## 1. Introduction

*y*and time (i.e., steady). The cross-shore momentum equation becomes a one-dimensional (1D) balance between the cross-shore pressure gradient and the total (wind plus wave) cross-shore forcing

*F*

_{x}(e.g., Longuet-Higgins and Stewart 1964):

*g*is gravitational acceleration,

*h*is the water depth,

*x*is the cross-shore coordinate, and

*η*

*F*

_{y}, bottom stress, and lateral mixing (e.g., Longuet-Higgins 1970):

*υ*

*c*

_{d}is a nondimensional drag coefficient, 〈〉 represents a time average over many wave periods, |

**u**| is the total instantaneous horizontal velocity vector, and

*υ*is the instantaneous alongshore velocity. Mean and wave-orbital velocities contribute to 〈|

**u**|

*υ*〉. The third term in (2) represents lateral mixing processes (

*ν*is an eddy viscosity) including shear dispersion (Svendsen and Putrevu 1994), shear waves (Slinn et al. 1998; Özkan-Haller and Kirby 1999), and small-scale turbulent mixing by breaking waves (Battjes 1975). The cross- and alongshore forcings are the sum of wind (

*τ*

^{wind}

_{x}

*τ*

^{wind}

_{y}

*ρ*is the water density and

*S*

_{xx}and

*S*

_{yx}are components of the radiation stress tensor (Longuet-Higgins and Stewart 1964).

The 1D setup [(1)] and alongshore current [(2)] dynamics are applicable to many laboratory and field situations (e.g., Bowen et al. 1968; Battjes and Stive 1985; Thornton and Guza 1986; and many others). Although simple, except for the pressure gradient term, the functional forms of the terms in (1) and (2) are not known and must be parameterized for use in models. Linear theory relates the wave forcing to the root-mean-square (rms) wave height *H*_{rms}, mean wave angle *θ**f**η**υ**υ*

The closure of 1D integrated alongshore momentum balances on cross-shore transects (Feddersen et al. 1998; Feddersen and Guza 2003) suggests that *c*_{d}〈|**u**|*υ*〉 adequately represents the bottom stress. A spatially constant *c*_{d} often has been used in models (Longuet-Higgins 1970; Thornton and Guza 1986; Özkan-Haller and Kirby 1999). However, within the surf zone *c*_{d} is elevated relative to seaward of the surf zone (Feddersen et al. 1998). A drag coefficient proportional to *h*^{−1/3} (*c*_{d} increases in shallower depths) improves 1D *υ**c*_{d} (Ruessink et al. 2001). The elevated surf zone or shallow-water *c*_{d} has been hypothesized to result from increased bottom roughness (e.g., Garcez-Faria et al. 1998) or breaking- wave-generated turbulence (e.g., Church and Thornton 1993), but the spatial variation of *c*_{d} is not understood.

The wave forcing, *c*_{d}, and the Reynolds stress terms are difficult to estimate directly, and therefore the quality of their parameterizations is not known. Instead, parameterizations are accepted or rejected by the accuracy of the model predictions. Parameterizations often can be tuned so that model predictions match a limited dataset and thus, rarely are rejected. Here, an inverse method is developed (section 2) that uses the setup and alongshore current observations and dynamics to solve for parameterized quantities, namely, the cross- and alongshore forcing and the drag coefficient. Specification of the measurement error variances and parameterized forcing and drag coefficient error covariances is required. Inverse solutions not consistent with the specified measurement and parameterization errors are considered spurious and are rejected. The inverse method is tested with a synthetic barred-beach example with known forcing and *c*_{d} (section 3), and works well given the number and quality of field observations typically available. The inverse method is applied to one case example each from the Duck94 and SandyDuck field experiments (section 4). The case example inverse results are discussed in the context of wave rollers and possible drag coefficient dependence on breaking-wave- generated turbulence and bed roughness (section 5). The results are summarized in section 6.

