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## Abstract

Surveys of the subaerial beach (e.g., landward of approximately the MSL depth contour) are widely used to evaluate temporal changes in sand levels over large alongshore reaches. Here, seasonal beach face volume changes based on full bathymetry beach profiles (to ~8 m in depth) are compared with estimates based on the subaerial section of the profile. The profiles span 15 years and 75 km of Southern California shoreline, where seasonal vertical fluctuations in near-shore sand levels of a few meters are common. In years with relatively low winter wave energy, most erosion occurs above the MSL contour, and subaerial surveys capture as much as 0.8 of the total (relatively small) seasonal beach face volume change. In response to more energetic winter waves, beach face erosion increases and occurs as deep as 3 m below MSL, and subaerial surveys capture as little as 0.2 of the total beach face volume change. Patchy, erosion-resistant rock and cobble layers contribute to alongshore variation of the subaerial fraction of beach face volume change.

## Abstract

Surveys of the subaerial beach (e.g., landward of approximately the MSL depth contour) are widely used to evaluate temporal changes in sand levels over large alongshore reaches. Here, seasonal beach face volume changes based on full bathymetry beach profiles (to ~8 m in depth) are compared with estimates based on the subaerial section of the profile. The profiles span 15 years and 75 km of Southern California shoreline, where seasonal vertical fluctuations in near-shore sand levels of a few meters are common. In years with relatively low winter wave energy, most erosion occurs above the MSL contour, and subaerial surveys capture as much as 0.8 of the total (relatively small) seasonal beach face volume change. In response to more energetic winter waves, beach face erosion increases and occurs as deep as 3 m below MSL, and subaerial surveys capture as little as 0.2 of the total beach face volume change. Patchy, erosion-resistant rock and cobble layers contribute to alongshore variation of the subaerial fraction of beach face volume change.

## Abstract

A data-adaptive directional-spectrum estimator is developed for “point” measurement systems such as the pitch and roll buoy and slope array. This estimator, unlike the much employed unimodal cosine power parameterization method of Longuet-Higgins and others, does not make *a priori* assumptions about the shape of the directional spectrum. Instead improved resolution is obtained with a maximum likelihood method similar to those successfully used with spatial arrays. The numerical algorithm is relatively simple and computationally fast. The capabilities and limitations of the new estimator are illustrated with a variety of synthetic directional spectra. The estimator is applied to field data obtained from a slope array in 9 m depth at Santa Barbara, California and is found to yield physically realistic directional spectra. It marginally resolves two directional modes that topographical features dictate should be separated by approximately 70 degrees.

## Abstract

A data-adaptive directional-spectrum estimator is developed for “point” measurement systems such as the pitch and roll buoy and slope array. This estimator, unlike the much employed unimodal cosine power parameterization method of Longuet-Higgins and others, does not make *a priori* assumptions about the shape of the directional spectrum. Instead improved resolution is obtained with a maximum likelihood method similar to those successfully used with spatial arrays. The numerical algorithm is relatively simple and computationally fast. The capabilities and limitations of the new estimator are illustrated with a variety of synthetic directional spectra. The estimator is applied to field data obtained from a slope array in 9 m depth at Santa Barbara, California and is found to yield physically realistic directional spectra. It marginally resolves two directional modes that topographical features dictate should be separated by approximately 70 degrees.

## Abstract

Analytic and numerical models for longshore currents generated by obliquely incident random waves am compared with field observations. Five days of observations were selected during which the waves were narrow banded in both frequency and direction, in keeping with model assumptions. The extensive measurements included radiation stress and wave directional spectra in 9 m depth, and a closely spaced array of current and pressure sensors on a line perpendicular to shore. The longshore current models are based on balancing the gradient of the radiation stress with the alongshore bed shear and Reynold's stresses, assuming stationary wave conditions and straight and parallel bottom contours. The spatial variation of wave height, required to determine the gradient of the radiation stress, is modeled using linear random wave theory. Given *H*
_{rms} in 9 m depth, the model predicts *H*
_{rms} at shoreward locations with an average error of less than 9%. Using a nonlinear bottom shear stress formulation and the measured topography, a bed shear stress coefficient of *c _{f}
* = 0.006 gives optimal agreement between observed and predicted longshore currents. Eddy viscosity was found not to be important, at least for the nearly planar topography present during the observations.

