Nearshore bathymetric inversion from video using a fully non-linear Boussinesq wave model

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1 Journal of Coastal Research SI 64 pg - pg In press ISSN Nearshore bathymetric inversion from video using a fully non-linear Boussinesq wave model R. Almar, R. Cienfuegos, P. A. Catalán, F. Birrien, B. Castelle and H. Michallet College of Engineering Pontificia Universidad Catolica de Chile, Santiago, Chile ralmar@ing.puc.cl racienfu@ing.puc.cl Departamento de Obras Civiles Universidad Tecnica Federico Santa Maria Valparaíso, Chile patricio.catalan@usm.cl UMR EPOC (Université Bordeaux1-CNRS) Bordeaux, France f.birrien@epoc.u-bordeaux1.fr b.castelle@epoc.u-bordeaux1.fr UMR LEGI (UJF-INPG-CNRS) Grenoble, France Herve.Michallet@hmg.inpg.fr ABSTRACT Almar, R., Cienfuegos, R., Catalán, P.A., Birrien, F., Castelle, B. and Michallet, H Nearshore bathymetric inversion from video using a fully non-linear Boussinesq wave model. Journal of Coastal Research, SI 64 (Proceedings of the 11th International Coastal Symposium), pg pg. Szczecin, Poland, ISSN This paper presents a new depth inversion methodology from video imagery. The strength of the method is the use of a fully non-linear Boussinesq wave model in combination with a very complete video-derived laboratory wave observation that includes period, celerity and wave height. Compared to the previous use of wave dispersion formulas, here the better description of wave dynamics substantially improves bathymetry estimation. For the considered laboratory case, error on bathymetry is as small as 8 % whereas other formulations, shallow water or non-linear derived solution can only attain 24 % and 14 %, respectively. More in-depth analysis on the error shows a fair sensitivity on video-derived breaker height and describes the large contribution of nonlinearities. The recent possibility of using Serre s dispersion relation in combination with video-derived wave height provides a reasonable performance and should be further envisaged for one-dimensional depth inversion. Future extensions of this work involve the use of a two-dimensional Boussinesq model to include more hydrodynamics processes such as wave-driven circulation over three-dimensional surfzone sandbars. ADDITIONAL INDEX WORDS: depth inversion algorithms, laboratory experiment, wave celerity, remote sensing, beach morphodynamics, Serre equations INTRODUCTION In the nearshore area, waves and wave-induced currents can drive significant and sometimes drastic sandy beach changes Predictive understanding of the evolving beach bathymetry remains one of the primary objectives of the nearshore community. However, a direct measurement is costly and difficult, particularly considering that the maximum of nearshore morphodynamic variability occurs during energetic wave conditions (i.e. storms). Low-cost alternatives based on remote video imagery have evolved significantly as tool for estimating beach morphology, from initial qualitative bathymetry assessment to more indirect and complex quantitative estimation (e.g. van Dongeren et al., 2008). One approach of these methods relies on assessing local wave characteristics to invert a linking relationship with the underlying bathymetry. The obvious choice was the linear dispersion relation, or slightly non-linear variations, that requires low level wave information: wave length, or celerity, and period, which could be video derived. However, this approach is largely hampered by the fact that wave dynamics in the nearshore can be far from linear, particularly for large waves propagating over complex bathymetries. Thus, using linear inversion can lead to substantial errors (up to 20 %) on bathymetry estimation (e.g. Grilli, 1998, Catalan and Haller, 2008; Birrien et al., in this issue) when nonlinearity becomes significant. Non-linear inversion schemes might significantly improve this estimation but additional information on wave height evolution is required (Catalan and Haller, 2008). On the other hand, with the reduction of computational times, phase resolving non-linear wave models become a potentially more suitable alternative for inversion of wave characteristics into bathymetry. One of the limitations is the need of higher level local wave information, which is difficult to obtain from video images. In this paper the first non-linear bathymetric inversion scheme is proposed, one that combines various video-derived wave properties (including wave height estimation from video) with a fully non-linear one-dimensional (1D) wave model with application to a laboratory experiment. LABORATORY DATA A large scale laboratory experiment was undertaken in 2008 at the multidirectional basin of the SOGREAH (LHF facility, France; Michallet et al., 2010, Castelle et al., 2010). The basin extent is 30 m in both cross-shore (X = 0 at the wavemaker) and alongshore (Y = 0 at the camera basin border) directions and comprises 60 independently-controlled piston-type wavemakers (Figure 1.a). For the objectives of the experiment, only shorenormal waves were considered, where each run consisted of 1- hour period of irregular waves complying to a JONSWAP spectrum. The bathymetry consisted of a sandy moveable bed resulting in the formation of nature-like three-dimensional surfzone sandbars. In this study, we focus on a single run with a significant wave height H s = 18 cm and peak period T p = 3.5 s. Free surface displacements were measured using 18 capacitive gages deployed every 1 m on a movable structure extending in the cross-shore direction. Acquisition frequency was set to 50 Hz. We