## 2. Inverse modeling

### a. Prior model and prior solutions

The equation for the setup (1) is linearized (i.e., the still-water depth *h* is used instead of *h* + *η**η**h* are similar in water depths ≥ 0.3 m, where the case example observations were obtained. The parameterized cross- and alongshore forcings are denoted as the prior forcing *F*^{(pr)}_{x}*F*^{(pr)}_{y}*c*^{(pr)}_{d}*x* = *L*) prior boundary condition for the setup model (1) is *η**d**υ**dx* = 0 at the shoreline (*x* = 0) and offshore (*x* = *L*) boundaries. The quadratic velocity term in the bottom stress is parameterized with 〈|**u**|*υ*〉 = *B*(*υ**σ*_{T}*υ*^{2} + (*υ**σ*_{T})^{2}]^{1/2} (Feddersen et al. 2000), where *σ*^{2}_{T}*η*^{(pr)} and alongshore current *υ*^{(pr)}.

### b. Setup inverse modeling

*f*

_{x}(

*x*), attributed to error in the prior wave forcing, is allowed on the right-hand side of the cross-shore momentum equation [(1)]. The inverse forcing is given by

*F*

^{(i)}

_{x}

*F*

^{(pr)}

_{x}

*f*

_{x}. The forcing error (or correction)

*f*

_{x}is assumed to be a zero-mean continuous Gaussian random variable with covariance

*C*

_{fx}

*x,*

*x*′) =

*E*[

*f*

_{x}(

*x*)

*f*

_{x}(

*x*′)]. Similarly, zero-mean Gaussian error with prior variance

*σ*

^{2}

_{ηL}

*η*

*L*) = 0. The

*M*noisy

*η*

*d*

^{(η)}

_{m}

*m*= 1, … ,

*M*) consist of signal and measurement error so that

*d*

^{(η)}

_{m}

*η*

*x*

_{m}

*e*

^{(η)}

_{m}

*e*

^{(η)}

_{m}

*σ*

^{2}

_{ηd}

*η*

*f*

_{x}that incorporate dynamics and data are found by minimizing a cost function that is a combination of dynamical, boundary condition, and data errors (e.g., Bennett 1992):

*η*

^{(i)}and forcing

*F*

^{(i)}

_{x}

*C*

_{fx}

*x,*

*x*′) is defined so that

*δ*(

*x*) is the Dirac delta function. The

*η*

*λ*

_{η}

*f*

_{x}with

*C*

^{−1}

_{fx}

*J*[

*η*

*J*[

*η*

*η*

^{(i)}and

*f*

^{(i)}

_{x}

*F*

^{(i)}

_{x}

*J*[

*η*

*J*

_{min}is the minimum of the cost function (3) found by solving (6), and

*η*

*C*

^{(i)}

_{η}

*x,*

*x*′)]

^{−1}is interpreted as the inverse

*η*

*η*

*C*

^{(pr)}

_{η}

*x,*

*x*′) is related to the forcing error covariance by removing the data term from

*J*[

*η*

*η*

*η*

*C*

^{(i)}

_{η}

*x,*

*x*′) is then given by

*C*

^{(i)}

_{η}

### c. Alongshore current inverse modeling

*f*

_{y}(

*x*) is allowed on the right-hand side of (2), and represents error in the forcing, bottom stress, and lateral mixing. Because the forcing is considered to have the largest uncertainty and with the drag coefficient solved for separately,

*f*

_{y}is ascribed to forcing error. Corrections to lateral mixing are neglected. The inverse alongshore wave forcing

*F*

^{(i)}

_{y}

*F*

^{(i)}

_{y}

*F*

^{(pr)}

_{y}

*f*

_{y}, and the forcing error (or correction)

*f*

_{y}is assumed to be a zero-mean Gaussian random variable with covariance

*C*

_{fy}

*x,*

*x*′). Errors in the prior slip boundary conditions are assumed to be zero- mean Gaussian random variables with variance

*σ*

^{2}

_{υx0}

*σ*

^{2}

_{υxL}

*x*= 0 and

*x*=

*L.*The inverse method also allows for drag coefficient deviations from the prior

*c*

^{(pr)}

_{d}

*υ*

*c*

_{d}error is considered to be a zero-mean Gaussian random variable with prior covariance