## Abstract

Analytic and numerical models for longshore currents generated by obliquely incident random waves am compared with field observations. Five days of observations were selected during which the waves were narrow banded in both frequency and direction, in keeping with model assumptions. The extensive measurements included radiation stress and wave directional spectra in 9 m depth, and a closely spaced array of current and pressure sensors on a line perpendicular to shore. The longshore current models are based on balancing the gradient of the radiation stress with the alongshore bed shear and Reynold's stresses, assuming stationary wave conditions and straight and parallel bottom contours. The spatial variation of wave height, required to determine the gradient of the radiation stress, is modeled using linear random wave theory. Given *H*
_{rms} in 9 m depth, the model predicts *H*
_{rms} at shoreward locations with an average error of less than 9%. Using a nonlinear bottom shear stress formulation and the measured topography, a bed shear stress coefficient of *c _{f}
* = 0.006 gives optimal agreement between observed and predicted longshore currents. Eddy viscosity was found not to be important, at least for the nearly planar topography present during the observations.

## Abstract

Wavenumber-frequency spectra of the infragravity (periods 20-200 sec) wave velocity field in the surf zone of two California beaches are estimated. Because the longshore arrays of biaxial electromagnetic current meters are relatively short (comparable to the wavelengths of interest), high resolution spectrum estimators are required. Model testing provides insight into the limits, capabilities and reliability of the estimators used in this paper. On all 15 days analyzed, between 42% and 88% of the longshore current variance at the array is contributed by low mode (*n*≤2) edge waves. (Percentage estimates are not made at a few frequencies because the array is positioned near nodes.) The low mode signal in the cross-shore velocity at the arrays is usually masked by unresolvable high mode and/or leaky waves. The percentage of cross-shore current variance at the array estimated unresolvable high mode is less than 35%, with one exception for which approximately 50% of the variance is mode 0 across a substantial portion of the infragravity band. On average, low mode (*n*≤2) edge waves constitute 69% (17%) of the variance of the longshore (cross-shore) infragravity velocities at both arrays. There are days at both beaches that show factors of 3 asymmetry in the energy of up and downcoast progressive edge waves of a particular mode number, but the ratio of up and downcoast energy of up and downcoast progressive edge waves of a particular mode number, but the ratio of up and downcoast energy is usually within 1±0.1 On 8 of the 15 days, the spectrum of swash motions on the beach face is measured with a run-up meter. The swash spectrum, an estimate of the one-dimensional (summed over all wavenumbers) infragravity shoreline elevation spectrum, is compared to the edge wave shoreline elevation variances inferred from the velocity measurements at the array. As much as 50% of the variance in the present dataset, low mode edge waves contribute significantly to both the longshore velocity and run-up components of the nearshore infragravity wave field. Daily fluctuations in the shoreline elevation variance of individual low mode edge waves are regressed against the total wind and swell wave variance (periods 3–20 sec) measured outside the surf zone. The correlations are statistically significant at one beach, but not the other. Distortions of the observed edge wave dispersion curved (from a plane beach solution) because of beach concavity and mean longshore currents are small but detectable.

## Abstract

Wavenumber-frequency spectra of the infragravity (periods 20-200 sec) wave velocity field in the surf zone of two California beaches are estimated. Because the longshore arrays of biaxial electromagnetic current meters are relatively short (comparable to the wavelengths of interest), high resolution spectrum estimators are required. Model testing provides insight into the limits, capabilities and reliability of the estimators used in this paper. On all 15 days analyzed, between 42% and 88% of the longshore current variance at the array is contributed by low mode (*n*≤2) edge waves. (Percentage estimates are not made at a few frequencies because the array is positioned near nodes.) The low mode signal in the cross-shore velocity at the arrays is usually masked by unresolvable high mode and/or leaky waves. The percentage of cross-shore current variance at the array estimated unresolvable high mode is less than 35%, with one exception for which approximately 50% of the variance is mode 0 across a substantial portion of the infragravity band. On average, low mode (*n*≤2) edge waves constitute 69% (17%) of the variance of the longshore (cross-shore) infragravity velocities at both arrays. There are days at both beaches that show factors of 3 asymmetry in the energy of up and downcoast progressive edge waves of a particular mode number, but the ratio of up and downcoast energy of up and downcoast progressive edge waves of a particular mode number, but the ratio of up and downcoast energy is usually within 1±0.1 On 8 of the 15 days, the spectrum of swash motions on the beach face is measured with a run-up meter. The swash spectrum, an estimate of the one-dimensional (summed over all wavenumbers) infragravity shoreline elevation spectrum, is compared to the edge wave shoreline elevation variances inferred from the velocity measurements at the array. As much as 50% of the variance in the present dataset, low mode edge waves contribute significantly to both the longshore velocity and run-up components of the nearshore infragravity wave field. Daily fluctuations in the shoreline elevation variance of individual low mode edge waves are regressed against the total wind and swell wave variance (periods 3–20 sec) measured outside the surf zone. The correlations are statistically significant at one beach, but not the other. Distortions of the observed edge wave dispersion curved (from a plane beach solution) because of beach concavity and mean longshore currents are small but detectable.