2 Non-linear bathymetric inversion from video imagery using 29 ground survey points (Holland et al., 1997) after a correction of the lens radial distortion. Although varying somewhat throughout the field of view, the cross-shore horizontal footprint was less than 15 cm in the region of interest. Video timestack images were aligned with cross-shore rods (Figure 1.b). The lighting of this indoor experiment was chosen to be diffusive thus allowing the identification of the wave trajectory before and after breaking. Validation of video-derived T, C and Hb for this experiment is not presented here but was previously undertaken and detailed in Almar et al. (submitted). RMS errors on T and C are close to 10 % and around 25 % for Hb. INVERSION METHOD The two major progresses of the methodology are (1) the use of a non-linear wave phase-resolving model and (2) the use of a large variety of remotely-derived wave characteristics. Bathymetric inversion scheme is based on iterative convergence between video-obtained and model output wave data. Model features The SERR1D model (Cienfuegos et al., 2010) is used to estimate the wave properties. This one-dimensional model is based on a full non-linear version of the Boussinesq equations (Serre equations, Serre, 1953) and includes a parameterization to account for breaking-waves dissipation. Previous validation tests indicated a good model performance to represent nearshore hydrodynamics, including bathymetric-induced wave transformation. Video wave properties estimation methods Various estimates of wave characteristics were obtained from video timestacks images. Namely, the time-averaged wave period (T) and celerity (C) are obtained following the methodology detailed in Almar et al. (2008) where C is resolved by space-time cross-correlations. Recent developments (Almar et al., submitted) have made possible to access for the first time, wave-by-wave breaking characteristics such as the breakpoint position Xb and the breaker height Hb, as a key parameter. These parameters are derived from the optical intensity signature at the breaking inception. The resulting set of video-obtained wave data is the richest used to date and allows in-depth comparison with model output. Iterative convergence scheme Figure 1. (a) Schematic of the laboratory experiment setup. (b) Three-dimensional bathymetry. Black line indicates bar (Y = 18 m) and rip (Y = 19 m) cross-shore studied transects. (c) Oblique video image of the wave basin. Crosses stand for ground points used for image rectification. focus on two contrasted transects. The first is located at Y = 18 m, in a rather monotonic bathymetric area and referred herein to as a bar profile (Figure 1.c). The second is located in the rip channel, at Y = 12 m (referred as rip profile). A 720x576-pixel video camera was set up on the basin corner (Figure 1.c). Video grabbed wave-runs at 25 frames/sec rate and was post-synchronized with the in-situ free surface data. Rectification of images from pixel coordinates into real world coordinates was accomplished by direct linear transformation The model hydrodynamic outputs are compared with the video observations where the model input parameters are iteratively updated until convergence is obtained. Matching is performed on C and Hb within a specified zone of interest. The upgrading of input parameters concerns offshore wave height Ho and bathymetry h. As shown in Figure 2, the depth inversion algorithm first estimates wave properties (T, C, Xb and Hb) from video imagery using the methods previously described. A perfect estimation (i.e. accurate, unbiased) is considered in the method. The algorithm is initiated (part 2.1 in Figure 2) with Ho first approximated by Hb and bathymetry linearly estimated from the linear dispersion relation C = g tanh kh, k being the wavelength. Using these k parameters, the SERR1D model is started (part 2.2 in Figure 2) and h is updated iteratively until minimizing the error between Cmod and Cvid. Resulting h is set as input in a phase-averaged wave