*C*

^{(pr)}

_{cd}

*x,*

*x*′). The

*N*noisy alongshore current observations

*d*

^{(υ)}

_{m}

*d*

^{(υ)}

_{m}

*υ*

*x*

_{m}

*e*

^{(υ)}

_{m}

*e*

_{m}is zero-mean Gaussian measurement error with prior variance

*σ*

^{2}

_{υd}

*I*[

*υ*

*c*

_{d}] is defined as a combination of dynamical, boundary condition, drag coefficient, and data errors:

*I*[

*υ*

*c*

_{d}] with respect to

*υ*

*c*

_{d}to zero leads to the Euler–Lagrange equations for the cost function minimum:

*υ*

*λ*

_{υ}

*η*

*υ*

^{(i)},

*f*

^{(i)}

_{y}

*c*

^{(i)}

_{d}

*I*[

*υ*

*c*

_{d}] is rewritten as

*I*

_{min}is the minimum of

*I*[

*υ*

*c*

_{d}] found by solving (11), and

*υ*

*c*

^{′}

_{d}

*I*[

*υ*

*c*

_{d}] at the minimum {e.g., [

*C*

^{(i)}

_{υ}

*x,*

*x*′)]

^{−1}and [

*C*

^{(i)}

_{cd}

^{−1}} are interpreted as inverse covariances (appendix A). Similarly, the prior

*υ*

*C*

^{(pr)}

_{υ}

*υ*

*υ*

*C*

^{(i)}

_{υ}

*c*

^{(pr)}

_{d}

*c*

^{(i)}

_{d}

*C*

^{(pr)}

_{υ}

*υ*

^{(i)}used in

*dB*;cl

*d*

*υ*

*η*

*c*

_{d}covariance,

*υ*

*c*

_{d}covariance

*C*

_{υ,cd}

### d. Prior covariances

*x*−

*x*′) bell-shaped covariance often used in objective mapping (e.g., Brethereton et al. 1976):

*γ*is either

*f*

_{x},

*f*

_{y}, or

*c*

_{d}. The

*f*

_{x},

*f*

_{y}, and

*c*

_{d}variances (

*σ*

^{2}

_{fx}

*σ*

^{2}

_{fy}

*σ*

^{2}

_{cd}

*l*

_{fx}

*l*

_{fy}

*l*

_{cd}

*σ*

_{fx}

*F*

^{(pr)}

_{x}

*similar*to those using the homogeneous form (16). The homogeneous form was used for simplicity, because the form of the true covariances is unknown. Note that the homogeneous forcing error covariances, once filtered by the

*η*

*υ*

*η*

*υ*

*c*

_{d}[(15)] covariance.

## 3. Test of the inverse method

The ability of the inverse method to solve for the forcing and drag coefficient is tested with synthetic data. A true cross- and alongshore wave forcing (based on rollers) and a cross-shore variable *c*_{d} yield [through (1) and (2)] the true *η*^{(tr)} and *υ*^{(tr)}. Prior (nonroller) forcing and constant *c*_{d} similarly yield the prior *η*^{(pr)} and *υ*^{(pr)} and reflect the imperfect knowledge of the dynamics. The true values represent the dynamical information that the inverse method should reproduce, given the prior values, noisy data, and assumptions about the errors.

### a. True and prior conditions

Barred-beach bathymetry *h* from Duck, North Carolina (Lippmann et al. 1999), is used with a domain extending from the shoreline (*x* = 0 m) to 300 m offshore (Fig. 1a). The bar crest is located at *x* = 80 m and has a half-width of 15 m. At the offshore boundary, the wave height *H*_{rms} = 1.2 m, the wave period is 10 s, and the wave angle is 15° relative to shore-normal. The waves are transformed shoreward (Thornton and Guza 1983) over the barred bathymetry (Fig. 1b), yielding the (without rollers) prior wave forcing *F*^{(pr)}_{x}*F*^{(pr)}_{y}*σ*^{2}_{T}*F*^{(tr)}_{x}*F*^{(tr)}_{y}^{−4} m^{2} s^{−2} (roughly corresponding to a 14-kt alongshore wind) is added to the prior and true alongshore forcing.

