Journal of Coastal Research
SI 64
pg - pg
In press
ISSN 0749-0208
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
[email protected]
[email protected]
‡ Departamento de Obras Civiles
Universidad Tecnica Federico
Santa Maria
Valparaíso, Chile
[email protected]
∞ UMR EPOC
(Université Bordeaux1-CNRS)
Bordeaux, France
[email protected]
[email protected]
§ UMR LEGI
(UJF-INPG-CNRS)
Grenoble, France
[email protected]
ABSTRACT
Almar, R., Cienfuegos, R., Catalán, P.A., Birrien, F., Castelle, B. and Michallet, H. 2011. 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 0749-0208
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 1hour 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 Hs = 18 cm and peak period Tp = 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
Journal of Coastal Research, Special Issue 64, 2011
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
Journal of Coastal Research, Special Issue 64, 2011
Almar et al.
energy conservation model including breaking (based on Battjes
and Janssen, 1978). The value of the offshore boundary condition,
Ho, is iteratively updated until matching between the observed Hb
and modeled Hb at Xb. The new ‘guess’ for Ho 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 Hb = 12 cm at Xb =
12.8 m produces a correct adjustment of H over the profile, with
larger underestimation (30 %) in the surf zone than on Ho.
Difficulties in the estimation are partially due to video estimation
of Hb 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 (hNL)
and offshore wave height (Ho).
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 H0 and Tp.
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 Hb = 11 cm at Xb =12.34 m (step 2.3
– Figure 2) and is in agreement with the measured profile.
Overestimation of Ho 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
Journal of Coastal Research, Special Issue 64, 2011
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 Hb 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 Hb,
an analysis has been undertaken. Error on estimated h has been
computed for various Ho, values, varying from 2 cm to 20 cm.
Results show (Figure 5) that for both bar and rip profiles and for
all Ho values, our method is the most accurate. Lowest errors (7 8 %) are associated to measured Ho = 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 Ho
and to MSW for larger Ho, 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 Hb) 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 (Ho) 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., 2008.
Wave celerity from video imaging: a new method,
International Conference on Coastal Engineering, 1(5), 661673
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., 1978. Energy loss and setup due to breaking of random waves. International
Conference on Coastal Engineering, 569-578.
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., 2010. The kinematics and
stability of solitary and cnoidal wave solutions of the Serre
Journal of Coastal Research, Special Issue 64, 2011
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., 2010. Laboratory experiment on rip current
circulations over a moveable bed: drifter measurements.
Journal of Geophysical Research, 115, C12008,
doi:10.1029/2010JC006343.
Catalán, P., and Haller, M., 2008. Remote sensing of breaking
wave phase speeds with application to nonlinear depth
inversion, Coastal Engineering, 55, 93-111.
Cienfuegos, R., Barthelemy, E. and Bonneton, P., 2010. WaveBreaking Model for Boussinesq-Type Equations Including
Roller Effects in the Mass Conservation Equation. Journal of
Waterway, Port, Coastal, and Ocean Engineering, 136(1), 1026
van Dongeren, A., Plant, N., Cohen, A., Roelvink, D., Haller, M.
and Catalán, P., 2008. Beach Wizard: Nearshore bathymetry
estimation through assimilation of model computations and
remote observations, Coastal Engineering, 55(12), 1016-1027.
Grilli, S.T., and Skourup, J., 1998. Depth inversion for nonlinear
waves shoaling over a barred-beach, International Conference
on Coastal Engineering, 603-616.
Holland, K.T., Holman, R.A, Lippmann, T.C., Stanley, J. and
Plant, N., 1997. Practical use of video imagery in nearshore
oceanographic field studies. Oceanic Engineering 22, 1, 81-92.
Michallet H., Castelle B., Bouchette F., Lambert A., Berni C.,
Barthélemy E., Bonneton P., Sous D. 2010. 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, 379-386.
Serre, F. 1953. Contribution à l’étude des écoulements permanents
et variables dans les canaux. Houille Blanche, 8, 374-388.
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: 10.1016/j.euromechflu.2010.11. 003.
ACKNOWLEDGEMENT
The laboratory work was undertaken within the framework of
the Project MODLIT (RELIEFS/INU, SHOM-DGA). R.A. funded
by project FONDECYT Nº 3110030. B.C. acknowledges financial
support from BARBEC (ANR N° 2010 JCJC 602 01). R.C.
acknowledges financial support from ECOS-CONICYT Nº
C07U01.
Journal of Coastal Research, Special Issue 64, 2011