Correcting surface winds by assimilating high-frequency radar surface currents in the German Bight

Abstract

Surface winds are crucial for accurately modeling the surface circulation in the coastal ocean. In the present work, high-frequency radar surface currents are assimilated using an ensemble scheme which aims to obtain improved surface winds taking into account European Centre for Medium-Range Weather Forecasts winds as a first guess and surface current measurements. The objective of this study is to show that wind forcing can be improved using an approach similar to parameter estimation in ensemble data assimilation. Like variational assimilation schemes, the method provides an improved wind field based on surface current measurements. However, the technique does not require an adjoint, and it is thus easier to implement. In addition, it does not rely on a linearization of the model dynamics. The method is validated directly by comparing the analyzed wind speed to independent in situ measurements and indirectly by assessing the impact of the corrected winds on model sea surface temperature (SST) relative to satellite SST.

Appendix: Analysis update for non-linear observation operators

The background estimate x ^b and the observations y ^o (involving a possibly non-linear observation operator h(·)) are assumed to be an unbiased estimate of the true state x ^t.

$$ E\left[ {\bf x}^b \right] = {\bf x}^t $$

(14)

$$ E\left[ {\bf y}^o \right] = h({\bf x}^t) $$

(15)

The error of the background estimate x ^b and the observations y ^o are assumed to be independent. One seeks an analysis scheme of the following form,

$$ {\bf x}^a = {\bf x}^b + {\bf K} \left({\bf y}^o - h\left({\bf x}^b\right)\right) $$

(16)

where K is a matrix to be defined by requiring that the analysis has a minimal error. On average, the analysis should be unbiased.

$$ E\left[ {\bf x}^a \right] = {\bf x}^t + {\bf K} \left(h\left({\bf x}^t\right) - E\left[ h\left({\bf x}^b\right) \right]\right) $$

(17)

This is the case if:

$$ \label{eqn:testbias} E\left[ h\left({\bf x}^b\right) \right] = h({\bf x}^t) $$

(18)

The error of the analysis is given by:

$$ {\bf x}^a - {\bf x}^t = {\bf x}^b - {\bf x}^t + {\bf K} \left({\bf y}^o - h\left({\bf x}^t\right)\right) - {\bf K} \left(h({\bf x}^t) - h\left({\bf x}^b\right)\right) $$

(19)

Since the background estimate is independent from the observations, h(x ^b) is also independent from the observations ($\mbox{cov}({\bf y}^o,h({\bf x}^b)) = 0$). The error covariance of the analysis yields:

$$ \begin{array}{rll} {\bf P}^a &=& {\bf P}^b - {\bf K} \mbox{cov}\left(h\left({\bf x}^b\right),{\bf x}^b\right) - \mbox{cov}\left({\bf x}^b,h\left({\bf x}^b\right)\right) {\bf K}^T \\ && + {\bf K} \mbox{cov}\left(h\left({\bf x}^b\right),h\left({\bf x}^b\right)\right) {\bf E}^T + {\bf K} {\bf R} {\bf K}^T \end{array}$$

(20)

Covariances are approximated by the ensemble covariances:

$$ {\bf P}^a = {\bf P}^b - {\bf K} {\bf E} {\bf S}^T - {\bf S} {\bf E}^T {\bf K} + {\bf K} \left( {\bf E} {\bf E}^T + {\bf R} \right) {\bf K}^T $$

(21)

The optimal Kalman gain K is obtained by minimizing the total error $\mbox{tr}({\bf W} {\bf P}^a)$ where W is an arbitrary weighting matrix.

$$ J({\bf K}) = \mbox{tr}({\bf W} {\bf P}^a) $$

(22)

The minimum is obtained by:

$$\begin{array}{rll} \delta J &=& J({\bf K}+ \delta {\bf K}) - J({\bf K}) \\ &=& - 2 \mbox{tr}\left({\bf W} \delta {\bf K} {\bf E} {\bf S}^T\right) + 2 \mbox{tr}\left({\bf W} \delta {\bf K} \left({\bf E} {\bf E}^T + {\bf R}\right) {\bf K}^T\right) = 0 \end{array}$$

(23)

δJ is zero for any δ K if,

$$ {\bf K} = {\bf S} {\bf E}^T \left({\bf E} {\bf E}^T + {\bf R}\right)^{-1} $$

(24)

In summary, the analysis update with a non-linear observation operator is not guaranteed to be unbiased. The condition of Eq. 18 has thus to be verified separately. The Kalman gain can then be derived without additional assumptions (compared to the case of a linear observation operator).

In the context of the extended Kalman filter, a non-linear observation operator h(·) is linearized around the model forecast. Such linearization is not necessary here because the covariance matrices in the extended Kalman filter H P ^b H ^T and P ^b H ^T are directly derived from the ensemble ($\mbox{cov}(h({\bf x}^b),h({\bf x}^b))$ and $\mbox{cov}({\bf x}^b,$ h(x ^b))).