## Abstract

This is Part 1 of a study of nonlinear effects on natural wind waves. Array measurements of pressure at the sea floor and middepth, collected 30 km offshore in 13-m depth, are compared to an existing theory for weakly nonlinear surface gravity waves. In this depth, free surface waves (obeying the linear dispersion relation) an weakly attenuated at the sea bed at sea and swell frequencies (0.05–0.3 Hz) but very strongly attenuated at frequencies higher than about 0.35 Hz. Only nonlinearly driven motions can reach the sea floor at these high frequencies. Nonlinear interactions between free (primary) waves of about the same frequency, travelling in nearly opposing directions, theoretically excite long-wavelength, double-frequency forced (secondary) waves that are only weakly attenuated at the sea door and form a mechanism for the generation of microseisms at great depth. In 13-m depth, wind-generated free waves and corresponding long-wavelength, high-frequency forced waves can be simultaneously observed on the sea floor, and the coupling, between the two examined in some detail.

Bottom-pressure spectra observed over a 4-day period show large [O(10^{2})] fluctuations in high-frequency (0.35–0.6 Hz) forced-wave energy levels at the sea floor occurring in only a few hours. Correspondingly rapid changes in estimates of the free-wave frequency-directional spectrum show that forced-wave energy levels are weak in unidirectional seas and increase dramatically in response to nearly opposing seas, consistent with the theoretical generation mechanism. On one occasion, directionally opposing seas, and a corresponding double-frequency forced-wave peak, followed a rapidly veering wind. However, comparable increases in forced-wave energy levels were observed in response to the arrival of nonlocally generated seas with directions much different than local winds and seas.

Although the accuracy of theoretical forced-wave predictions is limited by the directional resolution of the small aperture (20 m × 20 m) middepth array, predicted and observed forced-wave energy levels agree within about a factor of 2. The observed weak decay between middepth and sea-floor wave pressure at double sea frequencies is also consistent with theoretically expected long wavelengths. Wavelengths, propagation directions, and phase coupling between free and forced waves are examined using the bottom-pressure array data in Part 2.

## Abstract

This is Part 1 of a study of nonlinear effects on natural wind waves. Array measurements of pressure at the sea floor and middepth, collected 30 km offshore in 13-m depth, are compared to an existing theory for weakly nonlinear surface gravity waves. In this depth, free surface waves (obeying the linear dispersion relation) an weakly attenuated at the sea bed at sea and swell frequencies (0.05–0.3 Hz) but very strongly attenuated at frequencies higher than about 0.35 Hz. Only nonlinearly driven motions can reach the sea floor at these high frequencies. Nonlinear interactions between free (primary) waves of about the same frequency, travelling in nearly opposing directions, theoretically excite long-wavelength, double-frequency forced (secondary) waves that are only weakly attenuated at the sea door and form a mechanism for the generation of microseisms at great depth. In 13-m depth, wind-generated free waves and corresponding long-wavelength, high-frequency forced waves can be simultaneously observed on the sea floor, and the coupling, between the two examined in some detail.

