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Keywords:

  • error correction;
  • forecast sensitivity;
  • observation impact

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Correction
  5. 3. Summary and discussion
  6. Acknowledgements
  7. References

The ensemble sensitivity method proposed by Liu and Kalnay (2008) to estimate the impact of observations on reducing forecast error is shown to have a slight error and is corrected here. The corrected formula captures the actual forecast error reduction better and removes the positive bias in the estimation introduced by the original formula. Copyright © 2010 Royal Meteorological Society


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Correction
  5. 3. Summary and discussion
  6. Acknowledgements
  7. References

Liu and Kalnay (2008, hereafter LK08) proposed an ensemble-based method to estimate the impact of any or all observations on a selected measure of short-range forecast in an ensemble Kalman filter, achieving the same goal as Langland and Baker (2004, hereafter LB04), but without using the adjoints of the forecast model and the data assimilation system. Although most formulae in LK08 are correct, there is a slight error. In this letter, we correct this error and show the impact of the correction on the accuracy of estimation.

2. Correction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Correction
  5. 3. Summary and discussion
  6. Acknowledgements
  7. References

In LK08, a cost function is defined to measure the forecast-error reduction at time t due to assimilating the observations at time t = 0, which is

  • equation image(1)

where equation image is the perceived error of the forecast equation image at time t started from the analysis at t = 0 and xt is a verifying state at time t. equation image is the corresponding error of the forecast equation image started from the analysis at t = −6 h. The difference between these two forecasts is due solely to the observations equation image assimilated at t = 0.

Substituting the definitions of et|0 and et|−6 into Eq. (1), we can rewrite the cost function as:

  • equation image(2)

or

  • equation image(3)

LK08 utilized the data assimilation formula of the Local Ensemble Transform Kalman Filter (LETKF, Hunt et al., 2007) to rewrite equation image as a function of the observation increments equation image at t = 0:

  • equation image(4)

where equation image is a matrix whose columns are the forecast ensemble perturbations at time t started from the analysis at t = −6 h,

  • equation image

is a K × P intermediate matrix used in the LETKF. Here K is the number of ensemble members and P is the number of observations. For a detailed derivation of Eq. (4) (i.e. Eq. (8) in LK08), please refer to LK08.

Substituting (4) into Eq. (2) or (3) we obtain

  • equation image(5a)

or

  • equation image(5b)

Note that the cost function in Eqs (5a,b) is already the total observation impact that we are seeking (i.e. the forecast-error reduction due to the observation increments v0) for the LETKF. Unfortunately, LK08 then carried out the unnecessary step of calculating the sensitivity of the cost function J in (5b) to the observation increments v0 as

  • equation image(6)

and wrote the observation impact as

  • equation image(7)

which are Eqs (11) and (12) of LK08.

Unlike LB04 where two cost functions Jf and Jg are defined separately as the forecast error norm of et|0 or et|−6 (A3 and A4 in LB04) and therefore the inner product

  • equation image

(A10 in LB04) denotes the forecast-error difference, the cost function J defined here in Eq. (1) is already the forecast-error difference (J = {JfJg}/2). Therefore the computation of ∂J/∂v0 in LK08 is akin to taking a second derivative of the forecast error to observations and the resulting J′ is no longer the forecast-error difference due to assimilating the observations at t = 0, (the total observation impact) that we are seeking. This can be easily proven since Eqs (6) and (7) result in

  • equation image(8)

which is inconsistent with the correct cost function J given in Eq. (5b). The factor of 1/2 missing from Eq. (8) compared to (5b) is due to the unnecessary use of the derivative ∂J/∂v0.

To correct the erroneous observation impact formula (7), we simply omit Eqs (6) and (7) and rewrite Eqs (5a,b) using a scalar inner product formula as

  • equation image(9a)

or

  • equation image(9b)

Either (9a) or (9b) can be used to calculate the impact of a given subset of observations. Note that formula (9b) is similar to formula (7), except the presence of the factor 1/2, thus we refer to it as the corrected LK08 formula. The other estimation formula (9a) includes the term et|−6 + et|0 explicitly, as that in LB04 which has been shown to be a third-order approximation to the actual error reduction by considering the two model forecast trajectories in et|−6 + et|0 rather than just one of them (Errico, 2007). For the same reason, formulae (9a) and (9b) are also the third-order approximations.

