(Note: The following tests are described in greater detail in: Orzech et al. (2013), A variational assimilation system for nearshore wave modeling, J. Atm. & Oceanic Tech., DOI: 10.1174/JTECH-D-12-00097.1.)
A limited version of SWANFAR® (stationary mode, no sources or sinks) is compared to a previously developed linear, stationary assimilation system based on an analytical adjoint (Walker, 2006). A twin experiment format is utilized on a measured bathymetry from Santa Rosa Island, Florida. Artificial "observed" spectra are generated by fully nonlinear forward SWAN at the four locations indicated in the map below (TA1, TA2, SAB, SIB).
In each test, the SWANFAR® and Walker (2006) systems are each provided with the observed spectra at one of the 4 locations. The systems are both forced to adopt a zero-energy "first guess" spectrum at the observation location (TA1) and then run to convergence, tasked with recapturing the observed quasi-spectra at all four observation locations as closely as possible. System performance is measured by (1) qualitative comparison of estimated spectra with observed values at each location, and (2) computing RMS skill scores to cumulatively compare energy levels in all spectral bins. The skill score is computed as follows:
in which E denotes spectral energy level in each bin, mod indicates SWANFAR® estimated values, and obs indicates SWAN-generated observations. A perfect spectral match will earn a skill score of 1, while a very poor comparison can result in a score of zero or less.
The figure below shows an example of the spectral results from one test case in which the numeric (SWANFAR®) and analytic (Walker) adjoint-based systems were initialized with data at location TA1. Final estimated spectra from each system are shown in the top two panels of each four-panel group. Difference spectra (observed minus estimated) are shown in the bottom two panels of each group. (Click on figure to see a larger version.)
A total of 64 tests of this type were conducted, using varied conditions. RMS skill scores for the limited SWANFAR® system ranged from 0.82 to 0.98, with a mean skill of 0.91. Skill scores for the Walker adjoint system ranged from 0.78 to 0.97, with the same mean skill of 0.91. This was considered a validation of the basic numerical-adjoint method used in construction of the limited SWANFAR® assimilation system.
Following the above comparisons to the analytically based assimilation system of Walker (2006), the limited SWANFAR® system was tested with a second set of twin experiments at Duck, NC. In this case, the system was tasked with recapturing actual measured wave spectra from up to five nearshore instruments. These instruments include four Nortek Acoustic Wave and Current profilers (AWACs) and the FRF's 8 m Array of fifteeen near-bottom pressure sensors (along with the 3km Waverider buoy used for forward model initialization) and are circled on the following map (Figure courtesy of USACE Field Research Facility).
The assimilation system was initialized with data from different combinations of the five nearshore instruments, in a total of 9 different twin experiment cases, as tabulated below ("O.W." indicates over-weighting of specified instruments).
System performance was measured by comparing observed spectral wave properties at all five locations for all cases with SWANFAR®-estimated values. These results are summarized in the figure below (click on figure for enlarged version).
While correlations are fairly good for wave heights, directions, and periods, estimated and observed directional spreads are poorly correlated. The underestimation of larger wave heights may be due in part to nonlinear effects that were neglected in this more limited stationary, homogeneous system. Directional spreading errors, in contrast, stem largely from differences in how wave spectra are calculated at the AWACs versus the 8-m array. The following figure compares a spectrum from the FRF's 8-m array (top left) with a concurrent spectrum from the the 8-m-depth AWAC (bottom left). Also shown are assimilation system outputs at each location (middle panels) for Case 1 (i.e., system initialized at the 8-m array). A comparison of these results in frequency space alone is provided in the right two panels.
Although the two instruments are nearly collocated, the spectrum at the 8-m AWAC is directionally much broader than that at the 8-m array. This is a consequence of the different processing techniques used to obtain each spectrum. The assimilation system cannot account for such a difference. Thus, following initialization at the 8-m array, its estimated spectrum at the AWAC location (lower right panel) does not exhibit the spreading seen from the AWAC and instead closely resembles the nearby 8-m array spectrum.
It is not yet known how SWANFAR® might be modified to "homogenize" wave spectra assimilated from different instrument types. In such circumstances, the system may be limited to assimilating spectral statistics (e.g., wave height, mean period, etc.) or 1-D frequency spectra instead of complete 2-D frequency-directional spectra. As illustrated in the right column above, the system attains a much higher skill in recapturing the 1-D frequency spectrum at the 8-m AWAC (i.e., skill=0.79) than it did in matching the 2-D AWAC spectrum (i.e., skill=0.26).
When the actual measured spectra from each of the five instruments are replaced by quasi-observations, generated by fully nonlinear forward SWAN, the assimilation system performance in the nine twin experiment cases improves significantly, as shown below (click on image for larger version).
As would be anticipated, correlations for all statistics improve. Results for directional spread show the greatest improvement, likely a direct consequence of replacing multiple instrument types with a single, uniform pseudo-instrument at each location.