Bottom-pressure spectra observed over a 4-day period show large [O(10^{2})] fluctuations in high-frequency (0.35–0.6 Hz) forced-wave energy levels at the sea floor occurring in only a few hours. Correspondingly rapid changes in estimates of the free-wave frequency-directional spectrum show that forced-wave energy levels are weak in unidirectional seas and increase dramatically in response to nearly opposing seas, consistent with the theoretical generation mechanism. On one occasion, directionally opposing seas, and a corresponding double-frequency forced-wave peak, followed a rapidly veering wind. However, comparable increases in forced-wave energy levels were observed in response to the arrival of nonlocally generated seas with directions much different than local winds and seas.

Although the accuracy of theoretical forced-wave predictions is limited by the directional resolution of the small aperture (20 m × 20 m) middepth array, predicted and observed forced-wave energy levels agree within about a factor of 2. The observed weak decay between middepth and sea-floor wave pressure at double sea frequencies is also consistent with theoretically expected long wavelengths. Wavelengths, propagation directions, and phase coupling between free and forced waves are examined using the bottom-pressure array data in Part 2.

## Abstract

An improved method for estimating the directional spectrum of linear surface gravity waves from in Situ observations is presented. The technique, a refinement and extension of the inverse method of Long and Hasselmann, is applicable to multicomponent wave measurements at fixed locations in constant or slowly varying depth water. On a frequency band by frequency band basis, an estimate of the directional distribution of wave energy *S*(θ) is obtained by minimizing a roughness measure of the form ∫*d*θ[*d*
^{2}
*S*(θ)/*d*θ^{2}]^{2} subject to the constraints: (i) *S*(θ) is nonnegative with unit integral, (ii) *S*(θ) fits the data within a chosen statistical confidence level, and (iii) *S*(θ) is zero on any directional sectors where energy levels are always relatively low because of the influence of geographic surroundings. The solution to this inverse problem is derived through a variational formulation with Lagrange multipliers.

A series of simulations using the new estimator show the fundamental limitations of sparse array data and the importance of using all available data-independent information [i.e., constraints (i) and (iii)] for achieving optimal estimates. The advantages of smoothness optimization are illustrated in a comparison of the present and Long and Hasselmann methods. The present method yields smooth estimates where Long and Hasselmann obtained rough estimates with multiple spurious peaks. A smooth solution to the inverse problem that has only truly resolved features is both easier to interpret and more readily evaluated numerically than wildly spurious solutions. The examples also demonstrate the subjectivity of intercomparing estimation techniques.

A few illustrative examples are presented of *S*(θ) estimates obtained from a two-dimensional array (aperture 120 m × 96 m) of 14 pressure transducer in 6 m water depth. Estimates using the full array and no geographic constraints are smooth and exhibit the expected refractive columnation of shoreward propagating energy towards normal incidence. Additionally, reflection from the mildly sloping beach 310 m shoreward of the center of this array is very weak at wind wave and swell frequencies. Estimates of *S*(θ) made using only the sensors on a longshore line, and a constraint of no reflected energy, are very similar to *S*(θ) obtained with the full array and no constraint.

## Abstract

An improved method for estimating the directional spectrum of linear surface gravity waves from in Situ observations is presented. The technique, a refinement and extension of the inverse method of Long and Hasselmann, is applicable to multicomponent wave measurements at fixed locations in constant or slowly varying depth water. On a frequency band by frequency band basis, an estimate of the directional distribution of wave energy *S*(θ) is obtained by minimizing a roughness measure of the form ∫*d*θ[*d*
^{2}
*S*(θ)/*d*θ^{2}]^{2} subject to the constraints: (i) *S*(θ) is nonnegative with unit integral, (ii) *S*(θ) fits the data within a chosen statistical confidence level, and (iii) *S*(θ) is zero on any directional sectors where energy levels are always relatively low because of the influence of geographic surroundings. The solution to this inverse problem is derived through a variational formulation with Lagrange multipliers.

A series of simulations using the new estimator show the fundamental limitations of sparse array data and the importance of using all available data-independent information [i.e., constraints (i) and (iii)] for achieving optimal estimates. The advantages of smoothness optimization are illustrated in a comparison of the present and Long and Hasselmann methods. The present method yields smooth estimates where Long and Hasselmann obtained rough estimates with multiple spurious peaks. A smooth solution to the inverse problem that has only truly resolved features is both easier to interpret and more readily evaluated numerically than wildly spurious solutions. The examples also demonstrate the subjectivity of intercomparing estimation techniques.

