## 1. Introduction

Wind-generated gravity waves drive nearshore processes, including beach erosion, along- and cross-shore material transport, and oceanfront flooding, during high water levels. High-resolution nearshore wave predictions support beach and boater safety, coastal risk assessment, and nearshore process research. Wave prediction errors in coastal areas sheltered by islands and complex bathymetry are sensitive to details of offshore wave directions that are not well resolved by directional buoy observations (Ochoa and Delgado-González 1990) or global wave models (O’Reilly and Guza 1993; Rogers et al. 2007; O’Reilly et al. 2016). However, observations sheltered from varying directions can be included to increase resolution (O’Reilly and Guza 1998), similar to applications in acoustic and optics of inhomogeneous media (Borcea et al. 2002). Though current operational wave models—for example, Simulating Waves Nearshore (SWAN) or WAVEWATCH III (WW3)—do not yet assimilate local wave observations, progress is being made. Recent developments (Veeramony et al. 2010; Orzech et al. 2013) use variational methodologies, relying on an adjoint model to propagate prediction misfits at local observation sites backward to the model boundary. An analytical adjoint (Walker 2006), developed originally for synthetic aperture radar satellite observations, has been extended to wave spectra (Veeramony et al. 2010); however, simulations were limited to linear wave processes (shoaling and refraction) and stationary conditions with homogeneous boundary forcing. Numerical methods have been developed to estimate an adjoint by linearizing the SWAN model (Orzech et al. 2013); however, initial tests are restricted to linear wave processes and stationary conditions. More recently, adjoint-free techniques have been developed, though initial testing has been restricted to synthetic data (Panteleev et al. 2015).

Linear wave propagation assumptions are valid for low-frequency wave energy in many nearshore regions where local fetch is limited and dissipation on a narrow shelf is negligible, for example, the Southern California Bight (SCB) (O’Reilly and Guza 1993; O’Reilly et al. 2016). However, in regions as large as the SCB (

This linear approach is limited to wave conditions, frequencies, and distances where dissipation and nonlinear spectral energy transfers are negligible. Offshore GWM predictions in Southern California (spatial scales of a few 100 km) suggest that linear wave propagation assumptions are valid for swell frequencies

An assimilation method is developed to estimate offshore directional spectra from directional and nondirectional wave buoy spectral observations. The assimilation treats each frequency band independently, but it could be extended to use all frequency bands simultaneously because of the method’s low computational costs. This would allow for assimilation of bulk parameters to the extent that non-swell-band energy is negligible (e.g., satellite altimeter observations and high-frequency radar during swell-dominated conditions).

Wave buoy observations are discussed in section 2, followed by details of the linear forward problem in section 3. The assimilation methodology (section 4) and applications to synthetic and real observations (section 5) are followed by a discussion of the model constraints and implications for buoy array design (section 6). The paper concludes with a summary (section 7).

## 2. Observations

*i*th buoy mooring location, at each frequency

*f*, the observed moments

Deep-water and shallow-water buoys site CDIP and NOAA identification numbers and mooring depth.

Buoy locations in the SCB (Fig. 1) were determined by user needs (e.g., harbor entrances) and logistics (e.g., mooring depth constraints). Shallow- (

## 3. Forward problem

*E*by

*K*contains transfer coefficients for the observed moment

*j*, location

*i*, frequency

*f*, and offshore direction

*θ*.

*θ*are ray directions, and

At sheltered locations the relationship between *K* contains many relatively narrow peaks and valleys (individual gray curves in Fig. 2b). Transfer coefficients are generated at fine frequency (0.0005 Hz) and directional (

(a) Nearshore ray angle vs offshore ray angle for 0.07 Hz. Rays are traced backward to deep water from buoy sites D3 and D8. Slope of the

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

(a) Nearshore ray angle vs offshore ray angle for 0.07 Hz. Rays are traced backward to deep water from buoy sites D3 and D8. Slope of the

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

(a) Nearshore ray angle vs offshore ray angle for 0.07 Hz. Rays are traced backward to deep water from buoy sites D3 and D8. Slope of the

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

*θ*now refers only to offshore wave directions. Additionally, the linear forward problem is solved independently at each frequency, and

*f*is dropped for brevity. Nearshore directional buoy moments are related to offshore wave spectra by

*i*locations, and the vector

(a) Time-invariant transfer coefficient vs offshore direction at deep-water SCB buoy sites (labeled) at the 0.07-Hz band for energy,

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

(a) Time-invariant transfer coefficient vs offshore direction at deep-water SCB buoy sites (labeled) at the 0.07-Hz band for energy,

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

(a) Time-invariant transfer coefficient vs offshore direction at deep-water SCB buoy sites (labeled) at the 0.07-Hz band for energy,

