Variational assimilation of surface wave data for bathymetry reconstruction. Part I: algorithm and test cases

R. A. Khan; N. K.-R. Kevlahan

doi:10.1080/16000870.2021.1976907

Variational assimilation of surface wave data for bathymetry reconstruction. Part I: algorithm and test cases

Tellus

Volume 73 (2021): Issue 1

By: R. A. Khan and N. K.-R. Kevlahan

Open Access

|Jan 2021

Figures & Tables

(a–c) Three test cases for bathymetry $β (x)$ and free surface perturbation initial conditions $η (x, 0)$ for the data assimilation scheme. The green circles represent the locations of the observations, with *N_obs* = 5 and $y_{1}^{(o)} = 0.1 L .$ Note that while the spatial distribution is correct, amplitude of the initial conditions $\hat{η},$ amplitude of the bathymetry $\hat{β},$ and average depth H are not to scale in these diagrams, as $\hat{η}$ was restricted to 1% of $\hat{β}$ across most of the numerical tests. Plots (d–f) show the propagating free surface wave at t = 1.95 with flat bottom (blue) and bathymetry (red) for each Cases I, II and III, respectively, to highlight the effect of bathymetry on surface wave propagation.

Table 1.

Notation used in the derivation of data assimilation scheme of the SWE to find the optimal bathymetry, using same format as given in Table 1 of Kevlahan et al. (2019).

SymbolDefinition

η (x, t)

General solution for the height perturbation

ϕ (x)

General initial condition, i.e.

ϕ (x) : = η (x, 0)

\hat{η}

Amplitude of the initial conditions

ϕ (x)

η^{(t)} (x, t)

True solution for the height perturbation

η (x, t)

β^{(t)} (x)

True bathymetry

\hat{β}

Amplitude of the true bathymetry

β^{(t)} (x)

β^{(g)} (x)

Starting guess for bathymetry

β^{(n)} (x)

Approximate bathymetry at iteration n of the assimilation algorithm

β^{(b)} (x)

Best approximation to the bathymetry (e.g. fixed point of iterations)

y^{(o)} (t)

Observations of the true height perturbation at positions

{x_{j}}, j = 1, \dots, N_{obs}

η^{(f)} (x, t)

Approximate (‘forecast’) solution generated by approximate bathymetry

J^{(n)}

Cost function at iteration n

{(\cdot)}^{*}

Adjoint

Table 2.

Cases considered for data assimilation Algorithm 1.

CaseBathymetryInitial conditionsIGaussianGaussianIISandbarGaussianIIIGaussianSinusoidal

[i] These bathymetry/initial conditions cases are illustrated in Fig. 1.

Results for iterative data assimilation scheme outlined in Algorithm 1, with $\nabla J \in L^{2} (Ω) .$ Only in Case I do we consider that the assimilation has reconstructed the bathymetry with sufficient accuracy ( $< 10 %$ relative error). (a) Convergence of the kappa test for the three cases. (b) Relative reduction in the cost function after 500 iterations. (c) Relative error in the reconstructed bathymetry. (d–f) Optimal reconstructed bathymetry for each case. We observe noise in the reconstruction for each case, especially in Case II.

The gradient of the cost function $J (β),$ obtained after one iteration for Case III, for H ¹ and H ² Sobolev smoothing compared to the (unsmoothed) L ² gradient.

Results for assimilation scheme with Sobolev H ² smoothing applied to $\nabla^{L^{2}} J .$ (a) Convergence of the kappa test. (b) Convergence of the cost function. (c) Relative L ² error $∥ β^{(t)} - β^{(n)} ∥_{L^{2}}^{2}$ / $∥ β^{(t)} ∥_{L^{2}}^{2}$ between the exact and reconstructed bathymetry at each iteration. (d–f) Reconstructed bathymetry with H ² smoothing and the exact bathymetry for cases I, II and III, respectively. Convergence is improved compared to results without smoothing given in Fig. 2.

Table 3.

Analysis of six experiments for Case I where $\hat{η} / \hat{β}$ and $\hat{β}$ are varying orders of magnitude.

\frac{\hat{η}}{\hat{β}} = O (10^{- 1})

\frac{\hat{η}}{\hat{β}} = O (10^{- 2})

\frac{\hat{η}}{\hat{β}} = O (10^{- 3})

\hat{β} = O (10^{- 1})

| κ (ε) - 1 | = 3 \times 10^{- 2}

| κ (ε) - 1 | = 2.7 \times 10^{- 2}

| κ (ε) - 1 | = 4.2 \times 10^{- 4}

Failed to convergeL² Error

= 1.3 \times 10^{- 3}

L² Error

= 4.6 \times 10^{- 3}

\hat{β} = O (10^{- 3})

| κ (ε) - 1 | = 2.5 \times 10^{- 3}

| κ (ε) - 1 | = 3 \times 10^{- 2}

| κ (ε) - 1 | = 2.7 \times 10^{- 3}

L² Error

= 7.4 \times 10^{- 3}

Failed to convergeFailed to converge

[i] The results show the convergence error in the kappa test and the error reconstruction error $‖ β^{(t)} - β^{(b)} ‖_{L^{2} (Ω)}^{2}$ / $‖ β^{(t)} ‖_{L^{2} (Ω)}^{2} .$ Entries highlighted in red denote non-convergent cases.