A few illustrative examples are presented of *S*(θ) estimates obtained from a two-dimensional array (aperture 120 m × 96 m) of 14 pressure transducer in 6 m water depth. Estimates using the full array and no geographic constraints are smooth and exhibit the expected refractive columnation of shoreward propagating energy towards normal incidence. Additionally, reflection from the mildly sloping beach 310 m shoreward of the center of this array is very weak at wind wave and swell frequencies. Estimates of *S*(θ) made using only the sensors on a longshore line, and a constraint of no reflected energy, are very similar to *S*(θ) obtained with the full array and no constraint.

## Abstract

This is Part 2 of a study of nonlinear effects on natural wind-generated surface gravity waves in 13-m depth, 30 km offshore of Virginia. At the sea floor in this depth, free surface gravity waves are only weakly attenuated at sea and swell frequencies (0.05–0.30 Hz) but are very strongly attenuated at frequencies higher than about 0.35 Hz. Hence, above 0.35 Hz, relatively long wavelength forced waves, excited by nonlinear interactions between directionally opposing free wind waves, are exposed at the sea floor. An array of pressure transducers at middepth was used to estimate the frequency-directional spectrum of (free) primary sea and swell waves, and the associated (forced) secondary pressure fluctuations were measured with an array on the sea floor. In Part 1, it was shown that forced-wave energy levels at the sea floor increase sharply in response to directionally opposing wind waves, in agreement with weakly nonlinear theory. In Part 2, wavelengths, propagation directions, and non-Gaussian phase coupling between free and forced waves are examined on three occasions with relatively high forced-wave energy levels.

A root-mean-square wavenumber magnitude and a vector-averaged mean wave propagation direction (both functions of frequency) can be expressed accurately in terms of the pressure array cross-spectra. The wavenumber estimates at the sea floor show the theoretically expected sharp transition between a 0.05–0.30 HZ frequency range dominated by free sea and swell waves and a 0.35–0.60 Hz range dominated by forced waves with wavelengths that are long relative to free waves of the same frequency. In the “free-wave frequency range,” wavenumber estimates are usually well within 10% of the linear dispersion relation and wave direction estimates are in excellent agreement with the directional spectra extracted from the middepth array. In the “forced-wave frequency range,” wavenumber and direction estimates agree with nonlinear theory predictions, confirming that the observed forced waves have the sum vector wavenumber of the interacting directionally opposing wind waves.

Phase coupling between free and forced waves is examined through third-order statistics of the sea floor pressure data. Consistent with theory, the normalized bispectrum has small imaginary parts scattered approximately randomly about zero and relatively large negative real parts at frequencies that correspond to directionally opposing seas and swell. Estimates of the bispectrum integrated for constant sum frequency confirm that nearly all the energy at double sea frequencies is nonlinearly coupled to directionally opposing wind waves. In qualitative agreement with nonlinear theory predictions, bispectral levels are sometimes significantly reduced by directional spreading of the interacting free waves.

## Abstract

This is Part 2 of a study of nonlinear effects on natural wind-generated surface gravity waves in 13-m depth, 30 km offshore of Virginia. At the sea floor in this depth, free surface gravity waves are only weakly attenuated at sea and swell frequencies (0.05–0.30 Hz) but are very strongly attenuated at frequencies higher than about 0.35 Hz. Hence, above 0.35 Hz, relatively long wavelength forced waves, excited by nonlinear interactions between directionally opposing free wind waves, are exposed at the sea floor. An array of pressure transducers at middepth was used to estimate the frequency-directional spectrum of (free) primary sea and swell waves, and the associated (forced) secondary pressure fluctuations were measured with an array on the sea floor. In Part 1, it was shown that forced-wave energy levels at the sea floor increase sharply in response to directionally opposing wind waves, in agreement with weakly nonlinear theory. In Part 2, wavelengths, propagation directions, and non-Gaussian phase coupling between free and forced waves are examined on three occasions with relatively high forced-wave energy levels.

