1. Introduction |
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In Var., background error covariances are represented as a set of control variable transforms. Any approach to modelling the transforms in this way leads to approximations which come to light in analyses of the implied covariances. A good background error covariance model should reproduce the position and scale dependencies of forecast errors. The background error covariance model used currently in Var. is unable to achieve this task well, but it is supposed that the scope of the model to capture both position and scale dependencies simultaneously is improved by using a waveband (or wavelet-like) approach, called the "waveband summation" (WS) method.
In [VWP19], a number of WS models are discussed, and are summarized at the start of section 6 in [Bannister]. In the rest of [Bannister], the formulation of one particular waveband model - the simplest - is studied in detail. This includes the transform definition, its pseudo-inverse (it has no exact inverse) and possible procedures for determining the many coefficients in the transforms, which can be estimated by running pairs of forecasts (or NMC runs). In preparation for a publication of this work, this document presents results from a simple cut-down and off-line version of the WS transform run at Reading. Investigations have been performed into the ability of the WS transform to capture the position and scale dependencies of forecast errors and the accuracy of the pseudo-inverse transform. The results are compared to other representations of background errors - the exact approach (the explicit B-matrix for the cut-down system) and the current Met Office spatial transforms (sections 4.2 and 4.3 of [Bannister]). The structure of this document is as follows. In section 2 we review the version of the WS transform used, and in section 3 we describe the experiments used to diagnose the properties of this model. Section 4 presents the results and section 5 describes them. References
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2. The WS transform, the data and the methodology |
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In the following lists, equation, figure and section numbers refer to [Bannister], unless indicated otherwise.
The transform used
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3. The experiments |
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What representations of B are used?
![]() The WS transform has no exact inverse. What tests have been done to assess the validity of the approximate inverse (section 6.3)?
(*) The configuration expected to be 'best' is used for experiments E and F. The 'optimal' number of bands is four (J=3) using top-hat functions for calibration (T-transform) and triangular functions for the implied covariance diagnostics (U-transform). |
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4.1 The results |
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The following documents show the results. Each of the first two links below are divided into two. The upper parts show the exact results (experiments A and B) - these are what we are trying to reproduce with the WS scheme - and the lower parts show the remaining experiments. The results are discussed briefly below.
Implied diagnostics results for streamfunction. Implied diagnostics results for temperature. Inverse transform tests for streamfunction. Inverse transform tests for temperature. Discussion of the diagnostics for streamfunctionThe exact diagnostics for streamfunction (Figs. A.1 and A.2) The 'exact' diagnostics show a strong variance of streamfunction of 0.225 units in the southern hemisphere centred around 50 degrees south and level 28 (Fig. A.1). Mirrored in the northern hemisphere is a much weaker variance structure, but is more spread out in latitude and joins the weak peak of variance close to the tropical tropopause. There is a lot of variance in the tropical and northern stratosphere. There is some variation of vertical scale with horizontal scale (Fig. A.2). Substantial correlations extend up to level 42 which peaks at a wavenumber roughly kmax/4. There is some negative correlations with the top and bottom of the model at roughly 2/3 kmax.The MetO diagnostics for streamfunction (Figs. C.1 and C.2) The current MetO scheme does a reasonable job at reproducing these. The variances have peaks in the right places with the right strengths (Fig. C.1), but the horizontal scale variation of the vertical scale is too weak. This strong position dependence and weak, not non-zero, scale dependence is what is expected and is a known property of the MetO transforms. Note that the scale dependence is non-zero because the MetO scheme is not exactly separable. The WS diagnostics for streamfunction, 4 bands, top-hat(T), triangular(U) (Figs. E.1 and E.2) The WS transform with four bands, which has been calibrated using top-hat band-pass functions, but used in the diagnostics with triangular functions does not perform well. The variances (Fig. E.1) peak in the right places but they are far too weak and are spread out. The scale variation (Fig. E.2) is still very weak. These are surprising results. It was expected that there would be some degradation of spatial scale with WS, but it was not expected to be so bad. It was expected that the scale dependence would be superior to the MetO scheme, but it is found to be comparable. The WS diagnostics for streamfunction, 4 bands, top-hat(T), top-hat(U) (Figs. G.1 