Following Church and Thornton (1993), the true drag coefficient *c*^{(tr)}_{d}*c*_{d}, just offshore of the bar crest and near the shoreline, occur where breaking-wave dissipation is maximum. This *c*_{d} is hypothetical and is used only to test the inverse method. A *c*_{d} that depends inversely on water depth [e.g., the Manning–Strickler equation used by Ruessink et al. (2001)] gives qualitatively similar *c*_{d} variation. The spatially constant *c*^{(pr)}_{d}*υ**ν* = 0.5 m^{2} s^{−1} was used to model *υ**ν* (0.1–0.9 m^{2} s^{−1}) suggested by Özkan- Haller and Kirby (1999). With this eddy viscosity, the modeled magnitude of lateral mixing is small relative to the forcing (Ruessink et al. 2001).

These inputs are used within the setup [(1)] and alongshore current [(2)] models to generate true and prior *η**υ**η*^{(tr)} and the main *υ*^{(tr)} peak are moved onshore from the prior locations due to the roller and (for *υ*^{(tr)}) by the reduced *c*^{(tr)}_{d}*υ**η**υ**d*^{(η)}_{m}*d*^{(υ)}_{m}^{−1}, representative of setup measurement (Raubenheimer et al. 2001) and electromagnetic current meter (Feddersen and Guza 2003) error, respectively. The eight data locations are typical of the cross-shore instrumented transects at Duck during the Duck Experiment on Low-Frequency and Incident-Band Longshore and Across-Shore Hydrodynamics (DELILAH), Duck94, and SandyDuck field experiments.

### b. Prior covariances

The prior covariances of the forcing, drag coefficient, boundary condition, and data errors also must be specified. The data errors *σ*_{ηd}*σ*_{υd}^{−1} are those used to create the synthetic data. The magnitude of the forcing error is constrained by the prior forcing magnitude. Because the prior forcing is believed to be qualitatively correct, the forcing errors *σ*_{fx}*σ*_{fy}*F*^{(pr)}_{x}*F*^{(pr)}_{y}*σ*_{fx}*σ*_{fy}*c*_{d} error *σ*_{cd}*c*^{(pr)}_{d}*σ*_{cd}*c*_{d} range (0.001–0.004). The prior *c*^{(pr)}_{d}*σ*_{cd}*c*_{d} (Fig. 1e). Because the bathymetry strongly controls the wave properties, the forcing and *c*_{d} error length scales are chosen to match the sandbar half-width (*l*_{fx}*l*_{fy}*l*_{cd}*η**σ*_{ηL}*υ**σ*_{υx0}^{−1} and *σ*_{υxL}^{−1}, allowing for typical boundary shear of 1 m s^{−1} over 20 m at *x* = 0 m and 0.2 m s^{−1} over 20 m at *x* = *L.* The prior *η**υ**C*^{(pr)}_{η}*C*^{(pr)}_{υ}*η**υ**η**υ*

### c. Inverse solution

With all the ingredients, the inverse method yields the inverse setup *η*^{(i)} (Fig. 3a), inverse alongshore current *υ*^{(i)} (Fig. 4a), and their covariances *C*^{(i)}_{η}*C*^{(i)}_{υ}*η*^{(tr)} and *υ*^{(tr)} and are significant improvements over the prior solutions (Figs. 2a,b). The rms differences between inverse solutions and data are 2.4 mm and 2.6 cm s^{−1} for *η**υ**η**υ**x* < 200 m), the inverse uncertainty increases.