A root-mean-square wavenumber magnitude and a vector-averaged mean wave propagation direction (both functions of frequency) can be expressed accurately in terms of the pressure array cross-spectra. The wavenumber estimates at the sea floor show the theoretically expected sharp transition between a 0.05–0.30 HZ frequency range dominated by free sea and swell waves and a 0.35–0.60 Hz range dominated by forced waves with wavelengths that are long relative to free waves of the same frequency. In the “free-wave frequency range,” wavenumber estimates are usually well within 10% of the linear dispersion relation and wave direction estimates are in excellent agreement with the directional spectra extracted from the middepth array. In the “forced-wave frequency range,” wavenumber and direction estimates agree with nonlinear theory predictions, confirming that the observed forced waves have the sum vector wavenumber of the interacting directionally opposing wind waves.

Phase coupling between free and forced waves is examined through third-order statistics of the sea floor pressure data. Consistent with theory, the normalized bispectrum has small imaginary parts scattered approximately randomly about zero and relatively large negative real parts at frequencies that correspond to directionally opposing seas and swell. Estimates of the bispectrum integrated for constant sum frequency confirm that nearly all the energy at double sea frequencies is nonlinearly coupled to directionally opposing wind waves. In qualitative agreement with nonlinear theory predictions, bispectral levels are sometimes significantly reduced by directional spreading of the interacting free waves.

## Abstract

Statistics of the nearshore velocity field in the wind–wave frequency band estimated from acoustic Doppler, acoustic travel time, and electromagnetic current meters are similar. Specifically, current meters deployed 25–100 cm above the seafloor in 75–275-cm water depth in conditions that ranged from small-amplitude unbroken waves to bores in the inner surf zone produced similar estimates of cross-shore velocity spectra, total horizontal and vertical velocity variance, mean currents, mean wave direction, directional spread, and cross-shore velocity skewness and asymmetry. Estimates of seafloor location made with the acoustic Doppler sensors and collocated sonar altimeters differed by less than 5 cm. Deviations from linear theory in the observed relationship between pressure and velocity fluctuations increased with increasing ratio of wave height to water depth. The observed covariance between horizontal and vertical orbital velocities also increased with increasing height to depth ratio, consistent with a vertical flux of cross-shore momentum associated with wave dissipation in the surf zone.

## Abstract

Statistics of the nearshore velocity field in the wind–wave frequency band estimated from acoustic Doppler, acoustic travel time, and electromagnetic current meters are similar. Specifically, current meters deployed 25–100 cm above the seafloor in 75–275-cm water depth in conditions that ranged from small-amplitude unbroken waves to bores in the inner surf zone produced similar estimates of cross-shore velocity spectra, total horizontal and vertical velocity variance, mean currents, mean wave direction, directional spread, and cross-shore velocity skewness and asymmetry. Estimates of seafloor location made with the acoustic Doppler sensors and collocated sonar altimeters differed by less than 5 cm. Deviations from linear theory in the observed relationship between pressure and velocity fluctuations increased with increasing ratio of wave height to water depth. The observed covariance between horizontal and vertical orbital velocities also increased with increasing height to depth ratio, consistent with a vertical flux of cross-shore momentum associated with wave dissipation in the surf zone.

## Abstract

Inverse methods are used to assimilate wave observations into numerical predictions of ocean swell (0.04–0.12 Hz surface waves) propagating over complex continental shelf bathymetry. Model predictions of swell on the shelf can be degraded by the limited accuracy and resolution of the initializing offshore (unsheltered deep water) frequency–directional spectra *S*
_{
o
}, usually derived from buoy measurements or meteorological hindcasts. The authors use a spectral refraction wave propagation model to find improved estimates of *S*
_{
o
} that are consistent with both unsheltered and sheltered (altered by coastal bathymetry) observations, and vary smoothly in direction and time.

In Part I, linear and nonlinear inverse assimilation methods are developed. Their potential and limitations are illustrated qualitatively using a scenario where no a priori knowledge of *S*
_{
o
} is used in the inverse estimates. Inverse estimates of *S*
_{
o
} based solely on sheltered wave data routinely collected in the Southern California Bight are compared to meteorological hindcasts of peak offshore swell directions for two time periods dominated by swell arrivals from a distant storm. Robust model–hindcast agreement for the peak direction of an energetic, unimodal North Pacific swell event demonstrates that offshore directional information can be inferred solely from sheltered measurements. The poor model–hindcast agreement for a south swell event is markedly improved by the a priori assumption that *S*
_{
o
} is unimodal with a prescribed parametric form, but assumptions about the shape of *S*
_{
o
} severely reduce the generality of the approach. The authors conclude that conventional (low directional resolution) measurements from a few sheltered sites cannot be used to routinely resolve *S*
_{
o
}, and offshore measurements or hindcasts of *S*
_{
o
} are needed as additional constraints in practical applications.