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

In (5) wave energy travel times are neglected but are as large as 12 h in the

(a) Schematic illustrating homogeniety assumption and time lag estimation between sites. (b) Time lag vs offshore direction for travel time between nearshore buoys D3 and D8 to offshore site D1 for 0.07-Hz energy. Note directions in Earth-relative compass coordinates.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

(a) Schematic illustrating homogeniety assumption and time lag estimation between sites. (b) Time lag vs offshore direction for travel time between nearshore buoys D3 and D8 to offshore site D1 for 0.07-Hz energy. Note directions in Earth-relative compass coordinates.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

(a) Schematic illustrating homogeniety assumption and time lag estimation between sites. (b) Time lag vs offshore direction for travel time between nearshore buoys D3 and D8 to offshore site D1 for 0.07-Hz energy. Note directions in Earth-relative compass coordinates.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

Offshore wave energy is assumed to be homogeneous in the along-crest (alongfront) direction (Fig. 4a), a validated assumption in the SCB for frequencies

*τ*depends on

*θ*). The time-dependent forward problem is discretized at 1-h time steps, and similar to (6) in matrix notation is

## 4. Inverse problem

We posit that WW3 accurately predicts incoming wave directions because storm locations are well defined by satellite wind observations that drive global wave models. In contrast, model swell energies and detailed directional properties depend on many modeling approximations, and they may have significant error. Therefore, before specifying *f* are adjusted to D1 observations.

*α*is a constant

Synthetic example of the assimilation technique: (a) energy vs direction at 0.06 Hz for plausible bimodal wave conditions. Synthetic observations are created with true spectra (black) run through the time-invariant forward problem [Eq. (6)] and perturbed by expected uncertainty in a 3-h record (see the appendix). Model prior (dashed–dotted line) and associated

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

Synthetic example of the assimilation technique: (a) energy vs direction at 0.06 Hz for plausible bimodal wave conditions. Synthetic observations are created with true spectra (black) run through the time-invariant forward problem [Eq. (6)] and perturbed by expected uncertainty in a 3-h record (see the appendix). Model prior (dashed–dotted line) and associated

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

Synthetic example of the assimilation technique: (a) energy vs direction at 0.06 Hz for plausible bimodal wave conditions. Synthetic observations are created with true spectra (black) run through the time-invariant forward problem [Eq. (6)] and perturbed by expected uncertainty in a 3-h record (see the appendix). Model prior (dashed–dotted line) and associated

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

WW3 time (solid line, left axis) and direction (dashed line, right axis) decorrelation length scales vs frequency. Decorrelation length scales are determined by best fit to Gaussian [Eq. (11)] of the WW3 autocorrelation function.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

WW3 time (solid line, left axis) and direction (dashed line, right axis) decorrelation length scales vs frequency. Decorrelation length scales are determined by best fit to Gaussian [Eq. (11)] of the WW3 autocorrelation function.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

WW3 time (solid line, left axis) and direction (dashed line, right axis) decorrelation length scales vs frequency. Decorrelation length scales are determined by best fit to Gaussian [Eq. (11)] of the WW3 autocorrelation function.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

*O*

*k*, where

The minimum solution to (21) with (22) was slightly negatively biased at the offshore buoy site D1 because event peak energies are consistently underpredicted owing to the smoothness constraint and proportionality of observed energy uncertainty to itself (see the appendix). The bias is corrected with the additional constraint that the event-averaged offshore inverse and observed energy (D1) match. This constraint is applied only at offshore site D1 because forward model errors and observational uncertainty preclude satisfaction at all buoy locations.

## 5. Case studies

### a. Synthetic observations

A synthetic case study of the time-invariant forward problem (for simplicity) demonstrates that the minimum solution to (21) with (22) behaves as expected, given the forward model assumptions (e.g., linear wave propagation, along-crest homogeneity, and no wave reflection) are satisfied. A plausible offshore directional spectrum (inspired by WW3 predictions) is used to represent the true offshore energy distribution for an example frequency band, 0.06 Hz (solid black in Fig. 5a). The forward problem,

### b. Real observations

Case studies with real buoy observations demonstrate the assimilation method’s skill in practice, where forward model assumptions are necessarily violated to some extent. Six moderate to large Pacific Northwest (NW) wave events were selected (Table 2), and various combinations of buoys were used in the assimilation. For simplicity, the additionally available offshore buoy observations, San Nicolas Island and Point Loma buoys (Fig. 1), used to drive CDIP’s operational wave predictions (O’Reilly et al. 2016), are not included. Each 4-day period, bracketing the event, was analyzed at 1-h and 1° resolutions. The first and last 12 h of solutions to (21) are truncated to allow for complete propagation across the domain (e.g., 12-h travel time). For each wave event, inverse offshore spectra (INV) were found that minimize (21) with (22) using varying sets of regional buoy observations and adjusted WW3 predictions at buoy D1.

