## 1. Introduction

Objective atmospheric data analysis is widely used in observational studies, numerical weather prediction (NWP), and for verifying new models and theories. In many analysis schemes, observations are optimally used together with a background field in a statistical sense. At most NWP centers, a particular implementation of an Optimal Interpolation (OI) (Eliassen 1954; Gandin 1963) or a variational data assimilation (VAR) (Lorenc 1986) is used. In the OI and VAR formulations, the observation and background error covariance matrices, **R****B****R****B**^{5} × 10^{5} to 10^{7} × 10^{7} for realistic atmospheric applications (Courtier 1997). Special considerations are needed in handling of **R****B**

In most OI implementations **B****HBH**^{T}, where **H****HBH**^{T} + **R****HBH**^{T} + **R**

In most VAR implementations there is no OI-type data selection. The most common assumption about **R****R****B****B****B****B****B**

In this paper the first problem is addressed by modifying the standard variational analysis scheme, so that the inversion is not needed. The second problem is then addressed by formulating a new scheme that uses a filter to replace the use of the matrix so that **B**

This paper is organized as follows. The standard, the modified, and the new schemes are described in section 2 together with comparisons to other related schemes. An illustrative example is given in section 3 using real 2-m temperature observations, followed by another more realistic example in section 4. The conclusions are summarized in section 5.

## 2. Analysis methods

### a. VAR: Variational analysis

**x**is the analysis vector (on model grid points),

**x**

^{b}is the background vector (on model grid points),

**y**is the observation vector (on observation locations),

*H*is the (nonlinear) observation operator that converts model variables to observations,

**B**

**R**

^{T}indicates transpose, and ( · )

^{−1}indicates inversion. In most practical implementations a penalty term for noise control is included in addition to

*J*

_{b}and

*J*

_{o}, but it is omitted here. The notations closely follow Ide et al. (1997).

**x**

^{0}, normally

**x**

^{0}=

**x**

^{b}, the final analysis

**x**

^{VAR}=

**x**

^{∞}can be obtained using a minimization algorithm. The simplest method is the so-called steepest descent, which can be expressed as

**x**

^{n+1}

**x**

^{n}

*α*

**∇**_{x}

*J,*

*α*is a constant (

*α*> 0), and

**∇**_{x}

*J*is the gradient of the cost function

*J*with respect to

**x**:

**∇**_{x}

*J*

**B**

^{−1}

**x**

**x**

^{b}

**H**

^{T}

**R**

^{−1}

*H*

**x**

**y**

**H**

*H.*

Given *N* model grid points and *M* observation locations, then **x** and **x**^{b} are vectors of length *N,* **y** is a vector of length *M,* **B***N* × *N* matrix, and **R***M* × *M* matrix.

### b. VAN: Variational analysis with no inversion of **B**

**B**

**B**

**v**is introduced, defined as

**v**

**B**

^{−1}

**x**

**x**

^{b}

*J*can now be written as

The transform from **x** to **v** using (3) and the redefinition of the cost function with respect to **v** were proposed by Lorenc (1988) and Derber and Rosati (1989) to speed up the convergence of descent algorithms such as the steepest descent and conjugate-gradient methods. In this note, VAN is used to provide guidance for the formulation of a new scheme. VAN is also used as a reference when the new scheme is tested.

**v**as the control variable, the minimization scheme (1) is now rewritten as

**v**

^{n+1}

**v**

^{n}

*α*

**∇**_{v}

*J,*

**∇**_{v}is the gradient of

*J*with respect to

**v**:

**∇**_{v}

*J*

**B**

^{T}

**v**

**H**

^{T}

**R**

^{−1}

*H*

**B**

**v**

**x**

^{b}

**y**

**v**

^{0},

**v**

^{0}

**B**

^{−1}

**x**

^{0}

**x**

^{b}

*J*is reached at

**v**

^{∞}and the final analysis

**x**

^{VAN}becomes

**x**

^{VAN}

**x**

^{b}

**B**

**v**

^{∞}

**B**

^{−1}is only needed to provide the initial guess value for the control variable. Consequently, if we assume

**v**

^{0}

**B**

**B**

The preconditioning has not been discussed. As the optimal preconditioning is the inverse of the Hessian matrix, which is (**B**^{T} + **B**^{T}**H**^{T}**R**^{−1}**HB**^{−1} for VAN, some simplifications are needed to really avoid the explicit inversion of **B**

### c. VAF: Variational analysis using a filter

In NWP applications, the matrix **B***N* × *N* ∼ 10^{14}) is too large to be stored in memory by present-day computers and therefore has to be calculated every time it is needed. In VAN, **B****∇**_{v}*J,* reveals that **B****B****B**

**x**by the

**B**

**x*** has its elements

*b*

_{ij}are the elements of

**B**

*i*= 1,

*N*;

*j*= 1,

*N*). In the case of a uniform grid, (6) could be considered as filtering

**x**with a spatial filter with the filter coefficients

*b*

_{ij}and the filter span covering the whole model domain. Therefore the VAN scheme can be written by using a filter to model

