and bias were 0.37 m and −0.07 m. For a temporally constant bathymetry (“initial state”), the RMSE and bias were 0.61 m and −0.05 m, respectively. The EnOI method outperforms the initial state and interpolated approximations in all cross-shore locations for RMSE (Figure 4d) and bias (Figure 4e) with the exception of near the shoreline.

3.2. Two-Dimensional Synthetic Experiment

For SYNTH2, the overall EnOI analysis RMSE and bias were 0.13 m and 0.02 m, respectively. The overall RMSE and bias for the initial state were 0.23 m and 0.03 m, and for the temporally interpolated estimate, they were 0.21 m and 0.02 m, respectively. Figure 5 displays the EnOI analysis and PBT reference bathymetry for three assimilation steps: Step 2/13 (Figure 5a–d), Step 7/13 (Figure 5e–h), and Step 12/13 (Figure 5i–l). Figure 5a,e,i show the reference bathymetry, and Figure 5b,f,j show the EnOI analysis.

Overall, the EnOI method qualitatively tracks morphological change well, with the exception of the shore-parallel contours at times near the midpoint of SYNTH2, where it fails to capture alongshore uniformity (Figure 5f). The highest errors for both EnOI and interpolated estimate occur at this point in the experiment, and the EnOI error metrics are uniformly lower than those for the interpolated estimate across the domain (Figure 5g,h). As in SYNTH1, the EnOI reduction in error relative to the interpolated estimate is the greatest in the vicinity of the sandbar while negligible in deeper water ($x>300$ m), where little to no change occurs in the reference bathymetry (RMSE—Figure 5m; bias—Figure 5n).

3.3. One-Dimensional Field Experiment

For FIELD1, the RMSE and bias were 0.42 m and −0.20 m, respectively. For the temporally interpolated estimate, the overall RMSE and bias were 0.52 m and 0.10 m. Figure 6a–f show the EnOI analysis and reference bathymetry on six different days for which survey data were available. The EnOI constrained the position of the sandbar crest and tracked the depth of the bar trough (Figure 6g,h), which contributed to the greatest improvement in skill when compared with the interpolated and initial state comparisons. The EnOI analysis underestimated the amplitude of the sandbar ($x\approx 250$ m in Figure 6g,h). With only three observations in cross-shore locations, the EnOI RMSE was approximately 0.1 m lower than the interpolated estimate and 0.2 m lower compared with the initial state (Figure 6g). The initial state and interpolated estimate were high-biased inshore at approximately $x=225$ m and low-biased between $x=225$ and $x=300$ m. The EnOI was low-biased at all cross-shore locations, but in most cases, the absolute bias was smaller with respect to both the initial state and the interpolated estimate (Figure 6h).

3.4. Two-Dimensional Field Experiment

For FIELD2, the overall RMSE and bias for the EnOI analysis were 0.30 m and 0.12 m, respectively. Figure 7 displays the EnOI analysis for the final assimilation step (b) and the corresponding reference bathymetry (a). The figure also presents a comparison of the EnOI analysis and the initial state along a single profile at $y=815$ m (c) and difference plots with respect to the reference bathymetry for the initial state (d), interpolated estimate (e), and EnOI analysis (f). In the vicinity of the bar crest ($x\approx 175$ m), the EnOI RMSE is lower than that of the initial state and interpolated estimate (g), and the EnOI bias is negligible, while it is nonzero for the initial state and interpolated estimate (h). Figure 7i–k show altimetric data (full-resolution and time-averaged data) and the reference bathymetry at three altimeter locations. Each of the three cases shown in subplots i, j, and k indicates offsets between the altimetric data and the survey-derived reference bathymetry.

4. Discussion

The four experiments described in Section 2.4, whose results are presented in Section 3 suggest that assimilating point bathymetry measurements with the EnOI method is useful for tracking morphological change between surveys. In this section, we will discuss the effectiveness of the method and how some of the implementation assumptions might affect its performance in a more general setting.