Mean RMSE of

Tests with

(left) Offshore directional spectra and (right) energy observations and predictions at buoy sites vs time at the 0.06-Hz swell-band frequency for a wave event during January 2008 (case 4). Gray shading indicates observational uncertainty at the 95% confidence level. (a) WW3 spectra, (b) WW3 spectra are shifted 1 h backward in time and adjusted to match total energy observed at buoy D1, and (c) INV spectra generated using all buoy observations (left) and adjusted WW3 spectra in the minimization of (23) with

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

(left) Offshore directional spectra and (right) energy observations and predictions at buoy sites vs time at the 0.06-Hz swell-band frequency for a wave event during January 2008 (case 4). Gray shading indicates observational uncertainty at the 95% confidence level. (a) WW3 spectra, (b) WW3 spectra are shifted 1 h backward in time and adjusted to match total energy observed at buoy D1, and (c) INV spectra generated using all buoy observations (left) and adjusted WW3 spectra in the minimization of (23) with

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

(left) Offshore directional spectra and (right) energy observations and predictions at buoy sites vs time at the 0.06-Hz swell-band frequency for a wave event during January 2008 (case 4). Gray shading indicates observational uncertainty at the 95% confidence level. (a) WW3 spectra, (b) WW3 spectra are shifted 1 h backward in time and adjusted to match total energy observed at buoy D1, and (c) INV spectra generated using all buoy observations (left) and adjusted WW3 spectra in the minimization of (23) with

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

INV spectra with smoothing constants as above and all available buoy observations (availability varies with the case study; Table 2) are generated for all cases at all swell-band frequencies. Frequency-integrated and time-averaged observed and INV-predicted energies (Fig. 8) show good agreement in many cases, but they highlight a particularly severe bias (misfits up to three standard deviations) at buoys D2 and D3, both located in the eastern half of the Santa Barbara Channel (Fig. 1). These deep-water buoys are highly sheltered from NW swell by Point Conception, and we hypothesize forward model errors are large. Relaxation of smoothness constraints—for example,

Predicted vs observed *E*, integrated across swell frequency bands and temporally averaged, for each case study. Predictions with INV solutions use all available buoy observations. Some observations are not well fit by the INV methods and indicate forward model errors, e.g., D2, D3, S1, S2, and S4.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

Predicted vs observed *E*, integrated across swell frequency bands and temporally averaged, for each case study. Predictions with INV solutions use all available buoy observations. Some observations are not well fit by the INV methods and indicate forward model errors, e.g., D2, D3, S1, S2, and S4.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

Predicted vs observed *E*, integrated across swell frequency bands and temporally averaged, for each case study. Predictions with INV solutions use all available buoy observations. Some observations are not well fit by the INV methods and indicate forward model errors, e.g., D2, D3, S1, S2, and S4.

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

## 6. Discussion

### a. Assimilation method skill

Nearshore prediction skill can likely be improved by corrections to the forward model physics, but only boundary condition corrections are considered here. Buoys D2 and D3 are excluded from the following analysis because of large forward model error. Assimilation of these sites may degrade boundary condition corrections and increase prediction errors at other locations. The skill of INV solutions with selected assimilated sites (D1, D5, D6, D8) is assessed with comparisons of predictions and observations at nonassimilated (validation) buoy locations. By design, improvement is expected at assimilated buoy sites (Fig. 7); however, skill improvement at validation sites suggests that the INV offshore spectrum is increasingly accurate and that prediction skill improvement is regionwide. For all cases, assimilation improves skill for swell-band-integrated bulk wave parameters,

### b. Model prior,

The model prior and model covariance uncertainty select a solution from a theoretically infinite set by imposing desired characteristics, for example, smoothness and similarity to a preferred model. Smoothness is set by

INV solutions were generally insensitive to variations over the following parameter ranges:

The model prior uncertainty

The assumption of Gaussian statistics throughout the inversion framework makes the search for a solution within a large model space computationally feasible. Although buoy-measured moments [(1)] are

### c. Implications for buoy array design

An optimal operational array balances buoy value and cost. Buoy costs depend on ease of servicing, mooring depth, exposure to shipping and fishing, and other factors. Buoy value depends on site-specific local factors (e.g., nearby port), and more generally on the improvement in local and regional model skill. Offshore buoys are of primary importance to regional prediction, providing unsheltered deep-water observations free from forward model errors and several hours before waves arrive at the coast. Energy travel time lags in the SCB vary with wave direction, and both a southerly located buoy (e.g., Point Loma) and a northerly located buoy (e.g., Harvest) are needed to provide nowcast and short-term forecast predictions (O’Reilly et al. 2016). In smaller regions one offshore buoy may be sufficient. Nearshore buoy locations have historically been determined by funding sources and local community needs. Assimilation methods estimate the overall added predictive skill and help assess optimal buoy placement.