**B**

**B**

**B**

*σ*

_{b}is the standard deviation of the background error,

*r*

_{ij}is the distance between model grid points

*i*and

*j,*and

*L*is a length scale that can be determined theoretically or by observations. If the

**B**

The motivation for using a filter to model **B****B****B****B****B****B**

*N*

_{λ}×

*N*

_{ϕ}=

*N, λ,*and

*ϕ*denote coordinates in the west–east and south–north directions, respectively. A two-dimensional filter is chosen for

**x*** is a filtered vector with its elements

*x*

^{*}

_{i}

*g*

_{ij}is the filter coefficient,

*w*

_{ij}is the Lanczos window,

*I*and

*J*are the filter orders in the

*λ*and

*ϕ*directions, Δ

*λ*and Δ

*ϕ*are grid distances in the

*λ*and

*ϕ*directions, and

*L*

_{f}is a length scale related to

*L*in

*b*

_{ij}. Note that the expression in the curly brackets is the same as

*b*

_{ij}except for the length scale. In practice,

*L*is determined by statistics while

*L*

_{f}is based on

*L*and modified due to the filter truncation and window selection.

*I*+ 1)(0.5

*J*+ 1) and the computation required is <(

*I*+ 1)(

*J*+ 1) ×

*N*(< is used here because the filter does not take on values outside of the analysis domain). The choice of

*I*and

*J*depends on the characteristics of

**B**

Take a typical two-dimensional regional analysis domain, 100 × 100 grid points, with the grid distance Δ*λ* = Δ*ϕ* = 30 km and the error covariance scale *L* = 200 km. The dimension of **B****B***N* × *N* = 10^{8} and 2 × 10^{8}, respectively. The required storage size for the filter is approximately (0.5*I* + 1)(0.5*J* + 1) ≈ 10^{2} and the filter multiplications required by each VAF iteration is 2 × (*I* + 1)(*J* + 1) × *N* ≈ 8 × 10^{6}. (For a VAR scheme using the spectral technique, the storage required for *b*^{*}_{ii}*N* = 10^{4}.)

If the background error at every analysis grid point is correlated with all other grid points, 2(*N*_{λ} − 1) and 2(*N*_{ϕ} − 1) should be chosen for *I* and *J.* In this special case, the VAF scheme without the Lanczos window should give exactly the same solution as the VAN scheme, which makes this particular case useful for checking a VAF implementation.

If the background constraint is removed from the VAF scheme, that is, take away the first term **v** from the gradient of the cost function **∇**_{v}*J,* the scheme becomes a form of successive correction (Bergthorsson and Döös 1955; Cressman 1959; Barnes 1964), which could lead to an analysis fitting observations exactly when the iterations converge. However, if the background contains useful information, as it does for most NWP applications, the constraint should remain. An in-depth discussion on the relations among different analysis schemes can be found in Lorenc (1986).

**B**

**B**

**v**,

**B**

**v**

**x**

**x**

^{b}

**B**

**B**

**B**

**B**

**B**

**B**

**B**

**B**

**B**

**B**

**B**

### d. BLUE: The Best Linear Unbiased Estimation

**H**

**B**

^{−1}

**x**

**x**

^{b}

**H**

^{T}

**R**

^{−1}

**H**

**x**

**y**

**x**

**x**

^{b}

**B**

^{−1}

**H**

^{T}

**R**

^{−1}

**H**

^{−1}

**H**

^{T}

**R**

^{−1}

**y**

**H**

**x**

^{b}

**x**

^{BLUE}

**x**

^{b}

**BH**

^{T}(

**HBH**

^{T}

**R**

^{−1}

**y**

**H**

**x**

^{b}

**x**

^{BLUE}is the BLUE solution, which in the linear case should be the final solution of variational schemes and therefore can be used as a reference.

The numerical solution for BLUE depends on the accuracy of (**HBH**^{T} + **R**^{−1}, which could be problematic, especially when *M* is large. The size of the matrix **BH**^{T} is *N* × *M.* The elements of this matrix are the background error covariance between all analysis grid points and observation locations. The size of **HBH**^{T} is *M* × *M.* The elements of this matrix are the background error covariances between all observation locations. Simplifications to **BH**^{T} (dimension *N* × *M*) and (**HBH**^{T} + **R**^{−1} lead to the widely used OI scheme. In most OI scheme implementations (e.g., Lönnberg and Shaw 1987), different data selection strategies are used to reduce *M* and *N.* With the reduced *M,* both **HBH**^{T} and **BH**^{T} are approximated and directly coded.

### e. PSAS: Physical-space Statistical Analysis Scheme

**B**

**w**exactly, which needs the inversion of (

**HBH**

^{T}+

**R**

**w**through minimizing the following cost function:

**x**

^{PSAS}

**x**

^{b}

**BH**

^{T}

**w**

^{∞}

**w**

^{∞}is the minimization result.

Although PSAS does not invert (**HBH**^{T} + **R***M* × *M* matrix could still lead to significant computational expenses. In the implementation of Cohn et al. (1998), simplifications are used to make the matrix sparse, which in a way mimics the OI data selection.