In Part II, the inverse methods are generalized to include hindcasts or unsheltered directional buoy data as primary constraints and sheltered observations are used to enhance the directional resolution and stability of *S*
_{
o
}. Initialized with these *S*
_{
o
}, the wave model yields improved predictions of regional swell conditions. The value of assimilating coastal observations into regional wave predictions is demonstrated with a comprehensive field study.

## Abstract

Inverse methods are used to assimilate wave observations into numerical predictions of ocean swell (0.04–0.12 Hz surface waves) propagating over complex continental shelf bathymetry. Model predictions of swell on the shelf can be degraded by the limited accuracy and resolution of the initializing offshore (unsheltered deep water) frequency–directional spectra *S*
_{
o
}, usually derived from buoy measurements or meteorological hindcasts. The authors use a spectral refraction wave propagation model to find improved estimates of *S*
_{
o
} that are consistent with both unsheltered and sheltered (altered by coastal bathymetry) observations, and vary smoothly in direction and time.

In Part I, linear and nonlinear inverse assimilation methods are developed. Their potential and limitations are illustrated qualitatively using a scenario where no a priori knowledge of *S*
_{
o
} is used in the inverse estimates. Inverse estimates of *S*
_{
o
} based solely on sheltered wave data routinely collected in the Southern California Bight are compared to meteorological hindcasts of peak offshore swell directions for two time periods dominated by swell arrivals from a distant storm. Robust model–hindcast agreement for the peak direction of an energetic, unimodal North Pacific swell event demonstrates that offshore directional information can be inferred solely from sheltered measurements. The poor model–hindcast agreement for a south swell event is markedly improved by the a priori assumption that *S*
_{
o
} is unimodal with a prescribed parametric form, but assumptions about the shape of *S*
_{
o
} severely reduce the generality of the approach. The authors conclude that conventional (low directional resolution) measurements from a few sheltered sites cannot be used to routinely resolve *S*
_{
o
}, and offshore measurements or hindcasts of *S*
_{
o
} are needed as additional constraints in practical applications.

In Part II, the inverse methods are generalized to include hindcasts or unsheltered directional buoy data as primary constraints and sheltered observations are used to enhance the directional resolution and stability of *S*
_{
o
}. Initialized with these *S*
_{
o
}, the wave model yields improved predictions of regional swell conditions. The value of assimilating coastal observations into regional wave predictions is demonstrated with a comprehensive field study.

## Abstract

Inverse models are developed that use data and dynamics to estimate optimally the breaking-wave-driven setup and alongshore current, as well as the cross-shore forcing, alongshore forcing, and drag coefficient. The inverse models accurately reproduce these quantities in a synthetic barred-beach example. The method is applied to one case example each from the Duck94 and SandyDuck field experiments. Both inverse solutions pass consistency tests developed for the inverse method and have forcing corrections similar to a roller model and significant cross-shore variation of the drag coefficient. The inverse drag coefficient is related to the wave dissipation, a bulk measure of the turbulence source, but not to the bed roughness, consistent with the hypothesis that breaking-wave-generated turbulence increases the drag coefficient. Inverse solutions from a wider range of conditions are required to establish the generality of these results.

## Abstract

Inverse models are developed that use data and dynamics to estimate optimally the breaking-wave-driven setup and alongshore current, as well as the cross-shore forcing, alongshore forcing, and drag coefficient. The inverse models accurately reproduce these quantities in a synthetic barred-beach example. The method is applied to one case example each from the Duck94 and SandyDuck field experiments. Both inverse solutions pass consistency tests developed for the inverse method and have forcing corrections similar to a roller model and significant cross-shore variation of the drag coefficient. The inverse drag coefficient is related to the wave dissipation, a bulk measure of the turbulence source, but not to the bed roughness, consistent with the hypothesis that breaking-wave-generated turbulence increases the drag coefficient. Inverse solutions from a wider range of conditions are required to establish the generality of these results.