Of particular interest to the nearshore processes community is the ability to construct surfaces that accurately reflect bathymetric evolution on relevant time scales. In the absence of reliable information pertaining to these changes, wave and current simulations often rely on static bathymetric boundary conditions. For time scales of days to months (sometimes years), the assumption of a static boundary may result in inaccuracies in modeled quantities related to waves and currents. In a forecast modality, a static bathymetric surface from a previous survey may be the only available option. Even in reanalysis, in the absence of other sources of information, it may be difficult to improve on an interpolated surface between surveys. For periods of relatively little bathymetric change, these approaches may be sufficient. During highly active periods, however, the actual bathymetry may bear little resemblance to either the initial state or an interpolated estimate. We discuss the EnOI results in comparison with these commonly used, simpler estimates.

4.1. Relative Performance

From a quantitative standpoint, the EnOI analysis reduces the overall RMSE with respect to the initial state in each of the four cases examined and in all but the FIELD2 experiment with respect to the temporally interpolated estimate. Overall bias is not similarly reduced, but, as Figure 4, Figure 5, Figure 6 and Figure 7 indicate, small overall biases often result from large positive errors in one region of the domain offsetting large negative errors in another. For example, in Figure 4e, high biases greater than $0.5$ m in the 150 and 200 m bins and low biases of similar magnitude in the 250 and 300 m bins contribute to a bias near zero over the entire domain. The relative reductions in the RMSE for the EnOI analysis with respect to the initial state are 0.68 m (SYNTH1), 0.42 (SYNTH2), 0.30 (FIELD1), and 0.23 (FIELD2). With respect to the interpolated estimate, the relative reductions are 0.47 (SYNTH1), 0.35 (SYNTH2), and 0.19 (FIELD1). For FIELD2, there is a relative increase in the overall RMSE of 0.12 for the ENOI analysis with respect to the temporally interpolated estimate. We note, however, that the difference in RMSE in the latter case (0.03 m) is within the standard uncertainty of the reference bathymetry (0.20 m).

4.2. Tracking Morphological Change

In addition to reducing error magnitudes, the EnOI approach demonstrates the ability to resolve realistic bathymetric change on the time scale of days. The SYNTH1 results (Figure 4) best illustrate this. Figure 4 conveys the large departure between the initial and final states and the inability of temporally interpolated estimates to accurately approximate intermediate states. Each daily analysis closely approximates the corresponding ground-truth realization with only minor discrepancies at finer-scale resolutions. The cross-shore position of the bar peak, which has an overall displacement of 100.0 m, is accurate to within 5.0 m. Data of this type are not available except in the most dedicated field experiments (e.g., Duck94 [41]), which are still used 30 years later today for numerical experiments. We did not examine the Duck94 experiment in the current study because our primary focus was the use of altimetric measurements, which were not available for that time.

The SYNTH2 example features the transition from a crescentic to an alongshore uniform configuration followed by a return to a longer-wavelength crescentic pattern, while varying the median cross-shore bar crest location. The EnOI analysis reproduces some but not all of the features that arise during the transition. Along cross-shore transects where observations are present, the analysis accurately reflects the reference bathymetry. Also, the analysis captures the direction and magnitude of the changes over the transition for much of the domain. While it does not reproduce the alongshore uniformity of the intermediate states, it accurately represents the new crescentic configuration that appears toward the end of the simulation, even where previous updates have altered the original crescentic pattern. Since the initial and final states share more similarities than either does with the alongshore uniform intermediate states, the temporally interpolated estimate does not offer a material improvement over the static initial state. The bottom panel of Figure 5 (subplots m and n) illustrates the similarity in the spatial distribution of errors between the initial state and temporally interpolated estimates.

The October 2015 BathyDuck experiment, which underlies the FIELD1 example, provides a field data example of departure from the initial state followed by evolution toward the original state. In this case, a terrace feature ($y=940$ m) transitions to a well-defined bar–trough system over approximately 15 days. After maintaining a consistent bar–trough pattern for 20 to 25 days, the trough begins to fill, and the beach profile begins to more closely resemble the initial state. Using the initial state as a predictor for the end-state results in an overall high bias of ≈0.75 m for trough depth and an overall low bias of ≈0.7 m for bar height over the course of the simulation. The temporally interpolated estimate offers some improvement over the entire domain but displays error magnitudes comparable to the initial state in the vicinity of the trough ($150\le x\le 250$ m in each profile shown in Figure 6). The EnOI analysis only modestly underestimates the trough and qualitatively captures the overall offshore migration trend followed by the upward shift toward the initial state. The EnOI underestimates the bar height for the FIELD1 simulation due to an absence of observations near the bar peak.