The ability of the inverse method to reproduce the cross- and alongshore forcing is examined by comparing the inverse forcing corrections [*f*^{(i)}_{x}*f*^{(i)}_{y}*f*^{(tr)}_{x}*F*^{(tr)}_{x}*F*^{(pr)}_{x}*f*^{(tr)}_{y}*f*^{(i)}_{x}*f*^{(i)}_{y}*f*^{(tr)}_{x}*f*^{(tr)}_{y}*f*^{(tr)}_{x}*x* = 110 m. Onshore of the last data point (*x* = 20 m), without information (data) for the inverse, *f*^{(i)}_{x}*f*^{(i)}_{y}*c*^{(i)}_{d}*c*_{d} covariance and qualitatively reproduces *c*^{(tr)}_{d}*x* = 20 m and offshore of *x* = 150 m, the *c*^{(i)}_{d}*c*_{d} = 0.0015 both because of the data sparseness and because the inverse method can adjust *f*_{y} to match the data with less cost. In the bar-trough region, the *c*^{(i)}_{d}*η**υ*

### d. Choosing covariance parameters

*σ*

_{fx}

*σ*

_{fy}

*σ*

_{cd}

*x*= 20–200 m) where data are concentrated,

*L*is the integration distance (180 m), and

*α*is any inverse or prior solution [e.g.,

*F*

^{(i)}

_{x}

For the *η**σ*_{fx}*σ*_{fx}*F*^{(pr)}_{x}*F*^{(i)}_{x}*η*^{(i)} misfits are reduced with increasing *σ*_{fx}*σ*_{fx}*F*^{(i)}_{x}*η*^{(i)} misfits are 33%–50% and 25%–33% (respectively) of their prior misfits. The largest %*σ*_{fx}*σ*_{fx}*σ*_{fx}*f*^{(i)}_{x}*C*_{fx}*σ*_{fx}*F*^{(i)}_{x}*η*^{(i)} misfits are small, near the minimum misfits for all %*σ*_{fx}*σ*_{fx}*σ*_{fx}*f*^{(i)}_{x}

For the *υ**σ*_{fy}*σ*_{fy}*F*^{(pr)}_{y}*σ*_{cd}*c*^{(pr)}_{d}*F*^{(i)}_{y}*F*^{(pr)}_{y}*σ*_{cd}*σ*_{fy}*σ*_{cd}*F*^{(i)}_{y}*c*^{(i)}_{d}*c*_{d} error, the *υ*^{(i)} misfit reduction is dramatic {*χ*[*υ*^{(i)}] is 20% of the *υ*^{(pr)} misfit}. The *c*^{(i)}_{d}*χ*[*c*^{(i)}_{d}*χ*[*c*^{(pr)}_{d}*c*^{(i)}_{d}*c*^{(tr)}_{d}*η**σ*_{fy}*χ*[*υ*^{(i)}]. Consistent solutions with larger %*σ*_{fy}*σ*_{cd}*F*^{(i)}_{y}*σ*_{fy}*σ*_{cd}*f*^{(i)}_{y}*c*^{(i)}_{d}*σ*_{fy}*σ*_{cd}

## 4. Case examples

The inverse method is applied to observations from two field experiments at Duck: Duck94 (Elgar et al. 1997; Feddersen et al. 1998; Ruessink et al. 2001) and SandyDuck (Elgar et al. 2001; Raubenheimer et al. 2001; Feddersen and Guza 2003; Noyes et al. 2004). Bathymetries are smoothed with a 10-m cutoff wavelength to remove bedforms that dominate the variance in the 1–5-m wavelength band (Thornton et al. 1998). All wave, setup, and alongshore current observations are based on hourly averages. In both cases the bathymetry is alongshore uniform, and the mean alongshore currents are consistent with 1D dynamics (Feddersen et al. 1998; Ruessink et al. 2001; Feddersen and Guza 2003).