and G.2) Using the same number of bands, but top-hat band-pass functions throughout (both calibration and diagnostics stages) does not help (Figs. G.1 and G.2). The band boundaries are visible in places in Fig. G.2. The WS diagnostics for streamfunction, 4 bands, triangular(T), triangular(U) (Figs. I.1 and I.2) Using the same number of bands, but triangular band-pass functions throughout (Figs. I.1 and I.2) gives worse results, especially for the variances. The WS diagnostics for streamfunction, 1 bands, top-hat(T), triangular(U) (Figs. K.1 and K.2) We revert back to the use of top-hat functions when calibrating and triangular functions when computing the diagnostics, but we change the number of bands. Using just one band (Figs. K.1 and K.2) gives excellent results for the spatial variation of variances. These are almost identical to the exact results. There is virtually no variation of vertical scale with horizontal scale though in the vertical correlations. This is expected for this limit. Note that in this limit, the results are not expected to be the same as the MetO scheme as here we have rotated the vertical modes from a vertical 'EOF' index, as used at the MetO scheme, back to height. There is thus here no means of associating vertical and horizontal scales. The WS diagnostics for streamfunction, 10 bands, top-hat(T), triangular(U) (Figs. M.1 and M.2) The case with many (ten) bands (Figs. M.1 and M.2) gives a very similar picture to that in the case with just four bands. This is surprising as it is expected to have further degraded position dependencies and improved scale dependencies. Please see the note below about vertical eigenvalues of vertical covariances matrices for streamfunction. Discussion of the diagnostics for temperatureThe exact diagnostics for temperature (Figs. B.1 and B.2) The 'exact' diagnostics show a strong variation of variance of temperature with latitude (Fig. B.1). there are double-peaked structures in each hemisphere at around 50 degrees, but the variance in the northern hemisphere extends to the pole. The strongest variances are in the southern hemisphere. the lower peak has a maximum of 25 units at level 18 and the upper peak has a maximum of about 45 units at level 41. There is some variance in the stratosphere. There is a distinct variation of vertical correlation scale with horizontal scale (Fig. B.2). The zero correlation line varies from level 28 at 0 wavenumber to level 24 at kmax with strong negative correlations above this line.The MetO diagnostics for temperature (Figs. D.1 and D.2) The current MetO scheme does a reasonable job at reproducing these. The variances (Fig. C.1) have peaks in the right places, including the double peaks, but the variances are too weak. The bands of positive and negative correlations (Fig. D.2) are present, but much of the variation of the correlation pattern is lost. In particular, the zero correlation line is flat. HERE The WS diagnostics for streamfunction, 4 bands, top-hat(T), triangular(U) (Figs. E.1 and E.2) The WS transform with four bands, which has been calibrated using top-hat band-pass functions, but used in the diagnostics with triangular functions does not perform well. The variances (Fig. E.1) peak in the right places but they are far too weak and are spread out. The scale variation (Fig. E.2) is still very weak. These are surprising results. It was expected that there would be some degradation of spatial scale with WS, but it was not expected to be so bad. It was expected that the scale dependence would be superior to the MetO scheme, but it is found to be comparable. The WS diagnostics for streamfunction, 4 bands, top-hat(T), top-hat(U) (Figs. G.1 and G.2) Using the same number of bands, but top-hat band-pass functions throughout (both calibration and diagnostics stages) does not help (Figs. G.1 and G.2). The band boundaries are visible in places in Fig. G.2. The WS diagnostics for streamfunction, 4 bands, triangular(T), triangular(U) (Figs. I.1 and I.2) Using the same number of bands, but triangular band-pass functions throughout (Figs. I.1 and I.2) gives worse results, especially for the variances. The WS diagnostics for streamfunction, 1 bands, top-hat(T), triangular(U) (Figs. K.1 and K.2) We revert back to the use of top-hat functions when calibrating and triangular functions when computing the diagnostics, but we change the number of bands. Using just one band (Figs. K.1 and K.2) gives excellent results for the spatial variation of variances. These are almost identical to the exact results. There is virtually no variation of vertical scale with horizontal scale though in the vertical correlations. This is expected for this limit. Note that in this limit, the results are not expected to be the same as the MetO scheme as here we have rotated the vertical modes from a vertical 'EOF' index, as used at the MetO scheme, back to height. There is thus here no means of associating vertical and horizontal scales. The WS diagnostics for streamfunction, 10 bands, top-hat(T), triangular(U) (Figs. M.1 and M.2) The case with many (ten) bands (Figs. M.1 and M.2) gives a very similar picture to that in the case with just four bands. This is surprising as it is expected to have further degraded position dependencies and improved scale dependencies. Please see the note below about vertical eigenvalues of vertical covariances matrices for streamfunction. Discussion of the inverse tests for streamfunctionDiscussion of the inverse tests for temperature |