Because we only have surveys at the endpoints of the FIELD2 example, it is not possible to characterize EnOI performance in tracking bathymetric change over the entire month-long period. However, Figure 7 indicates substantial offshore movement of material, with a terrace evolving toward a bar–trough system over the course of the assimilation. In the absence of observations where the nearshore trough forms, the error profile for the EnOI analysis resembles the initial state in this region. All three of the estimates (EnOI, initial state, and interpolated) underestimate the depth of the trough by up to 1 m. Where observations are available (offshore at $x=175$ m), however, the EnOI analysis constrains the RMSE to less than $0.20$ m, which is the standard uncertainty of the reference bathymetry for FIELD2.

4.3. Scales of Resolution

An important consideration for implementing an EnOI-based approach is the appropriate spatial distance between observations. This will depend on the characteristic length scales of the features to be resolved, which in turn will depend on the application. In general, characteristic bathymetric length scales are not only site-specific but also time-dependent. For example, the profile examined in FIELD1 is approximately sinusoidal with a wavelength of 200 m on 15 September, but after 45 days, the wavelength is approximately 100 m. Similar changes may be expected in the alongshore direction. The BathyDuck period, in particular, exhibits a highly alongshore variable trough formation with alongshore feature lengths of 100 m or less.

At the FRF, survey transects are typically spaced at approximately 50 m in the alongshore direction, and this spacing allows alongshore features greater than 100 m to be resolved [2]. This spacing should be sufficient to resolve sandbars whose alongshore length scales are approximately 100 m to kilometers [2]. However, there may be other morphological features of interest that the 50 m transect spacing could not be expected to resolve. For example, MacMahan [42] reports rip channel and shore-connected shoal widths as small as 30 m over a three-year study of rip channel stability and persistence at the FRF. MacMahan [42] also observes that rip current spacing “is not predisposed to a particular length scale”. Coarser resolution sampling also results in aliasing, such that higher resolution features (e.g., beach cusps) contaminate the resolution of larger-scale features [2]. For the practical purposes of this work, the sampling issue is confined to the alongshore direction; survey point data collected at the FRF are generally dense (<5 m) in the cross-shore direction and adequate for resolving cross-shore morphological features.

FRF altimeter transects are located at $y=769,861$, and 940 m in the local coordinate system, which is an alongshore spacing close to twice the survey transect spacing. At this spacing, it would be unreasonable to expect that the altimeter array alone could resolve features smaller than 160 m. This would include the smaller end of the range of bar formations and virtually all rip channel and cusp formations. In general, this inherent limitation with respect to spatial resolution should also extend to the use of altimetric measurements to update a prior bathymetry.

We observe examples of the limitations imposed by the observation spacing in the FIELD1 and FIELD2 examples. The cross-shore length scales in FIELD1 are both smaller and less uniform than those represented in the SYNTH1 problem. As mentioned earlier in this section, the approximate cross-shore wavelength associated with the bar–trough system changes from 200 m to 100 m over the 45-day period. Given the longer (approximately 200 m) wavelength observed in the beginning of the FIELD2 time window, EnOI updates based on altimetric observations at $x=150$ and $x=200$ m agree closely with the reference bathymetry in the vicinity of the observations. When the wavelength decreases to 50 m (on 30 October 2015, for example), however, the EnOI analysis fails to resolve the trough between the observations at $x=150$ and $x=200$ m.

4.4. Error and Uncertainty

It is an assumption of the EnOI approach used in this paper that observations are unbiased. Specifically, we assume that an observation represents a theoretical mean of possible observations of the unknown true state and specify a scale parameter for the distribution. In practice, observations are subject to bias from a number of sources related to how data are collected and processed. If it is not possible to correct an observational bias, it will propagate to the analysis. We observe the propagation of bias in the FIELD2 example. Figure 7 displays the difference between the EnOI analysis and reference bathymetry and also the altimetric data collected during the test for three locations. The altimetric plots show both the instantaneous data and daily average as well as the reference bathymetry at the nearest location. There is an apparent bias in each of the altimeters shown and a corresponding offset between the analysis and reference bathymetry. For example, there is a slight deep bias for the altimeter at $x,y=(150,769)$ m, and the analysis is deeper than the reference bathymetry in the vicinity of the observation. Similarly, the analysis is shallower than the reference bathymetry in the vicinity of observations at $x,y=(250,940)$ and $(350,769)$ m, where the altimeters display a shallow bias.

