kalmanFilter.blend

Blends ensemble covariance with a climatological covariance

Syntax



obj = obj.blend(C, Ycov)


obj = obj.blend(C, Ycov, weight)


obj = obj.blend(C, Ycov, weight, whichBlend)


[C, Ycov, weight, whichBlend] = obj.blend


obj = obj.blend(‘reset’)

Description



obj = obj.blend(C, Ycov)

Blends the ensemble covariance with a climatological covariance using a weighting of 50% of each covariance. When covariance is blended, the Kalman filter first calculates the usual covariance between the ensemble and the observation estimates. Next, the filter computes the weighted sum of this ensemble covariance and a fixed climatological covariance. The weighted sum of the two covariances is then used to update the ensemble. This technique is often used with offline evolving priors: the resulting covariance reflects both the evolving covariance from the ensemble, while the climatological covariance can help reduce sampling errors for small ensembles. Blending is applied after inflation and localization are applied to the ensemble covariance. Neither inflation nor localization are applied to the climatological covariance.

The first input provides the climatological covariance between the state vector elements and the observation sites. It should be a 3D array with one row per state vector element, and one column per observation site. The second input provides the climatological covariance between the observation sites and one another. It should be a 3D array with one row and one column per observation site. Each element along the third dimension of Ycov must be a symmetric matrix.

The two inputs must have the same number of elements along the third dimension. If the third dimension has one element (i.e. both inputs are matrices), then blends the same climatological covariances in all time steps. Otherwise, the third dimension must have one element per time step (although see the thrid syntax for relaxing this requirement). C may contain NaN elements, although the filter will return a NaN update at any state vector rows with a NaN covariance. Ycov cannot include NaN elements.



obj = obj.blend(C, Ycov, weight)

Specify the blending weight to use for each climatological covariance. By default, uses a blending weight of 0.5. The blending weight for the ensemble covariance is calculated as 1 - weight. Blending weights must be on the interval 0 < weight < 1. If you provide a single weight, uses the same weight for each set of covariances. Otherwise, the number of weights must match the number of sets of covariances.



obj = obj.blend(C, Ycov, weight, whichBlend)

Indicate which set of climatological covariances to use in each time step. This syntax allows the number of climatological covariances to differ from the number of assimilation time steps.



[C, Ycov, weight, whichBlend] = obj.blend

Returns the current climatological covariances and blending weights for the Kalman Filter.



obj = obj.blend(‘reset’)

Deletes any existing climatological covariances and blending weights from the Kalman filter.

Input Arguments

C

numeric 3*D* array [nState x nSite x 1|*nTime*|*nLoc*]
The climatological covariance between the state vector elements and observation sites. If using a single set of covariances, should have a single element along the third dimension. If the number of elements along the third dimension matches the number of time steps, uses the indicated set of covariances in each time step. If the number of elements is neither 1 nor the number of time steps, use the whichBlend input to indicate which climatological covariance to use in each time step. May include NaN values, but the filter will return a NaN update for state vector elements with NaN covariances.

Ycov

numeric 3*D* array [nSite x nSite x 1|*nTime*|*nLoc*]
The climatological covariance between the observation sites and one another. A 3D array with one site and one column per observation site. The number of elements along the third dimension must match the number of elements for the C input. Cannot include NaN values. Each element along the third dimension must be a symmetric matrix and the diagonal elements of these matrices must all be positive.

weight

numeric vector [1 | nLoc]
The blending weight to use for the climatological covariance. Must be on the interval 0 < weight < 1. The blending weight for the ensemble covariance is calculated as 1 - weight. If you provide a single weight, uses the same blending weight for all covariances. Otherwise, must have one element per covariance. By default, the blending weight is set to 0.5.

whichBlend

vector, linear indices [nTime]
Indicates which set of covariances to blend in each time step. Must have one element per assimilation time step. Each element is the index of one of the sets of climatological covariances.

Output Arguments

obj

scalar kalmanFilter object
The kalmanFilter object with updated blending options

C

numeric 3*D* array [nState x nSite x 1|*nTime*|*nBlend*] | []
The current blending covariances between the state vector elements and observation site for the filter. If blending has not been implemented, returns an empty array.

Ycov

numeric 3*D* array [nSite x nSite x 1|*nTime*|*nLoc*] | []
The current blending covariances between the observation sites and each other for the filter. If blending has not been implemented, returns an empty array.

weight

numeric vector [nLoc] | []
The current blending weight for each climatological covariance. If blending has not been implemented, returns an empty array.

whichBlend

vector, linear indices [nTime] | []
Indicates which set of covariances will be blended in each time step. If there is a single set of covariances, returns an empty array.