Journal of Oceanography
February 2020 , Volume 76, Issue 1, pp 29–41 | Cite as
An application study on adjoint-based variational wave assimilation scheme in German Bight with low spatial observation coverage
Q. Song
R. Mayerle
Y. Yu
C. Li
First Online: 08 October 2019
A variational assimilation scheme was modified for practical application of wave simulation in German Bight. Sensitivity analysis showed that responses of sparsely distributed observing sites to the errors of boundary conditions on wave boundaries crucially depend on observing locations and wave conditions. Therefore, a scheme designed for low spatial observation coverage is proposed. The scheme assumes that control variables can be expressed by a set of spatially uniform variables (referred to as basic control variable) adding varying perturbations. The basic control variables are assumed containing all the errors, so assimilation can proceed through adjusting them only. Twin experiments with pseudo-observations were conducted to assess the feasibility of the scheme with basic control variables. The proposed scheme using a single observational location shows a comparable assimilation effect to the original scheme without the basic control variable assumption using 25 observational locations, reducing the spectrum root mean square errors by approximately 50% throughout the whole computation domain. The practical experiment was performed over 1 day, considering both wave boundary conditions and wind fields as control variables. The spectrum root mean square errors at the validation buoy decreased by more than 60% via the proposed assimilation scheme with observations from a single in situ buoy. Overall, although the basic variable assumption will overestimate the spatial correlation of the errors and consequently adjust the control variables improperly in some areas, the scheme remains a good option for nearshore wave simulation where sea states are strongly correlated when observations are spatially limited.
Adjoint-based variational assimilation Wave modelling Sensitivity Observation coverage
We gratefully acknowledge financial support from the China Scholarship Council. In addition, we are grateful to BSH, who kindly provided the in situ spectral data from buoys in the study area.
Original assimilation scheme
When choosing spatially varying control variables, \(J_{1}\) in Eq. (3) reads
$$\begin{aligned} \begin{aligned}&J_{1}= \frac{1}{M_\mathrm{obs}} \int (\pmb {E}-\pmb {E_{a}})^\mathrm{T}\pmb {W_1}(\pmb {E}-\pmb {E_{a}})\mathrm {d}\theta \mathrm {d}f \mathrm {d}t, \end{aligned} \end{aligned}$$
where \(\pmb {E}\) is \(\left[ \begin{array}{c} E_1\\ E_2\\ ...\\ E_\mathrm{m} \end{array}\right] \) (E is the 2D wave spectrum density in frequence f and direction \(\theta \) domain and subscript 1-m is the location index in space domain) and the subscript a denotes the observations. \(\pmb {W_{1}}\) reflects the uncertainty of the observations. In this case, \(\pmb {W_{1}}\) is the inverse spatial error covariance matrix of the observed wave energy assuming that the observed spectrum density has the same spatial correlations at all frequencies and in all directions. \(M_\mathrm{obs}\) indicates the total number of observations used for assimilation over space and time. \(J_{2}\) is the wind deviation, and it reads as follows:
$$\begin{aligned} \begin{aligned} J_{2}=&\,\frac{1}{A_\mathrm{D}}\int [(\pmb {u}-\pmb {u_\mathrm{1st}})^\mathrm{T}\pmb {W_{2u}}(\pmb {u}-\pmb {u_\mathrm{1st}})\\&+(\pmb {v}-\pmb {v_\mathrm{1st}})^\mathrm{T}\pmb {W_{2v}}(\pmb {v}-\pmb {v_{1st}})] \mathrm {d}\mathbf {x}\mathrm {d}t, \end{aligned} \end{aligned}$$
where \(\pmb {u}\) and \(\pmb {v}\) are the space vectors of the wind speed components and the subscript 1st indicates the first guess value. \(\pmb {W_{2u}}\) and \(\pmb {W_{2v}}\) are the inverse error covariance matrix of \(\pmb {u}\) and \(\pmb {v}\), respectively. \({A_\mathrm{D}}\) is the area of the computational domain. Similarly, \(J_{3}\) reads as follows:
$$\begin{aligned} J_{3}=\frac{1}{L_b}\int (\pmb {E_\mathrm{b}}-\pmb {E_{\mathrm{b}_\mathrm{1st}}})^\mathrm{T}\pmb {W_{3}} (\pmb {E_\mathrm{b}}-\pmb {E_{\mathrm{b}_\mathrm{1st}}}) \,\mathrm {d}\theta \mathrm {d}f\mathrm {d}\mathbf {x_b}\mathrm {d}t, \end{aligned}$$
where the subscript b denotes the input wave spectra on the open boundaries and \(L_b\) is the length of the boundaries. Analogously, \(\pmb {W_{3}}\) is the inverse spatial covariance matrix of the first guess input wave energy on the open boundaries.