5. Conclusions

This paper explores ensemble optimal interpolation (EnOI) as a method to update nearshore bathymetry by using altimetric observations. While many implementations of ensemble-based data assimilation methods construct the background ensemble by sampling from a specified covariance function, we generate realizations via Karhunen–Loève expansion of historical anomalies. This approach eliminates the need for fitting covariance functions as is typically required for traditional optimal interpolation. We also provide justification for a site-specific covariance localization length scale by examining the spatial autocorrelation of historical anomalies.

To evaluate our approach, we consider four experiments, two synthetic ones and two with field data, which demonstrate the ability of the method to track morphological change on submonthly time scales. The root-mean-square error (RMSE) and bias for the EnOI analysis are reduced relative to a static bathymetry estimate in each of the four cases and relative to a temporally interpolated estimate in three out of four cases. In the single case where the temporally interpolated estimate is more accurate, the difference in RMSE is within the standard uncertatinty of the reference bathymetry. These results indicate that the proposed framework offers promise as a solution for updating nearshore bathymetry at sites with spatially dense but temporally sparse survey data.

Author Contributions

Conceptualization, T.J.H.; software, M.P.G.; investigation, M.P.G. and K.R.D.; resources, A.S.B. and P.J.D.; data curation, P.J.D.; writing—original draft preparation, M.P.G.; writing—review and editing, M.W.F. and K.L.B.; visualization, M.P.G. and K.R.D.; supervision, A.S.B.; project administration, A.S.B.; funding acquisition, A.S.B. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Dataset available on request from the authors.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. A detail of the Field Research Facility site showing the area of interest for this study. The brown dots (which appear as lines) represent the FRF monthly survey transects, the green and blue dots are the location of altimeters, and the red box is the study site for this paper.

Figure 2. Scree plot (left) and the first three modes of the SVD of the historical anomaly matrix for survey-derived bathymetry from 17 October 2018 (right three panels).

Figure 3. The bar positions used as inputs to the PBT overlain on the PBT-derived bathymetry for (a) Realization 0 (the initial step), (b) Realization 7, and (c) Realization 13 (the last step). The shoreline is set to x = 125 m, and the bar is determined by (5).

Figure 4. Fourteen PBT realizations representing offshore bar migration (a). EnOI analysis for each assimilation step (b). Linear interpolations in time between initial and final PBT realizations (c). Root-mean-square error (d) and bias (e) by offshore distance for EnOI analysis, initial state, and temporally interpolated estimate. “All” refers to the entire domain, [Forumla omitted. See PDF.] m.

Figure 5. Comparison of EnOI analysis and reference bathymetry for steps 2/13 (a–d), 7/13 (e–h), and 12/13 (i–l) of SYNTH2. Root-mean-square error (m) and bias (n) by cross-shore distance for EnOI analysis, initial state, and interpolated estimate. “All” refers to the entire domain, [Forumla omitted. See PDF.] m. Contour lines are −5.0 to −1.0 m (solid) and −4.5 to −0.5 (dashed) in 1.0 m increments. Black dots indicate observation locations.

Figure 6. Comparison of EnOI analysis and reference bathymetry for dates on which surveys were conducted for FIELD1 (a–f). Root-mean-square error (g) and bias (h) by cross-shore distance for EnOI analysis, initial state, and interpolated estimate. “All” refers to the entire domain, [Forumla omitted. See PDF.] m. The initial state for assimilation was derived from survey data collected on 15 September 2015.

Figure 7. Reference bathymetry (a), final EnOI analysis (b), and profile on 20 November 2018 (c); difference plots with respect to reference bathymetry for EnOI analysis (d), initial state (e), and temporally interpolated estimate (f); RMSE (g) and bias (h) summaries by cross-shore distance; altimetric data (full-resolution and time-averaged data) and reference bathymetry at three altimeter locations (i–k). Contour lines are −5.0 to −1.0 m (solid) and −4.5 to −0.5 (dashed) in 1.0 m increments. Black dots in (a,e) indicate altimeter locations.

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