The relative error in the bathymetry reconstruction $∥ β^{(t)} - β^{(b)} ∥_{L^{2} (Ω)}$ / $∥ β^{(t)} ∥_{L^{2} (Ω)}$ (where $Ω = [- L, L]$ ), shown for different amplitudes $\hat{β} / H,$ with amplitude of initial conditions $\hat{η} = 0.01 %$ of H. Note that Case III has barely converged for any value of bathymetry amplitude.

Relative cost function and relative L ² error for different numbers of observation points.

Reconstructed bathymetry for Cases I, II and III with *N_obs* = 5 and *N_obs* = 45, respectively.

The relative L ² error in the bathymetry reconstruction, shown for different amplitudes $\hat{β},$ and the corresponding relative L ² error in the propagating surface wave $η (x, t) .$ The amplitude of the initial conditions $\hat{η}$ is 0.001, and *N_obs* = 45.

The relative L ² error in the bathymetry reconstruction, shown for different values of *N_obs* , and the corresponding relative L ² error in the propagating surface wave $η (x, t) .$ The amplitude of bathymetry $\hat{β}$ is 0.1. The amplitude of the initial conditions $\hat{η}$ is fixed to be 1% of $\hat{β} .$

The relative L ² error in the bathymetry reconstruction and the corresponding relative L ² error in the propagating surface wave $η (x, t),$ as a function of the initial conditions amplitude $\hat{η} .$ The amplitude of the bathymetry is fixed to be 0.2.

Approximation of the density-based sensitivity indices for $\hat{β}$ (red), $\hat{η}$ (blue) and ψ (green). (a) Influence of each parameter on $Y_{β} .$ (b) Influence of each parameter on $Y_{η} .$

Approximation of the density-based sensitivity indices for $\hat{β} (red), \hat{η}$ (blue) and ψ (green), with confidence intervals derived using 700 re-samples. (a) Influence of each parameter on $Y_{β} .$ (b) Influence of each parameter on $Y_{η} .$

Table 4.

Width of the confidence interval $| T_{i}^{u b} - T_{i}^{l b} |,$ and mean index $T_{i}^{m}$ averaged over 700 bootstrap re-samples.

Input

| T_{i}^{u b} - T_{i}^{l b} |

(

Y_{β}

)

T_{i}^{m}

(

Y_{β}

)

| T_{i}^{u b} - T_{i}^{l b} |

(

Y_{η}

)

T_{i}^{m}

(

Y_{η}

)

\hat{β}

0.10810.22410.08500.8878

\hat{η}

0.08310.13350.08240.1265ψ0.08000.85830.08480.2899

[i] Results given for model output $Y_{β}$ and $Y_{η},$ respectively.

Convergence analysis for DBSA indices for (a) $Y_{β},$ and (b) $Y_{η},$ for re-samples of size N = 215 to N = 2375.

KS statistic with significance level 0.05. Values below the dotted red line are non-influential.

Table 5.

Model output $Y_{β} :$ indices T_i using different statistics in the definition of the sensitivity index (7.6).

InputMaxMeanMedian

\hat{β}

0.21140.13340.1336

\hat{η}

0.08470.06130.0610ψ0.85240.56080.5155

[i] Values highlighted in red are below the threshold value of 0.147 ( $c (α = 0.05) = 1.36$ ) and are non-influential.

Table 6.

Model output $Y_{η} :$ indices T_i using different statistics in the definition of the sensitivity index (7.6).

InputMaxMeanMedian

\hat{β}

0.89500.55820.5467

\hat{η}

0.07450.05600.0538ψ0.26240.17800.1626

[i] Values highlighted in red are below the threshold value of 0.147 ( $c (α = 0.05) = 1.36$ ) are non-influential.

Indices *T_i* for influence of $\hat{β}$ (red), $\hat{η}$ (blue) and ψ (green) on model outputs Y > M (left panel), and $Y \leq M$ (right panel) for (a) $Y_{β},$ and (b) $Y_{η} .$ Confidence intervals were calculated using 700 bootstrap samples.

Table 7.

Classification of each input parameter as influential or non-influential for the sub-regions of $Y_{β}$ and $Y_{η}$ in Fig. 15.

Input

Y_{β} > 0.15

Y_{β} \leq 0.15

Y_{η} > 0.001

Y_{η} \leq 0.001

\hat{β}

Non-influentialInfluentialInfluentialInfluential

\hat{η}

Non-influentialNon-influentialNon-InfluentialNon-influentialψInfluentialInfluentialInfluentialNon-influential

[i] Values below the threshold value of 0.147 ( $c (α = 0.05) = 1.36$ ) are non-influential. The entries highlighted in blue are the most influential parameter for $Y_{β}$ and $Y_{η},$ respectively. $\hat{β}$ is influential on bathymetry reconstruction error only when the error is larger than 0.15, and ψ is only influential on the surface wave error when the error is larger than 0.001.

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DOI: https://doi.org/10.1080/16000870.2021.1976907 | Journal eISSN: 3035-9554

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Language: English

Page range: 1976907 - 1976907

Published on: Jan 1, 2021

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

Keywords:

bathymetry estimation,

density-based sensitivity analysis (DBSA),

global sensitivity analysis (GSA),

shallow water equations,

tsunami modelling

© 2021 R. A. Khan, N. K.-R. Kevlahan, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 73 (2021): Issue 1

Variational assimilation of surface wave data for bathymetry reconstruction. Part I: algorithm and test cases

Figures & Tables

Fig. 1.

Table 1.

Table 2.

Fig. 2.

Fig. 3.

Fig. 4.

Table 3.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Table 4.

Fig. 13.

Fig. 14.

Table 5.

Table 6.

Fig. 15.

Table 7.

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