3 Almar et al. energy conservation model including breaking (based on Battjes and Janssen, 1978). The value of the offshore boundary condition, H o, is iteratively updated until matching between the observed H b and modeled H b at X b. The new guess for H o is then re-injected as input for the Boussinesq model and cycle (Steps 2.2 and 2.3) is achieved. This is generally obtained within a few iterations (4-5). The performance of the proposed algorithm is compared to the commonly used Modified Shallow-Water inversion celerity (MSW), C = 1. 2 gh, and with the non-linear dispersion relation for the Serre equations over horizontal bottom (Carter and Cienfuegos, in press). In the rip channel (Figure 3.c), matching of H b = 12 cm at X b = 12.8 m produces a correct adjustment of H over the profile, with larger underestimation (30 %) in the surf zone than on H o. Difficulties in the estimation are partially due to video estimation of H b but also may come from supposed significant wave-current interactions on this profile, because of the presence of an intense rip current (details on the measured rip current circulations are given in Castelle et al., 2010). The measured bathymetry (Figure 3.d) presents a barred profile that is weakly reproduced by the method while other methods totally smooth the bar. Similarly to the results across the bar profile, the method performance is better than results found by others. Mean RMS errors are 7, 24 and 13 % for our method, MSW and Serre respectively. For the bar case, the observed errors have the same order of magnitude. Figure 2. Description of the iterative scheme for bathymetry inversion. This method is fully remote video-based. Inputs are wave observations and outputs are non-linear bathymetry (h NL ) and offshore wave height (H o ). RESULTS AND DISCUSSION The method was applied to both, the rip and bar profiles. Results were compared to other bathymetric estimators and measured data. In order to reduce computation time and to conserve statistical validity of wave properties (JONSWAP spectrum), the Boussinesq model input wave time series approximated a sinusoidal wave with H 0 and T p. Figure 3 presents the estimations for wave height, H, and bathymetry, h. Over the bar (Figure 3.a), the wave height adjusted to match the video estimated H b = 11 cm at X b =12.34 m (step 2.3 Figure 2) and is in agreement with the measured profile. Overestimation of H o at X = 5 m is 8 %. Using these estimated wave characteristics, related h estimation is shown in Figure 3.b. The method is closer to the measured h than other estimators, MSW and Serre, with mean RMS errors of 8, 27 and 14 % respectively. Figure 3. Bar transect (Y=18 m): (a) Profiles of measured (+) and estimated (continuous) H. (b) Bathymetric profiles: measured (thick line), method output (+), Serre (solid), and MSW (dashed). Profiles in (c), H, and (d), h, are equivalent to (a) and (b) but correspond to the rip transect (Y = 12 m). In more detail was explored the behavior of the resulting errors on h and the supposed contribution of non-linearities. Of note, whereas predictors behave similarly in the shoaling zone for the bar and rip cases, a contrasting behavior is observed in the surf zone (Figure 4.a). Depending on the predictor, an over-estimation between 17 to 39 % is observed for the bar case and underestimation from, 13 to 35 % for the rip case. As previously mentioned, this may result from observed 2D circulation (Castelle et al., 2010), that is on- and offshore oriented flow across the bar and in the rip channel, respectively. As a reference, mean surface currents of 0.1 m/s (offshore oriented) were measured in the rip. While compared to C values of about 1 m/s typically measured for

4 Non-linear bathymetric inversion from video imagery Figure 4. (a) Cross-shore variation of error (% of measured h) for different estimators, MSW (dashed line), Serre (solid line) and our method (crosses), for both bar (thick) and rip cases (thin). In (b) is estimated the contribution of non-linear effects as difference (in % to h) between our method and linear theory. Thick lines stand for the bar case and thin ones for the rip case. theses depths, this represents a deviation of 10 % that is not taken into account by the depth estimator schemes. This argues in favor of a future complete two-dimensional bathymetric inversion method that would include wave-current interactions. In Figure 4.b the evolution of the difference on h between linear and fully non-linear methods is depicted. The cross-shore evolution of this difference points out the local contribution of non-linearities. For the presented laboratory case, differences can be large with a contrasting behavior between shoaling (up to -17 %) and surf zone (up to 75 %). This is consistent with real beach observations (Tissier et al., in press) where large H/h values in the inner surf zone were associated to much larger bore velocities (up to 2 times) than predicted by linear theory. Video-derived H b is associated to a 25 % uncertainty (Almar et al., submitted). This uncertainty is further propagated to the inverted h. In order to assess the sensibility of our method to H b, an analysis has been undertaken. Error on estimated h has been computed for various H o, values, varying from 2 cm to 20 cm. Results show (Figure 5) that for both bar and rip profiles and for all H o values, our method is the most accurate. Lowest errors (7-8 %) are associated to measured H o = 10.5 cm mean