Bathymetry from inverse wave refraction

Article reviewed by

It is possible to determine the bathymetry of a certain area using radar data. On the Island of Sylt at the German Bight Coast, measurement are done during storm conditions. This data is processed based on inversion of the non-linear and linear wave theory. More about the area of investigation, data processing, result and discussion of the results can be found below.

Introduction

The determination of the bathymetry in coastal environments by utilizing the ocean wave-shoaling photographic imagery, and the observed reduction of ocean wave phase speed with decreased water depth, is used since the WW-II (Williams 1946)[1]. The last decade, with the expansion of different ground based instrumentations, mainly radar and video imagery, for the observation of the sea surface and the exponential increase of the computational power, several methodologies for the bathymetry reckoning have been published, e.g. Bell 1999[2], Seemann et al. 1999[3], Stockdon and Holman 2000[4], Dankert 2003[5], Bell et al. 2004[6], Catalan and Haller 2008[7], Senet et al. 2008[8]. The core of the previously mentioned methods is the inversion of the wave characteristics by assuming the validity of linear or non-linear models for the propagation of the wavefield over uneven sea bottom.

In the present investigation, twelve hourly radar datasets acquired during storm conditions are analyzed by two methods: The non-linear method of Bell et al. 2004[6] (henceforth BW04), which is based on the inversion of the non-linear wave dispersion equation of Hedges (1976)[9] and the Dispersive Surface Classificator (henceforth DiSC08), Senet et al. 2008[8], which is based on the inversion of the linear wave theory. The results are validated as bathymetric retrieving instruments and the two wave propagation theories are compared about their sensitivity to the local bathymetric relief. The two methods are compared under the assumption of fundamentally similar implemented algorithms.

Area of investigation and data acquisition

The radar was mounted close to the lighthouse List West on the Island of Sylt in the German Bight. The acquired radar images cover the tidal inlet at List West, the Lister Landtief (maximum depth 11m), and parts of the Lister Tief (maximum depth 40m). The coastal waters around List West were observed because of the extremely high morphodynamic activity. A sand bar between the Lister Landtief and the Lister Tief is in the process of breaking, which leads in a dramatically change of the local tidal-stream situation; the tidal range of the area is 2m.

Figure 1. Bathymetry of area of investigation acquired by multibeam echosounder.

The instrument used for the acquisition of sea-surface image sequences is a ground-based nautical X-band radar with horizontal polarization. The device utilized during the experiments is a software–hardware combination consisting of a commercial, navigational Furuno X-band radar antenna and radar device, a WaMoS II analog–digital converter and acquisition of radar image sequences software. The radar was mounted from 1996 to 2007; the image sequences of the present investigation were recorded hourly from 23:00 UTC on 26.08.2003 to 10:00 UTC on 27.08.2003. During the data acquisition the minimum and maximum wind speed were 12m/s and 17m/s, respectively and the minimum and maximum significant wave height, 1.2m and 1.6m respectively, as measured by deployed instruments in the area of investigation. For the determination of datum, tidal-gauge-measurements were also used. A dataset of the bathymetry of part of the radar-covered area was acquired by multibeam echo sounder (by KOK/GKSS) during 25th of August 2003 (Fig. 1); the spatial resolution of the bathymetric data set is 2m x 2m and has been averaged over the grid of the radar results.

Data analysis

The BW04 and DiSC methods have common steps of analysis which are presented shortly in the current paragraph. The analysis of both methods begins with the conversion from the polar coordinates in which the raw data is recorded to a georeferenced Cartesian grid. The location of the transmitter and hence the origins of the polar conversions were determined by DGPS. A Fourier transform was carried out on each pixel through time in the image sequences to isolate individual wave frequencies. Each frequency layer within the transform was then analyzed to map the variations in wavelength across the area viewed by the radars using a discrete 2-D Fourier transform technique that isolates the strongest wave signal in the sub-images. Small subsections of the Fourier layer were used for this analysis. The wavelengths calculated from this analysis were then used in a least squares fit to find the water depth at each pixel, calculated both from linear wave theory (DiSC) and from a non-linear wave dispersion equation (BW04).

Results

Figure 2. Averaged bathymetry over 12h occurred by BW04 method.
Figure 3. The time series of water level calculated with BW04, DiSC in the deepest part of the tidal channel against the measured water level.

The results of the analysis are 12 hourly bathymetries from both methods, each of them have been averaged (Fig. 2) to increase their significance. The spatial resolution for BW04 is 60m and for the DiSC is 40m. The time series of instant water level in the deepest point sub-tracted from the mean of each time series has been plotted against the tidal gauge (Fig. 3). The water depths mapped from this analysis were compared to the survey data by computing the relative error of the BW04 and DiSC method, Fig. 4 and Fig. 6 respectively.

Figure 4. Map of the relative error of the BW04 method.
Figure 5. Scatter plot of the BW04 depth compared to the surveyed depth.
Figure 6. Map of the relative error of the DiSC method.

Finally the depths from the two inversion meth-ods with the measured tidal level added com-pared to echosounder’s data and the results plot-ted against each other.

Figure 7. Scatter plot of the DiSC depth compared to the surveyed depth.

Discussion

The comparison of the hourly results with the tidal gauge demonstrates that although none of the two methods is so precise at the determination of the water level, the results BW04 has higher correlation with the measured water level, due to the used model. The Hedge’s dispersion function includes the significant wave height; therefore the “noise” of the actual wave conditions is reduced in comparison of the linear dispersion used in DiSC. The scatter plots for both methods demonstrate clustering of the results.

The first cluster lays between 4.5m (the lower limit depends on the available echosoundings bathymetry) and 10m for BW04 and 12m for DiSC. At this cluster both methods present significant correlation (1 by 1) with the surveyed bathymetry. The second cluster is formed from the depths higher than 12m, where there is a clear underestimation of the depth due to the actual wave length. The underestimation of the deeper areas is the main reason for the inclination of the trend lines from the y=x line. The spatial distribution of the error illustrates the main sources of the error. In both cases, the bathymetry for deeper areas than 12m is underestimated due to the lack of long waves. The bathymetric gradient is the second source of the error. Both methods fail to determine the bathymetry over the steep sides of the ship channel and at the areas with sand dunes, this is obvious by comparing the map of the high resolution survey (Fig. 1) with the error maps. The BW04 approach has higher accuracy to the shoals due to the adapted wave model, but the mean accuracy of the DiSC is higher (approximately 90%) than the BW04 (more than 80%).

The present investigation proved that in principle, the inversion of the wave field propagation for the determination of the depth is a reasonable method with significant results and it could be used in operational base at hot spots of crucial sediment motion, such as Sylt.