Improving strategies with constraints regarding non-Gaussian statistics in a three-dimensional variational assimilation method

Abstract

We assess validity of a Gaussian error assumption, the basic assumption in data assimilation theory, and propose two kinds of constraints regarding non-Gaussian statistics. In the mixed water region (MWR) off the east coast of Japan exhibiting complicated frontal structures, a probability density function (PDF) of subsurface temperature shows double peaks corresponding to the Kuroshio and Oyashio waters. The complicated frontal structures characterized by the temperature PDF sometimes cause large innovations, bringing about a non-Gaussianity of errors. It is also revealed that assimilated results with a standard three-dimensional variational (3DVAR) scheme have some issues in MWR, arising from the non-Gaussianity of errors. The Oyashio water sometimes becomes unrealistically cold. The double peaks seen in the observed temperature PDF are too smoothed. To improve the assimilated field in MWR, we introduce two kinds of constraints, J _c1 and J _c2, which model the observed temperature PDF. The constraint J _c1 prevents the unrealistically cold Oyashio water, and J _c2 intends to reproduce the double peaks. The assimilated fields are significantly improved by using these constraints. The constraint J _c1 effectively reduces the unrealistically cold Oyashio water. The double peaks in the observed temperature PDF are successfully reproduced by J _c2. In addition, not only subsurface temperature but also whole level temperature and salinity (T–S) fields are improved by adopting J _c1 and J _c2 to a multivariate 3DVAR scheme with vertical coupled T–S empirical orthogonal function modes.

Appendix: Three-dimensional variational method with vertical coupled T–S EOF modes

Here, we briefly describe our 3DVAR scheme with vertical coupled T–S EOF modes, which is originally introduced by Fujii and Kamachi (2003b). The cost function in MOVE system is expressed as follows:

$$ \begin{aligned} J({{\mathbf z}} ) & = \frac{1}{2}\sum_{l}{\mathbf z}_{l}^{T} {\mathbf B}_l^{-1} {\mathbf z}_l + \frac{1}{2} \left[ {\mathbf {Hx(z)-y}}^{\rm TS} \right]^{T} {\mathbf R}^{-1} \left[ {\mathbf {Hx(z)-y}}^{\rm TS} \right]\\ & \quad+ \frac{1}{2\sigma_{h}^2}\left[{\mathcal H} ({\mathbf {x(z)}})-{\mathbf y}^{\rm SSH}\right]^T \left[ {\mathcal H} ({\mathbf {x(z)}} )-{\mathbf y}^{\rm SSH}\right], \end{aligned} $$

(A1)

where ${\mathbf z}_l$ is the vector composed of amplitudes of the vertical coupled T–S EOF modes, and matrices ${\mathbf B}_l$ and ${\mathbf R}$ are the horizontal background correlation matrix and observation error covariance matrices, ${\mathbf y}^{\rm TS}$ and ${\mathbf y}^{\rm SSH}$ are observation vectors for T–S profile and SSH, respectively. The subscript l denotes the l-th subregion. Operators ${\mathbf H}$ and $\mathcal{H}$ are observation operators for T–S profile and altimeter-derived SSH. The nonlinear operator $\mathcal{H}$ calculates sea surface dynamic height from T–S field.

The control variable is amplitudes of the vertical coupled T–S EOF modes ${\mathbf z}.$ Thus the analysis increments of T–S field, $\Updelta {\mathbf x},$ is calculated by

$$ \Updelta {\mathbf x} = {{\mathbf {x(z)}}} - {{\mathbf x}}^{b} ={{\mathbf S}} \sum_{l} {w_{l}} {{\mathbf U}}_l {\bf {\Uplambda}}_l {{\mathbf z}}_l, $$

(A2)

where ${\mathbf x}^{b}$ is the first-guess, ${\mathbf S}$ is a diagonal matrix whose diagonal elements are composed of standard deviation of the background field, ${\mathbf U}_l$ is a matrix composed of dominant T–S EOF modes, Λ _l is a diagonal matrix whose diagonal elements are the singular values of T–S EOF modes, and w _l is weight for the l-th subregion which needs to satisfy ∑ _l w _l  = 1 (Fukumori 2002).

The background error covariance matrix, ${\mathbf B},$ is expressed with the horizontal background correlation matrix ${\mathbf B}_l$ and the statistics in (A2) as follows:

$$ {\mathbf B} = {\mathbf S} \left[ \sum_l {{\mathbf U}}_l {\bf{\Uplambda}}_l {\mathbf B}_l {\bf{\Uplambda}}_l {\mathbf U}_l^{T}\right] {\mathbf S}. $$

(A3)

The vertical correlation matrix of the background errors is thus modeled by ${\mathbf U}_l {\bf {\Uplambda}}_l .$ The horizontal correlation is modeled by the Gaussian function with area-dependent decorrelation scales, which are assigned according to Kuragano and Kamachi (2000). In MWR, it is about 100 km.