Numerical methods for Data Assimilation

Kalman filter

Optimality transfer for observations

  • We consider .

  • We set

  • Is it possible to update de minimum argument to get the minimum argument ?

  • Application. Observations are obtained repeatly and we want to update the estimate accordingly, without processing all the information from the beginning.

  • Note that from Theorem(19), the minimum covariance matrix for the estimation of

    and

FundamentalTheorem

For some constant , we have .

Therefore, ,

where .

Proof

Since minimizes , we have the Taylor expansion

.

But is quadratic, therefore,

, and we have

, which yields the result.

This Theorem shows how to update incrementaly the estimation for new coming observations. Note that the algorithm updates the solution and the covariance matrix.

Incremental least squares algorithm

We want to find that minimizes

.

The following algorithm finishes with .

Incremental least squares
  1. Set

  2. For k=0,2, ... N Do

  3. Compute

  4. Compute

  5. EndDo

Note that this algorithm may be unstable in the presence of rounding-errors (on a computer) and that alternative formulations based on orthogonal transformations are available.

Optimality transfer for model errors

  • We consider and .

  • We set .

  • Is it possible to update the minimum argument of to get the minimum argument of ?

  • Application. Now we get observations from an object that is moving. The position of the object at is , and we would like to know the current position at of the object "knowing" an approximate dynamics and noisy observations .

FundamentalTheorem

For some constant , some vector , and some matrix ,

we have .

Therefore, , where

and .

Proof

From bloc Gaussian elimination, we have

,

where .

We have

We denote , and get

.

The conclusion follows from the fact that the minimum for is obtained for , and Theorem(19) provides an expression for minimizing

.

Kalman filter

We want to find such that minimizes

.

The following algorithm finishes with .

Kalman filter
  1. Set

  2. For k=0,2, ... N Do

  3. Compute

  4. Compute

  5. Compute

  6. Compute

  7. EndDo

Again, algorithm may be unstable in the presence of rounding-errors (on a computer) an use orthogonal transformations recommended.

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HomepageHomepagePrintPrint S. Gratton and Ph. Toint, submitted to Open Learn. Res. Ed. INPT 0502 (2013) 6h Attribution - Share AlikeCreated with Scenari (new window)