Our case studies are not comprehensive, but initial analysis suggests that assimilation of nearshore buoy D8 yields the largest improvement in overall swell prediction (mainly

The (a) *x* axis).

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

The (a) *x* axis).

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

The (a) *x* axis).

Citation: Journal of Atmospheric and Oceanic Technology 34, 8; 10.1175/JTECH-D-17-0003.1

Accurate unbiased directional buoy observations have historically been difficult and expensive to acquire. Some buoy directional moments contained bias (e.g., the National Data Buoy Center 3-m discus; O’Reilly et al. 1996). More recently inexpensive GPS sensors have been shown to yield accurate energy spectra and second-order directional moments but poorly resolve vertical motions, resulting in biased first-order directional moments (Herbers et al. 2012). While the high value of accurate offshore directional moments for regional swell prediction is clear (O’Reilly et al. 2016; Crosby et al. 2016), our analysis suggests less value for nearshore directional information. RMSE errors for INV predictions generated with and without assimilation of nearshore directional buoy observations are similar (Fig. 9a). Note that directional observations from offshore buoy D1 are included in both inversions. Regional skill is most efficiently improved by D1 energy and directional observations, and D8 energy observations. The negligible improvement in regional skill by nearshore directional information may be attributed to 1) the lack of shoreline reflection in the forward model (shoreline incident energy is assumed to completely dissipate) that affects directional observations significantly more than energy (Herbers et al. 1999; Crosby et al. 2016); and/or 2) increasingly sheltered regions’ directional observations becoming nearly redundant to energy observations because waves arrive from a single known direction. The only useful information is the magnitude and timing of arriving energy; the arrival direction is already known. Although adding relatively little to regional swell prediction (given the nonreflective, linear forward model), directional nearshore observations may provide useful local wave information, for example, providing robust estimates of local radiation stress (Longuet-Higgins and Stewart 1964).

Theoretically, additional buoy observations add the most skill to regional swell prediction when their offshore transfer functions (Fig. 3) are most unique, providing additional constraints on the offshore spectrum. The mean array resolution (Fig. 9b) is estimated by averaging the res(

## 7. Summary

We developed methods to improve nearshore swell-band wave prediction by assimilating nearshore directional buoy observations of frequency spectra and offshore global wave model (GWM) spectra predictions. Assimilation yields accurate high-resolution (direction and time) offshore wave spectra and accurate high-resolution (space and time) coastal wave predictions. Methods use linear (shoaling and refraction) wave propagation assumptions, valid on narrow continental shelves—for example, the U.S. West Coast—at swell-band frequencies (0.04–0.09 Hz). The linear wave energy propagation model, generated with backward ray tracing techniques, requires minimal computation and enables relatively rapid assimilation (compared to 4DVAR techniques). Regional prediction skill improvements are relatively insensitive to user-determined constraints. Nearshore buoy observations poorly fit by assimilated offshore spectra identify sites where significant wave physics are likely missing from the linear forward model. Overall, regional swell prediction errors are reduced by approximately 30% when observations are assimilated, and incremental improvements become insignificant after two to three buoys are assimilated. Varying the buoy observations included in the assimilation provides insight into optimal array design for regional nearshore swell prediction. The Bayesian framework of the assimilation advantageously incorporates prior information easily, for example, offshore GWM spectra predictions. Future improvements to GWM predictions, and an improved understanding of their uncertainty, will ultimately improve regional prediction skill. Though methods are restricted to regions and frequencies where wave propagation is approximately linear, these computationally rapid methods could be incorporated into a larger assimilation scheme, providing estimates for low-frequency energy and/or a first guess for more sophisticated nonlinear techniques.

## Acknowledgments

This study was funded by the U.S. Army Corps of Engineers (W912HZ-14-2-0025), the California Department of Parks and Recreation (C1370032), Division of Boating and Waterways Oceanography Program, California Sea Grant Traineeship Program, the National Oceanic and Atmospheric Administration (IOOSREG-T-000-00), and the Southern California Coastal Ocean Observing System. Bruce Cornuelle was sponsored by NOAA CIMEC Award NA15OAR4320071. Buoy data were obtained from the Coastal Data Information Program (CDIP, http://cdip.ucsd.edu/), and C. B. Olfe (CDIP) assisted with data access.

## APPENDIX

### Observation Uncertainty

*z*) and horizontal (

*γ*is the signal coherence. At 0.005-Hz frequency resolution and 3-h smoothed temporal resolution,

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