It is clear that the linear assumption is made for the measurement function to obtain Eq. (4), as indicated by the linear operator h. To deal with the nonlinear measurement function in the EnKF, HM2001 proposed Eqs. (11) and (11) to directly evaluate the nonlinear measurement functions. Apparently, there is a gap here, i.e., the left hand sides of Eqs. (11) and (11) need the linear measurement function h, whereas their right hand sides directly use the nonlinear function h. HM2001 realized that there was this gap and used the equivalence sign "#cod#8801;" instead of the equality sign "=" in Eqs. (11) and (11). However, the equivalence is primarily based on intuition. It is necessary to examine the equivalence in a rigorous, statistical framework.
First, we re-examine the optimal Kalman gain, Eq. (4), using the general form of the Kalman gain, Eq. (15). When the dynamical model is linear, i.e., , where Q=E(#cod#951;t-1#cod#951;t-1 T). When the measurement function is linear, i.e., ,
where R=E(#cod#949;t#cod#949;t T). Then, the optimal Kalman gain is
Equation (19) is identical to Eq. (4). Therefore, Eq. (4) used in the KF, EKF, and EnKF is a special case of Eq. (15), under the assumption of a linear measurement function.
We now examine Eqs. (11) and (11) of HM2001 that have been widely used to treat the nonlinear measurement function in the EnKF. Emphasis is placed on the comparison of Eqs. (4), (11), and (11) against Eq. (15).
When the noise is additive, the nonlinear state-space equation [Eq. (1)] becomes
The assumption of additive noise is commonly used for the assimilation in Gaussian-based systems. For a non-additive noise system (e.g., multiplicative noise), Gaussian-based assimilation methods, such as the EnKF, are often invalid.
We started from Eq. (15), i.e., . If the estimate is unbiased and the ensemble size L is infinite, we can use the ensemble mean to represent the true value, i.e.,
where E[#cod#183;] denotes the expectation, i is the ensemble index and the overbar represents the mean over all the ensemble members. The terms #cod#951;t and #cod#962;t were added due to the random nature of the true states:
For a realistic ensemble system with finite ensemble size, can be written as:
Similarly,
Here, the assumption is that the noise terms #cod#949;t and #cod#951;t have zero mean and are uncorrelated with other variables. The variances of #cod#951;t and #cod#949;t are Q and R, respectively. Equation (22) represents the forecast error covariance estimated by the ensemble member. Compared to the standard EnKF [Eq. (9)], there appears to be one more item Q on the left hand side of Eq. (22). The absence of Q in the standard EnKF algorithm is because the forecast error is being defined with respect to the ensemble mean rather than to the true state. Thus, the random nature of the true states is ignored. The standard EnKF often systematically underestimates the error covariance and requires an inflation scheme to "adjust" the estimated error covariance.
The Kalman gain is written as
A comparison of Eqs. (11) and (11) with Eqs. (21) and (23) reveals that they are completely equivalent, if Eq. (12) holds true. From the linearization point of view, Eq. (12) holds true only if Eq. (13) also holds true. Conversely, when Eqs. (21) and (23) are used instead of Eqs. (11) and (11) in the EnKF, the modified Kalman gain form should be more rigorous in the statistical framework, which is equivalent to the general Kalman gain form, Eq. (15), without demanding any assumption of linearization. Clearly, when the noise is non-additive, the equivalence is no longer valid. However, in case of non-additive noise, all Kalman-based filters are invalid due to the non-Gaussian nature of the systems. Recently, (Ambadan and Tang, 2011) discussed the assimilation of a nonlinear system in a multiplicative noise environment and found that the intrinsic properties of the multiplicative noise challenge the current EnKF algorithms.
In the above derivations, we used the forecast measurement to represent the true measurement, i.e., , as indicated in Eqs. (23) and (24). One important assumption here is the unbiased nature of the forecast , i.e., the ensemble mean instead of the unknown true state. This unbiased assumption results in the prediction error of measurement, which may be serious in some cases. One solution used to reduce the impact of the unbiased assumption on the estimate of the Kalman gain is to directly use the actual observation yt, o to represent the true measurement; namely
yt= yt,o+#cod#x003b5;t. (26)
Thus, the Kalman gain can be written as
One important disparity between Eqs. (26) and (24) is the disappearance of R in Eq. (26). However, it is implicitly represented by the perturbed observation. In other words, the observation should be randomly perturbed when applying the general form of the Kalman gain.
In the above discussion, we only assume that the model forecast is unbiased, i.e., . If we further assume that the forecast of the measurement model is unbiased and random, i.e.,
we have
So we can see that Eqs. (28) and (29) are identical to Eqs. (11) and (11). Thus, another interpretation of Eqs. (11) and (11) is the application of the unbiased assumption to the measurement forecast, under which the linearization assumption, Eq. (12), can be removed. The assumption of Eq. (12) can clearly be seen as equivalent to the assumption of the unbiased nature of the measurement forecast. Thus, Eqs. (11) and (11) have a rigorous statistical foundation when the unbiased assumption is applied to both the model forecast and the measurement forecast.
In summary, there are three schemes used to estimate the Kalman gain in the EnKF while the measurement function is nonlinear. The similarity, disparity, and theoretical accuracy of the estimates are summarized in Table 1. Clearly, when the observation is appropriately perturbed, Scheme 3 should have the smallest estimated errors, followed by Scheme 2 and Scheme 1.