Reconstructing Bathymetry from Experimental Wave Data using PDE-Constrained Optimization
แนวคิดหลัก
PDE-constrained optimization can be used to reconstruct a Gaussian-shaped bathymetry in a wave flume from point measurements of the water surface elevation.
บทคัดย่อ
The paper presents a study on using PDE-constrained optimization to reconstruct the bathymetry of a wave flume from experimental data. The authors use the shallow water equations as the forward model and derive the continuous adjoint equations to compute the gradient for a gradient descent optimization.
The study is conducted in two parts:
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Reconstruction from simulated observations: The authors first test the approach using simulated data generated by the forward model. They analyze the reconstruction quality with and without added noise in the observations. The results show that the approach can provide a qualitatively correct reconstruction, with the maximum of the bathymetry placed approximately at the right position, even though the height is underestimated.
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Reconstruction from experimental observations: The authors then apply the approach to real experimental data obtained from a wave flume experiment. They use measurements from up to three sensors to reconstruct the bathymetry. The results are comparable to the simulated case, with the maximum of the bathymetry located correctly, but the overall shape being somewhat smeared out. The normalized root mean square error (NRMSE) of the reconstructions from experimental data is around 14%, which is in line with recent results from the literature using machine learning approaches.
The authors also analyze the sensitivity of the reconstruction quality to the number and placement of sensors, finding that using only two sensors can provide results almost as accurate as using three sensors.
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Bathymetry reconstruction from experimental data using PDE-constrained optimisation
สถิติ
The maximum differences in the water height with and without bathymetry are around 1.5 mm at sensor 2 and 3 mm at sensors 3 and 4.
The relative ℓ2-errors between simulated and measured time series at sensor positions 2, 3, 4 are around 1.7 × 10^-3, 1.7 × 10^-3 and 1.5 × 10^-3.
คำพูด
"While theoretical studies of the efficacy of such a PDE-constrained optimisation approach for bathymetry reconstruction exist, there seem to be few publications that study its application to data obtained from real-world measurements."
"As Khan et al. [9] state in their 2021 paper on bathymetry reconstruction, 'while the theoretical results from such models are promising, their applicability to real-world measurements has not been established'."
สอบถามเพิ่มเติม
How could the reconstruction approach be extended to handle more complex bathymetry shapes beyond the Gaussian-like profile studied here?
To handle more complex bathymetry shapes, the reconstruction approach could be extended by incorporating additional parameters or basis functions to represent the bathymetry. Instead of assuming a simple Gaussian-like profile, a more flexible representation could be used, such as a piecewise linear or polynomial function. This would allow the model to capture more intricate bathymetry features, such as multiple peaks, valleys, or irregular shapes. By increasing the complexity of the representation, the optimization algorithm would have more degrees of freedom to adjust the bathymetry to fit the observed data accurately.
What are the limitations of the shallow water equation model, and how could the approach be improved by using a more sophisticated flow model?
The shallow water equation (SWE) model, while computationally efficient, has limitations in capturing complex flow phenomena accurately. One major limitation is its inability to account for dispersive effects, which can lead to inaccuracies in wave propagation and reflection. Additionally, the SWE assumes hydrostatic pressure distribution and neglects vertical velocity variations, which may be significant in certain scenarios.
To improve the approach, a more sophisticated flow model, such as the Navier-Stokes equations or a higher-order shallow water model, could be used. These models would better capture the dynamics of the flow, including turbulence, viscosity, and non-hydrostatic effects. By incorporating a more detailed flow model, the reconstruction algorithm would be able to account for a wider range of physical processes and provide more accurate bathymetry reconstructions.
What other types of wave or flow measurements, beyond just surface elevation, could be incorporated into the optimization framework to further improve the bathymetry reconstruction?
In addition to surface elevation measurements, other types of wave or flow measurements could be incorporated into the optimization framework to enhance bathymetry reconstruction. Some examples include:
Velocity Measurements: Data on water velocity at different depths and locations could provide valuable information about flow patterns and dynamics, which could help refine the bathymetry reconstruction.
Pressure Measurements: Pressure sensors placed at various points in the water column could offer insights into the hydrostatic pressure distribution, which is influenced by bathymetry. Integrating pressure measurements into the optimization framework could improve the accuracy of the reconstructed bathymetry.
Acoustic Data: Sonar or acoustic measurements can provide information about the seafloor characteristics and depth, offering an alternative source of data for bathymetry reconstruction. By combining acoustic data with surface wave measurements, a more comprehensive picture of the underwater topography could be obtained.