### a. Duck94 example

During Duck94, there were no setup observations, so only the alongshore current inverse method is applied. Wave breaking occurs offshore of and on the crest (*x* = 110 m) of a well-developed sandbar (Figs. 7a,b). In the bar trough (40–80 m from the shoreline), the wave height remains constant. A tuned 1D wave model (without rollers) (e.g., Thornton and Guza 1983) accurately (rms error 2.2 cm) predicts the wave height evolution (solid curve in Fig. 7b). The wave model (initialized with offshore *H*_{rms} and *S*_{xy} estimated from an array of pressure sensors in 8-m water depth), together with *τ*^{wind}_{y}*F*^{(pr)}_{y}*c*^{(pr)}_{d}*υ*^{(pr)} peak magnitudes (Fig. 7d). The prior %*σ*_{fy}*σ*_{fy}*σ*_{cd}^{−4}, 30% of the prior *c*^{(pr)}_{d}*l*_{fy}*l*_{cd}*ν* = 0.5 m^{2} s^{−1} is that used by Ruessink et al. (2001) to model a larger dataset from which one of the case examples is drawn. Inverse solutions with *ν* ranging between 0.1 and 2 m^{2} s^{−1} were similar, with smoother inverse solutions for larger *ν* (not shown). For these *ν*, the magnitude of the lateral mixing term was small and did not qualitatively change the results. The prior *F*^{(pr)}_{y}*c*^{(pr)}_{d}*υ*^{(pr)} and its error bars (Fig. 7d). Typical of barred-beach *υ**υ*^{(pr)} rms errors (0.28 m s^{−1}) are substantial.

The *υ**σ*_{υd}^{−1}, yields solutions (Fig. 8) that pass the consistency tests and agree well with the *υ*^{−1}). The *υ**F*^{(i)}_{y}*x* ≥ 110 m), and is increased toward the trough (60 < *x* < 90 m), consistent with the concept of a wave roller (Fig. 8b). The slightly negative *F*^{(i)}_{y}*x* = 30 m indicates a reversal of forcing and, although consistent with the *f*_{y} error covariance, seems physically unrealistic (no mechanism for reversal is known). This may be the result of data noise mapped into the forcing correction. The *c*^{(i)}_{d}*c*^{(i)}_{d}*f*^{(i)}_{y}*f*^{(r)}_{y}*x* < 200 m) are quite similar (Fig. 8d), as is the increase in alongshore forcing in much of the bar trough (70 < *x* < 100 m). Within the %*σ*_{fy}*σ*_{cd}*f*^{(i)}_{y}*c*^{(i)}_{d}

### b. SandyDuck example

The SandyDuck case example does not have a well- developed bar (Fig. 9a). There is a steep slope region for 25 < *x* < 100 m and a nearly constant depth terrace for 100 < *x* < 200 m. Large waves begin breaking offshore of the terrace, have approximately constant height over the terrace, and then dissipate rapidly farther onshore on the steep slope (asterisks in Fig. 9b). A tuned 1D wave model (without rollers) accurately (rms error of 4 cm) predicts the wave height evolution (Fig. 9b). The wave model and observed wind give the prior *F*^{(pr)}_{x}*F*^{(pr)}_{y}*c*^{(pr)}_{d}*σ*_{fx}*σ*_{fy}*F*^{(pr)}_{x}*F*^{(pr)}_{y}*σ*_{cd}^{−4}. Large covariance parameters within the window of parameters giving consistent solutions are chosen (section 3d). Although there is no sandbar to set the length scales, *l*_{fx}*l*_{fy}*l*_{cd}*η*^{(pr)} and *υ*^{(pr)} with error bars (Figs. 9e,f). The errors in the (nonroller) prior model predictions of setup (rms error of 1 cm) and the alongshore current (rms error of 0.2 m s^{−1}) exceed those expected from instrument noise alone. The expected *η**σ*_{ηd}*σ*_{υd}^{−1}.

Consistent inverse solutions for *η*^{(i)} and *υ*^{(i)} (Figs. 10a and 11a) agree well with the data (rms errors of 3.9 mm and 3.3 cm s^{−1}, respectively) with significantly reduced uncertainties. Both the cross- and alongshore inverse forcing magnitudes are increased relative to the prior in the terrace region (100 < *x* < 200 m) and are reduced near *x* = 80 m (Figs. 10b and 11b). As with the Duck94 example, changes in the cross- and alongshore forcing are consistent with the roller concept. The inverse model makes *F*^{(i)}_{x}*x* < 100 m to match the setup observations in this region, consistent with observed wave shoaling at *x* = 100 m (Fig. 9b). The *c*^{(i)}_{d}*c*^{(i)}_{d}*x* < 90 m (Fig. 9b). The *c*^{(i)}_{d}*f*^{(r)}_{x}*f*^{(r)}_{y}*f*^{(i)}_{x}*f*^{(r)}_{x}*x* < 250 m (Fig. 10c). Offshore of *x* = 250 m, where there are no data, *f*^{(i)}_{x}*f*^{(r)}_{x}*f*^{(i)}_{y}*f*^{(r)}_{y}*x* < 200 m, although there is a factor of 2 difference in the magnitude of the *f*^{(i)}_{y}*f*^{(r)}_{y}*x* = 80 m (Fig. 11d).