The erroneous formula (7) for J′ was tested with the Lorenz 40-variable model (Lorenz and Emanuel, 1998) in LK08. Here we perform the same experiment shown in Figure 2 of LK08 but use both the original and the corrected formula (9b) to estimate the observation impacts. Since formula (9b) is similar to the original LK08 formula (except that the latter was missing a factor of 1/2), the implementation details for (9b) are exactly the same as those in LK08. Figure 1 shows the comparison of the estimated observation impact from the original incorrect formula (7) and from the corrected formula (9b) as well as the actual forecast-error reduction. It is clear that the estimated observation impact from the corrected formula is almost indistinguishable from the actual forecast-error reduction, while the impact from formula (7) has a positive bias because the term equation image in formula (9b) is always positive and therefore formula (7) doubles this positive effect by missing the factor of 1/2.

thumbnail image

Figure 1. Evolution of the spatially summed actual forecast-error reduction from the observations equation image assimilated at each analysis step (black line with open circles), the estimated observation impact from the incorrect J′ formula (7) (grey line with closed circles) and the correct J formula (9b) (black line with plus signs). Note that the black line with open circles and the grey line are exactly the same as those appearing in Figure 2 of LK08.

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thumbnail image

Figure 2. Evolution of the spatially summed forecast error differences and the estimated observation impact for the normal case between analysis cycle 5700 and 5780. The actual forecast error difference between the 24-hour and 30-hour forecasts: black line with open circles; the observation impact calculated from the adjoint method: black line with plus signs; the observation impact calculated from the ensemble method: grey line with closed circles; zero line: no impact

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3. Summary and discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Correction
  5. 3. Summary and discussion
  6. Acknowledgements
  7. References

LK08 proposed an ensemble-based method to estimate observation impact on forecast-error reduction. This method is efficient and easy to implement since it does not need the adjoints of the forecast model and the data assimilation required in the adjoint-based approach (LB04). Though most of the formulae in LK08 are correct, there was an error in the derivation of LK08 due to the use of sensitivity of forecast-error reduction with respect to observation increment rather than the error reduction itself. The corrected formula based on the error reduction J is tested with the Lorenz 40-variable model and found to remove the positive bias introduced by the incorrect formula J′ used in LK08, and to better capture the actual forecast error.

The localization length (Houtekamer and Mitchell, 2001) is an important parameter in the Ensemble Kalman Filter, and in particular in the LETKF, and is likely to affect the accuracy of the approximation in Eq. (4) and therefore the estimation given by Eqs (9a,b), since the forecast ensemble perturbations equation image at time t could propagate beyond the local patch defined at analysis time t = 0. For the low-order Lorenz-40 model, we find that the corrected formula with a large localization (a patch of 39) returns 99% of the actual reduction of errors (Figure 1), whereas with a patch size of 13 grid points this is reduced to 94% and, when using only 7 grid points, to 90%. The SPEEDY global atmospheric model (Molteni, 2003) with 30 ensemble members gives similar results (not shown). These indicate that with a strong localization, the LK08 method underestimates the actual observation impact. With more ensemble members and therefore larger local patches, this problem could be minor.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Correction
  5. 3. Summary and discussion
  6. Acknowledgements
  7. References

We are grateful to an anonymous reviewer for insightful comments on localization. This study is supported by the National Science Foundation grant 40975067 and 973 Program (2009CB421500). Authors Junjie Liu and Eugenia Kalnay are supported by DOE grants DEFG0207ER64337 and ER64437, and NASA grants NNX07AM97G and NNX08AD40G.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Correction
  5. 3. Summary and discussion
  6. Acknowledgements
  7. References