Correcting circulation biases in a lower-resolution global general circulation model with data assimilation

Abstract

In this study, we aim at developing a new method of bias correction using data assimilation. This method is based on the stochastic forcing of a model to correct bias by directly adding an additional source term into the model equations. This method is presented and tested first with a twin experiment on a fully controlled Lorenz ’96 model. It is then applied to the lower-resolution global circulation NEMO-LIM2 model, with both a twin experiment and a real case experiment. Sea surface height observations are used to create a forcing to correct the poorly located and estimated currents. Validation is then performed throughout the use of other variables such as sea surface temperature and salinity. Results show that the method is able to consistently correct part of the model bias. The bias correction term is presented and is consistent with the limitations of the global circulation model causing bias on the oceanic currents.

Appendix

One can show that the analysis using the average model state (13) provides the same analysed bias $\widehat {{\mathbf {b}^{a}}}$ as when the full trajectory is included in the estimation vector (6).

Using i=1,…,N to refer to the ensemble members, the forecast of the model trajectory can be defined as

$$ \begin{array}{c} \mathbf{x}^{\prime f}_{i} = \left[\begin{array}{c}\mathbf{x}^{f(1)}_{i}\\ \mathbf{x}^{f(2)}_{i}\\ {\vdots} \\ \mathbf{x}^{f(m_{\max})}_{i}\\ \widehat{{\mathbf{b}}}^f_{i} \end{array}\right], \end{array} \begin{array}{c} \mathbf{x}^{\prime a}_{i} = \left[\begin{array}{c}\mathbf{x}^{a(1)}_{i}\\ \mathbf{x}^{a(2)}_{i}\\ {\vdots} \\ \mathbf{x}^{a(m_{\max})}_{i}\\ \widehat{{\mathbf{b}}}^{a}_{i} \end{array}\right]. \end{array} $$

(37)

The analysis is provided by

$$ \mathbf{x}^{\prime a}= \mathbf{x}^{\prime f} + \frac{1}{N-1} \mathbf{X}^{\prime f} \underbrace{(\mathbf{X}^{\prime f})^{T} \mathbf{H}^{\prime T} (\mathbf{H}^{\prime} \mathbf{P}^{\prime f} \mathbf{H}^{\prime T} + R )^{-1} ({\mathbf{y}}^{o} -\mathbf{H}^{\prime} \mathbf{x}^{\prime f})}_{\mathbf{W}^{\prime}}, $$

(38)

where

$$ \begin{array}{c} \mathbf{x}^{\prime f} = \frac{1}{N} \sum\limits_{i=1}^{N} \mathbf{x}^{\prime f}_{i}, \end{array}\qquad\qquad \begin{array}{c} \mathbf{x}^{\prime a} = \frac{1}{N} \sum\limits_{i=1}^{N} \mathbf{x}^{\prime a}_{i}, \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} \mathbf{P}^{\prime f} & = &\frac{1}{N-1} \sum\limits_{i=1}^{N} (\mathbf{x}^{\prime f}_{i} - \mathbf{x}^{\prime f})(\mathbf{x}^{\prime f}_{i} - \mathbf{x}^{\prime f})^{T} \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} & = &\frac{1}{N-1} \mathbf{X}^{\prime f} (\mathbf{X}^{\prime f})^{T}. \end{array} $$

(41)

The observation operator $\mathbf {H}^{\prime }$ applied to the trajectory $\mathbf {x}^{\prime }$ also includes a time average and an extraction operator H of the observed part of the model state

$$\begin{array}{@{}rcl@{}} \mathbf{H}^{\prime} \mathbf{x}^{\prime} & = &\sum\limits_{m=1}^{m_{max}} \mathbf{H} \mathbf{x}^{(m)} = \mathbf{H} \overline{\mathbf{x}}, \end{array} $$

(42)

$$\begin{array}{@{}rcl@{}} \overline{\mathbf{x}} & = &\frac{1}{m_{max}} \sum\limits_{m=1}^{m_{max}} \mathbf{x}^{(m)}. \end{array} $$

(43)

Hence, the ensemble mean of the analysed bias correction term $\widehat {{\mathbf {b}^{\prime a}}}$ is contained in the analysed model trajectory $\mathbf {x}^{\prime a}$. One can also first take the time average of the trajectory, defined as

$$ \begin{array}{c} \mathbf{x}^{\prime\prime f}_{i} = \left[\begin{array}{c}\overline{\mathbf{x}}^{f}_{i} \\ \widehat{{\mathbf{b}}}^{f}_{i} \end{array}\right], \end{array}\qquad\qquad\qquad \begin{array}{c} \mathbf{x}^{\prime\prime a}_{i} = \left[\begin{array}{c}\overline{\mathbf{x}}^{a}_{i} \\ \widehat{{\mathbf{b}}}^{a}_{i} \end{array}\right]. \end{array} $$

(44)

The analysis is then given by

$$ \mathbf{x}^{\prime\prime a}\,=\, \mathbf{x}^{\prime\prime f} \!\,+\, \frac{1}{N\,-\,1} \mathbf{X}^{\prime\prime f} \underbrace{(\mathbf{X}^{\prime\prime f})^{T} \mathbf{H}^{\prime\prime T} (\mathbf{H}^{\prime\prime} \mathbf{P}^{\prime\prime f} \mathbf{H}^{\prime\prime T} \!\,+\, R )^{-1} ({\mathbf{y}}^{o} \,-\,\mathbf{H}^{\prime\prime} \mathbf{x}^{\prime\prime f})}_{\mathbf{W}^{\prime\prime}}, $$

(45)

where

$$ \begin{array}{c} \mathbf{x}^{\prime\prime f}= \frac{1}{N} \sum\limits_{i=1}^{N} \mathbf{x}^{\prime\prime f}_{i}, \end{array}\qquad\qquad\qquad \begin{array}{c} \mathbf{x}^{\prime\prime a}= \frac{1}{N} \sum\limits_{i=1}^{N} \mathbf{x}^{\prime\prime a}_{i}, \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} \mathbf{P}^{\prime\prime f} & = &\frac{1}{N-1} \sum\limits_{i=1}^{N} (\mathbf{x}^{\prime\prime f}_{i} - \mathbf{x}^{\prime\prime f})(\mathbf{x}^{\prime\prime f}_{i} - \mathbf{x}^{\prime\prime f})^{T} \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} & =& \frac{1}{N-1} \mathbf{X}^{\prime\prime f} (\mathbf{X}^{\prime\prime f})^{T}. \end{array} $$

(48)

The ensemble mean of the analysed bias correction term $\widehat {{\mathbf {b}^{\prime \prime a}}}$ is contained in the analysed mean model state $\mathbf {x}^{\prime \prime a}$. Given that

$$ \mathbf{H}^{\prime} \mathbf{x}^{\prime}= \mathbf{H}^{\prime\prime} \mathbf{x}^{\prime\prime}, $$

(49)

it follows that $\mathbf {W}^{\prime } = \mathbf {W}^{\prime \prime }$. Hence, $\widehat {{\mathbf {b}^{\prime \prime a}}} = \widehat {{\mathbf {b}^{\prime a}}}$, since they are both constrained by the same linear combination of $\widehat {{\mathbf {b}}}^{f}_{i}$.