## 5. Discussion

Overall, agreement between both the Duck94 and SandyDuck inverse forcing corrections and the roller model is remarkable, particularly because *f*^{(i)}_{x}*f*^{(i)}_{y}

Inferences can be drawn from the inverse-derived *c*_{d}. Assuming that the maximum (and minimum) *c*^{(i)}_{d}*c*_{d} is increased (reduced) from the prior *c*_{d} can be calculated to determine the statistical significance of *c*^{(i)}_{d}*c*_{d} is significantly increased (reduced) from the prior.

Hypotheses that *c*_{d} depends either on the bed roughness *k*_{rms} (e.g., Garcez-Faria et al. 1998) or on breaking- wave-generated turbulence (e.g., Church and Thornton 1993) are examined. Wave dissipation, a measure of the breaking-wave-generated turbulence source, is calculated from the modeled wave energy flux gradients in the region where *c*^{(i)}_{d}*c*_{d}. A relationship between wave dissipation and *c*^{(i)}_{d}*r* = 0.64 and *r* = 0.90 for Duck94 and SandyDuck, respectively), which is consistent with the hypothesis that increases in wave dissipation result in increased *c*_{d}. No explicit or implicit connection exists in the inverse method between *c*_{d} and wave dissipation. Bed roughness *k*_{rms} was estimated with eight fixed altimeters (Duck94) (Feddersen et al. 2003) and a towed altimeter (SandyDuck) (Gallagher et al. 2003). For the Duck94 example, *k*_{rms} varies between 1 and 7 cm, but for the SandyDuck example, the bed was smooth (*k*_{rms} < 2 cm). No relationship (i.e., statistically significant correlation) between *k*_{rms} and *c*^{(i)}_{d}*k*_{rms}/*h* and *c*^{(i)}_{d}*k*_{rms} have errors (Feddersen et al. 2003), the lack of a relationship suggests that bed roughness is not a primary factor in determining *c*_{d} (e.g., Feddersen et al. 2003).

The two inverse realizations presented here are insufficient to draw conclusions regarding forcing or *c*_{d} parameterizations. Many inverse realizations of the forcing correction and *c*_{d}, spanning a wide range of conditions, would allow statistical testing of wave-forcing or *c*_{d} hypotheses. Additional interpretations of the inverse solutions are possible. For example, the alongshore forcing error could be ascribed to tidal (e.g., Ruessink et al. 2001) or buoyancy (e.g., Lentz et al. 2003) forcing.

## 6. Summary

Uncertainties regarding wave-forcing and drag coefficient parameterizations in the nearshore have motivated development of an inverse method that combines dynamics and data to yield optimal estimates of the setup *η**υ**c*_{d}. The method also yields error bars (covariances) for the *η**υ**c*_{d} inverse solutions. Tests that determine the consistency of the inverse solutions with prior assumptions were presented. The inverse method was tested with a synthetic barred-beach example, and consistent inverse solutions reproduced well the specified true cross- and alongshore forcing and *c*_{d}.

The method was applied to two case examples from field experiments yielding inverse solutions that passed the consistency tests. The independently estimated cross- and alongshore inverse forcing corrections were similar to the modeled effect of wave rollers. The significant cross-shore variation of the inverse-derived *c*_{d} was related to variations in wave dissipation, but was not related to variation in the observed bed roughness. Although consistent with the hypothesis that breaking- wave-generated turbulence increases *c*_{d}, the two examples are not sufficient to examine this relationship statistically. Additional field cases spanning a wide range of nearshore conditions are needed to test hypotheses about the wave forcing and drag coefficient.