value. Interestingly, this analysis also reveals the utility of the analytic Serre formulation (Carter and Cienfuegos, in press) for depth inversion. Serre estimation converges to our method for small H o and to MSW for larger H o, which corresponds to larger surf zone extensions. CONCLUSIONS This paper shows the advantages of using a fully non-linear Boussinesq model to inverse bathymetry from video imagery. The use of such a model, together with recent progresses to obtain additional video-derived wave parameters (T, C and H b ) allows a better description of wave dynamics and for this reason provides satisfying bathymetry estimation. Figure 5. Sensibility of bathymetric estimators on offshore wave height (H o ) for MSW (dashed line), Serre (solid line) and our method (crosses), for both the bar (thick) and rip cases (thin). A new methodology is presented together with its validation for a three-dimensional beach laboratory experiment. Results for the two studied bar and rip transects indicate a better performance of the method (RMS error 8 %) than other estimators: the modified shallow water celerity (25 %) and the non-linear Serre dispersion relation (14 %). The improvement obtained by using the method is not substantial in the surf-zone but is large in the shoaling zone. Through additional analysis on the errors origin, is showed the weak sensibility on video-obtained breaker height uncertainty. Difference between linear and non linear depth estimations underlines a large contribution of non-linearities and clearly indicates its cross-shore variation. Among other results, using the Serre s dispersion relation provides a significant reduction of the error when compared to linear or weakly non-linear theories. Hence, using this relation should be further envisaged for onedimensional depth inversion. Future extensions of this work involve the use of a two-dimensional Boussinesq model to incorporate other hydrodynamic processes in the inversion method such as wave-current interaction. LITERATURE CITED Almar, R., Bonneton, P., Senechal, N. and Roelvink, D., Wave celerity from video imaging: a new method, International Conference on Coastal Engineering, 1(5), Almar, R., Cienfuegos, R., Catalan, P., Michallet, H., Castelle, B., Bonneton, P. and Marieu, V., submitted. A new breaking wave height direct estimator from video imagery. Submitted to Coastal Engineering. Battjes, J. A. and.janssen, J. P. F. M., Energy loss and setup due to breaking of random waves. International Conference on Coastal Engineering, Birrien, F., Castelle, B., Marieu, V., Almar, R. and Michallet, H., in this issue. Application of a data-model assimilation method to a 3D surf zone sandbar physical experiment. Journal of Coastal Research, SI 64 Carter, J.D. and Cienfuegos, R., The kinematics and stability of solitary and cnoidal wave solutions of the Serre

5 Almar et al. equations. Accepted for publication in the European Journal of Mechanics/B Fluids Castelle, B., Michallet, H., Marieu, V., Leckler, F., Dubardier, B., Lambert, A., Berni, C., Bonneton, P., Barthélemy, E. And Bouchette, F., Laboratory experiment on rip current circulations over a moveable bed: drifter measurements. Journal of Geophysical Research, 115, C12008, doi: /2010jc Catalán, P., and Haller, M., Remote sensing of breaking wave phase speeds with application to nonlinear depth inversion, Coastal Engineering, 55, Cienfuegos, R., Barthelemy, E. and Bonneton, P., Wave- Breaking Model for Boussinesq-Type Equations Including Roller Effects in the Mass Conservation Equation. Journal of Waterway, Port, Coastal, and Ocean Engineering, 136(1), van Dongeren, A., Plant, N., Cohen, A., Roelvink, D., Haller, M. and Catalán, P., Beach Wizard: Nearshore bathymetry estimation through assimilation of model computations and remote observations, Coastal Engineering, 55(12), Grilli, S.T., and Skourup, J., Depth inversion for nonlinear waves shoaling over a barred-beach, International Conference on Coastal Engineering, Holland, K.T., Holman, R.A, Lippmann, T.C., Stanley, J. and Plant, N., Practical use of video imagery in nearshore oceanographic field studies. Oceanic Engineering 22, 1, Michallet H., Castelle B., Bouchette F., Lambert A., Berni C., Barthélemy E., Bonneton P., Sous D Modélisation de la morphodynamique d une plage barrée tridimensionnelle, Proceedings of XIe Journées Nationales Génie Côtier Génie Civil, Editions Paralia CFL, Sables d Olonnes, France, June 2010, Serre, F Contribution à l étude des écoulements permanents et variables dans les canaux. Houille Blanche, 8, Tissier, M., Bonneton, P., Almar, R., Castelle, B., Bonneton, N. and Nahon, A., in press. Field measurements and non-linear prediction of wave celerity in the surf zone. Eur. J. Mech. B Fluids, doi: /j.euromechflu ACKNOWLEDGEMENT The laboratory work was undertaken within the framework of the Project MODLIT (RELIEFS/INU, SHOM-DGA). R.A. funded by project FONDECYT Nº B.C. acknowledges financial support from BARBEC (ANR N 2010 JCJC ). R.C. acknowledges financial support from ECOS-CONICYT Nº C07U01.