Time-dependent control variables such as wind fields or wave boundary conditions in a forward model are normally obtained by interpolating the inputs at the start and the end points of an input interval to match the computational time step. Therefore, a time-dependent control variable c can be expressed by a linear combination of multiple time-independent control variables as follows:
$$\begin{aligned} c(t)=\sum _{n=1}^{N}w_{n}(t)c_{n}, \end{aligned}$$
where \(w_{n}(t)\) denotes the interpolation weighting function with respect to time and \(c_n\) is the control variable at the start or end time point of an input interval. Normally, \(w_{n}(t)\) is a linear function of time. Thus, the time-dependent control variables \(E_\mathrm{b}\) and u, v in Eqs. (17) and (18) can be replaced by time-independent control variables \(E_{bn}\) and \(u_n\), \(v_n\). The expressions of \(J_2\) and \(J_3\) are changed to the following:
$$\begin{aligned} J_{2}= & {} \frac{1}{A_\mathrm{D}}\sum _{n=1}^{N} \int [(\pmb {u_n}-\pmb {u_{n_{1st}}})^\mathrm{T}\pmb {W_{2nu}}(\pmb {u_n}-\pmb {u_{n_{1st}}})\nonumber \\&+(\pmb {v_n}-\pmb {v_{n_{1st}}})^\mathrm{T}\pmb {W_{2nv}}(\pmb {v_n}-\pmb {v_{n_{1st}}})] \mathrm {d}\mathbf {x} \end{aligned}$$
$$\begin{aligned} J_{3}= & {} \frac{1}{L_b} \sum _{n=1}^{N}\int (\pmb {E_{bn}}-\pmb {E_{bn_{1st}}})^\mathrm{T}\pmb {W_{3n}}\nonumber \\&(\pmb {E_{bn}}-\pmb {E_{bn_{1st}}}) \mathrm {d}\theta \mathrm {d}f \mathrm {d}\mathbf {x_b}. \end{aligned}$$
Song and Mayerle (2017) assumed that waves can be regarded as linear superpositions of swell and wind sea contributions. Hence, following the Lagrange multiplier method, the resulting adjoint equations read as follows:
for wind sea:
$$\begin{aligned} \begin{aligned}&- \frac{\partial \lambda _\mathrm{w}}{\partial t}- \mathbf {C}\mathbf {\nabla }\lambda _\mathrm{w} = \lambda _{w}\frac{\partial S_\mathrm{w}}{\partial E}-\frac{\partial J_{1w}}{\partial N} \delta (\mathbf {x_b}-\mathbf {x_{a}}) \delta (t-t_\mathrm{a}), \end{aligned} \end{aligned}$$
for swell:
$$\begin{aligned} \begin{aligned}&- \frac{\partial \lambda _\mathrm{s}}{\partial t}- \mathbf {C}\mathbf {\nabla }\lambda _\mathrm{s} = \lambda _\mathrm{s}\frac{\partial S_{s}}{\partial E}-\frac{\partial J_{1\mathrm{s}}}{\partial N} \delta (\mathbf {x_b}-\mathbf {x_{a}}) \delta (t-t_\mathrm{a}) \end{aligned} \end{aligned}$$
where the \(\delta \) is a delta function with \(\delta (0)=1\). \(J_{1\mathrm{w}}\) and \(J_{1\mathrm{s}}\) are the wind sea and swell contributions of \(J_1\), respectively. The wind sea and swell contributions are separated and classified by the partition method (Hasselmann et al. 1997). \(S_\mathrm{w}\) and \(S_\mathrm{s}\) are the source/sink terms related to local winds and swells, respectively. After obtaining the Lagrange multipliers \(\lambda _\mathrm{w}\) and \(\lambda _\mathrm{s}\), by solving Eqs. (22) and (23), the gradients of the cost function with respect to the control variables can be calculated using the following formulas:
for wind fields:
$$\begin{aligned} \begin{aligned}&\mathrm {grad}_{u_{n}}J=\frac{\partial J_{2}}{\partial u_{n}}- \int \left[ \frac{\lambda _\mathrm{w}w_{n}(t)}{2\pi f} \frac{\partial S_\mathrm{w}}{\partial u}\right] \mathrm {d}\theta \mathrm {d}f\mathrm {d}t\\&\mathrm {grad}_{v_{n}}J=\frac{\partial J_{2}}{\partial v_{n}}- \int \left[ \frac{\lambda _{w}w_{n}(t)}{2\pi f} \frac{\partial S_\mathrm{w}}{\partial v}\right] \mathrm {d}\theta \mathrm {d}f\mathrm {d}t, \end{aligned} \end{aligned}$$
for wave boundaries:
$$\begin{aligned} \begin{aligned}&\mathrm {grad}_{E_{bn}}J=\frac{\partial J_{3}}{\partial E_{bn}}- \int \left[ \frac{w_\mathrm{n}(t)}{2\pi f} \lambda _{s} \mathbf {C}\mathbf {n_{b}}\right] \mathrm {d}t, \end{aligned} \end{aligned}$$
where \((u_n, v_n)\) are the input wind components, \(E_{bn}\) is the input wave spectra on open boundaries, and \(\mathbf {n_{b}}\) is the direction vector normal to the boundary. Because the gradients are accessible, the increments of the control variables are obtained by means of conjugate-gradient descent algorithm (Polak 1971) to update the forward wave model.
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© The Oceanographic Society of Japan and Springer Nature Singapore Pte Ltd. 2019
1.Institute of Space-Earth Environmental ResearchNagoya UniversityNagoyaJapan
2.FTZ-Westküste / Coastal Research LaboratoryChristian-Albrechts-Universität zu KielKielGermany
3.State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of OceanographyMinistry of Natural ResourcesHangzhouChina
4.National Marine Data and Information ServiceTianjinChina
Song, Q., Mayerle, R., Yu, Y. et al. J Oceanogr (2020) 76: 29. https://doi.org/10.1007/s10872-019-00526-5
Received 03 June 2019
Revised 18 September 2019
Accepted 21 September 2019
Publisher Name Springer Singapore
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