## Acknowledgments

This research was supported by ONR, NSF, NOPP, and WHOI. The Duck94 and SandyDuck instruments were constructed, deployed, and maintained by staff from the Integrative Oceanography Division of SIO. Britt Raubenheimer helped design and manage the SandyDuck field experiment and provided high-quality setup data. Tom Herbers provided wave observations, and Edie Gallagher provided SandyDuck bed roughness observations. The Field Research Facility, Coastal Engineering Research Center, Duck, North Carolina, provided excellent logistical support, the bathymetric surveys, and 8-m depth pressure array data. A summer course on inverse methods taught by A. F. Bennett served as a springboard for this work. Gene Terray provided important insight, and John Trowbridge, Dennis McGillicuddy, and two anonymous reviewers provided valuable feedback.

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## APPENDIX A

### Relating Cost-Function Weights to Covariances

*C*

^{−1}

_{fx}

*C*(

*x,*

*x*′) is decomposed into (cf. Courant and Hilbert 1953)

*ĉ*

_{l}and

*g*

_{l}(

*x*) are the eigenvalues (real and >0) and (orthogonal) eigenfunctions of

*C*(

*x,*

*x*′). Orthogonality is defined as

*L*is the domain size and

*δ*

_{lm}is the Kronecker delta function. For the set of continuous functions, the

*g*

_{l}(

*x*) provide a basis set. If

*C*(

*x,*

*x*′) were a function of

*x*−

*x*′ only (i.e., homogeneous), then the

*g*

_{l}are complex exponentials. The inverse of

*C*(

*x,*

*x*′) is

*f*(

*x*),

*f*(

*x*) and

*C*(

*x,*

*x*′) into a cost-function-type integral results in

*f̂*

_{l}with probability density function

*f*(

*x*) decomposed into an infinite sum of independent zero-mean Gaussian random variables

*f̂*

_{l}, it is straightforward to demonstrate that

*E*[

*f*(

*x*)] = 0:

*E*[

*f*(

*x*)] = 0 as well. The covariance of the random function

*f*(

*x*), defined as

*E*[

*f*(

*x*)

*f*(

*x*′)], is

*E*[

*f̂*

_{l}

*f̂*

_{m}],

*E*

*f*

*x*

*f*

*x*

*C*

*x,*

*x*

## APPENDIX B

### Consistency Checks with Prior Assumptions

*δ*

*υ*

*υ*

*η*

*δ*

*υ*

_{n}

*υ*

^{(i)}

*x*

_{n}

*d*

^{(υ)}

_{n}

*N*) samples from a zero-mean Gaussian random variable with prior data variance

*σ*

^{2}

_{υd}

*δ*

*υ*

*χ*

^{2}

_{N−1}

*y*) represents the location where the chi- squared cumulative distribution function (cdf) with

*N*− 1 degrees of freedom equals the probability

*y.*If this confidence interval does not contain the prior

*σ*

^{2}

_{υd}

*δ*

*υ*

*t*distribution with

*N*− 1 degrees of freedom. If the interval

*t*

_{M}(

*y*) is the location where the Student's

*t*cdf equals probability

*y*] does not contain zero, then the inverse solution is not consistent with the zero-mean data error and also should be rejected.

*c*

_{d}error. The continuous functions (e.g.,

*f*

_{x}) are decomposed into Fourier coefficients [e.g., (A1)] using the basis functions of their respective prior covariances. Each Fourier coefficient (

*f̂*

_{l}) is then a sample from a zero-mean Gaussian random variable with variance given by the prior covariance eigenvalue (i.e.,

*ĉ*

_{l}). If these hypotheses are correct, then the statistics (summed over the number of data

*N*)

^{N}

_{l=1}

*ĉ*

_{l}and a (

*N*− 1 degrees of freedom) chi-squared random variable, respectively. The significance tests described for the data are applied to test whether the inverse forcing error or

*c*

_{d}error are consistent with the prior assumptions. The sum is over

*N*(instead of ∞) because with finite data only a finite amount of information (i.e., approximately the first

*N*Fourier coefficients) is added.