Ensemble Data Assimilation of Concentration Measurements Following the Accidental Release of a Contaminant in the Ocean: Method Testing in an Idealized Setting

Abstract

The capabilities of the ensemble Kalman filter (EKF) data assimilation (DA) method to reduce errors in simulations of concentration distributions following the accidental release of a contaminant in the ocean were evaluated. The method was tested in an idealized setting where the contaminant was released in the ocean described by a simple linear Stommel model that includes the main features of two-dimensional (2D) wind-driven circulation in the ocean on the β-plane. The wind stress curl in the right-hand side of the equation for stream function was randomly perturbed to generate an ensemble of the perturbed fields of currents. The velocity fields obtained from the ensemble of stream functions were then used for the calculation of the ensemble of concentration fields following short-duration point release. On day 1000 of the simulation, correlation coefficients of the members of the concentration ensemble and the unperturbed concentration distribution fell to 0.087. The ensemble member with the maximum deviation from the unperturbed concentration distribution was selected to be used as “truth” in data assimilation experiments. Due to the high inhomogeneity of the concentration fields, the free regularization parameter had to be defined and tuned using the L-curve approach. Different DA scenarios were considered with different topologies of measurement networks and different source locations. In all cases, data assimilation gradually brought ensemble-averaged concentration fields close to the true distribution. The root mean square errors of the analyzed concentrations on day 1000 decreased by the factors varying from 3 to 4 in different DA scenarios as compared to the simulation without DA.

Appendix

1.1 Appendix A: Solution of the equation for the stream function

This Appendix describes a method for solving Eq. (1) with the r.h.s., defined with Eqs. (3)-(4). Here the suffix, standing for the realization of the stochastic process in Sect. 2.1.1, is omitted because any of the ensemble members were considered in the same way.

When F = F₀, then Eq. (1) is rewritten as:

$$\Delta \psi + \alpha \frac{\partial \psi }{{\partial x}} = \gamma \sin \left( {\frac{\pi y}{b}} \right)$$

(11)

where

$$\alpha = \frac{\beta H}{r},\,\,\,\gamma = \frac{{F_{0} \pi }}{rb}$$

The conditions $\psi = 0$ are imposed on rectangular basin boundaries.

The solution involves a solution for homogeneous part ${\psi }_{h}$ and a particular solution ${\psi }_{p}$ for the nonhomogeneous part (Stommel, 1948)

$$\psi = \psi_{h} + \psi_{p} .$$

(12)

The homogeneous part $\psi_{h}$ is

$$\psi_{h} = \sum\limits_{k = 1}^{\infty } {X_{k} Y_{k} } ,$$

(13)

where:

$$X_{k} = p_{k} \exp (A_{k} x) + q_{k} \exp (B_{k} x),$$

(14)

$$Y_{k} = \delta_{k1} \sin \left( {\frac{\pi y}{b}} \right),$$

(15)

$$A_{k} = - \frac{\alpha }{2} + \sqrt {\frac{{\alpha^{2} }}{4} + \lambda_{k}^{2} } ,\,\,\,A_{k} = - \frac{\alpha }{2} - \sqrt {\frac{{\alpha^{2} }}{4} + \lambda_{k}^{2} } ,\,\,\,\lambda_{k} = \frac{\pi k}{b},$$

(16)

$$p_{k} = n_{k} \frac{{1 - \exp (B_{k} a)}}{{(\exp (A_{k} a) - \exp (B_{k} a)}},\,\,\,q_{k} = n_{k} - p_{k} ,\,\,\,n_{k} = \gamma \left( {\frac{b}{\pi k}} \right)^{2}$$

(17)

Note that due to $Y_{k} = 0$ at $k > 1$ in the case of constant F₀, the homogeneous part is defined only by the first term of sum (13), i.e., $\psi_{h} = X_{1} Y_{1}$. The solution for the nonhomogeneous part is

$$\psi_{p} = - \gamma \left( {\frac{b}{\pi }} \right)^{2} \sin \left( {\frac{\pi y}{b}} \right).$$

(18)

In the case of non-constant F, it is convenient to expand the r.h.s. of Eq. (11), defined by Eq. (3), into a series:

$${\text{r.h.s.}} = (F_{0} + \delta F(y))\cos \left( {\frac{\pi y}{b}} \right) = \sum\limits_{k = 1}^{\infty } {F_{k} \cos \left( {\frac{\pi y}{b}} \right)} ,$$

(19)

The coefficients ${F}_{k}$ could be found by calculating the respective scalar products:

$$F_{k} = \frac{2}{b}\left\langle {r.h.s.,e_{k} } \right\rangle ,$$

where a system of functions

$$e_{k} = \cos \left( {\frac{k\pi y}{b}} \right),\quad k = 1, \ldots ,\infty$$

are orthogonal functions formed by the solution of the Sturm–Liouville problem.

The homogeneous part $\psi_{h}$ is expressed in the same form as (13)–(17), but with

$$n_{k} = \gamma_{k} \left( {\frac{b}{\pi k}} \right)^{2} ,\,\,\,\gamma_{k} = \frac{{F_{k} \pi k}}{rb}.$$

The solution of the nonhomogeneous part is obtained by summing the components:

$$\psi_{p} = \sum\limits_{k = 1}^{\infty } {\psi_{pk} } ,$$

(20)

where

$$\psi_{pk} = - \gamma_{k} \left( {\frac{b}{\pi }} \right)^{2} \sin \left( {\frac{\pi ky}{b}} \right).$$

Finally, the solution for first N₁ terms in expansion is

$$\psi (x,y) \approx \sum\limits_{k = 1}^{N_{1} } {(X_{k} } Y_{k} + \psi_{pk} ).$$

(21)

In the implementation of the presented approach, N₁ was set to N₁ = 15 based on preliminary numerical experiments that showed that the influence of the higher-order terms in the expansion is negligibly small. For each realization of the stochastic process $\delta F_{i} (y)$, $1 \le i \le N$, the coefficients F_k, $1 \le k \le 15$ entered in (19) were calculated and then used in formula (21) for calculation of the stream function in every node of the computational grid, as described in Sect. 2.1.2.