The major difference between PSAS and VAN (VAF) is where the control variable is based. In PSAS, the minimization is performed at the observation locations. The research efforts have been devoted to simplifying (**HBH**^{T} + **R****B****B**

## 3. An illustrative example

To demonstrate the use of the VAN and VAF schemes described in section 2, a small-sized two-dimensional univariate analysis problem, for which the BLUE solutions are easily obtained as references to evaluate the new schemes, is considered. (A numerical lab, ANALAB, for VAR, VAN, VAF, OI, and PSAS schemes, written in FORTRAN-77, has been developed by the author for educational purposes at the Danish Meteorological Institute. The complete ANALAB is available from the author. ANALAB is based on the OILAB, which has been used in undergraduate courses for many years at the Department of Meteorology, Stockholm University.)

### a. Grid, *y, x*^{b}, and **H**

**H**

A regular latitude–longitude grid is chosen. The number of analysis grid points is 21 × 21 (*N* = 441). The grid resolution is 0.3° in latitude and 0.6° in longitude. The analysis domain is shown in Fig. 1. The center of the model domain is located at (56.5, 14.0). Using this coordinate, the grid resolution is about 37 km in the east–west direction and 33 km in the north–south direction.

Real 2-m temperature observations are used in this illustrative example. The observations are shown in Fig. 1. They are collected mainly in Denmark and southern Sweden at 0300 UTC 3 March 1992. A total of 38 observations are in the database (*M* = 38). The observation locations are indicated by stars. The observed values in degrees Celsius are printed to the right of their locations.

The background temperature field is also shown in Fig. 1. It is clear that the large-scale features of the temperature field are captured by the background. It is the purpose of the analysis with a smaller error covariance scale to extract the small-scale information in the observations in a statistically optimal way.

The observation operator, **H**

### b. **R** , **B** , and G

**R**

**B**

In the following experiments, the observation errors are assumed noncorrelated. The matrix **R***σ*^{2}_{o}*σ*_{o} = 1°C.

The background errors are assumed correlated and the Gaussian function, defined in (8), is used to calculate the elements in **B***σ*_{b}, is also chosen to be 1°C. Using the analysis grid defined earlier, the error covariance between one arbitrary grid point and the neighboring grid points is calculated using (8) and plotted against distance in Fig. 2. In the figure, b200 and b100 are covariances using *L* = 200 km and *L* = 100 km, respectively. In the testing of the VAF code, the 40th-order Gaussian filter without the Lanczos window is used and the filter coefficients, g200_40_nowin, are identical to b200 plotted in Fig. 2.

From the figure, it is clear that the covariance becomes less than 10% only when two grid points are separated by more than 10 grid points when *L* = 200 km and by more than 5 grid points when *L* = 100 km. Using these numbers, a 20th-order filter and a 10th-order filter are chosen in the VAF experiments.

Due to the Lanczos window, the effective scale of the Gaussian filter is reduced. To compensate this scale reduction, *L*_{f} = 260 km is chosen for the 20th-order filter and *L*_{f} = 128 km is chosen for the 10th-order filter. (The choice of *L*_{f} will be discussed later.) The Gaussian filter coefficients, calculated using (9), as functions of distance are also plotted in Fig. 2, where g260_20 is for a 20th-order filter and g128_10 is for a 10th-order filter. As can be seen from the figure, these two filters closely resemble the two covariance functions.

### c. The BLUE solution

The BLUE analyses are shown in Fig. 3. They are obtained by using *L* = 200 km and *L* = 100 km, respectively. As has been discussed in section 2d, the BLUE solution is the final solution of the variational schemes assuming the observation operator is linear and the inversion of (**HBH**^{T} + **R**

### d. Minimization

To solve the variational problems discussed in section 2 an algorithm for minimizing the cost function is needed. In this study, for illustration purposes, the steepest descent minimization algorithm is used. There is no general rule for the choice of the minimization step size, *α,* used in (4). In the experiments described in this paper, *α* = 0.002 is chosen, which is approximately the largest value with which the steepest descent minimization method converges for all the experiments.

### e. VAN solutions

In the experiment using the VAN scheme, the parameters are set to be the same as those used to obtain the BLUE solution with *L* = 200 km. The cost functions *J*_{o}, *J*_{b}, and *J* at each iteration (up to 1000) are plotted in Fig. 4a. Note that a logarithmic scale is used.

The choice of the analysis grid, the number of observations, the observation, and back ground error covariances leads to *J*_{o}/*J*_{b} ∼ 5; that is, the difference between the cost functions of the analysis and observations is about five times larger than that between analysis and background. With different parameters this ratio can change. As can be seen from Fig. 4a, the largest changes in the cost function occur during the first iterations. Even 10 iterations are enough for practical purposes.

Since the BLUE solution could be considered as the final solution for VAN iterations, the rms difference between the VAN solution at each iteration and the BLUE solution is plotted in Fig. 4b. Using this measure, it is shown that the convergence to the final solution could take more time. However, the rms difference is less than 0.1°C after 10 iterations.

The VAN analyses after 1, 10, 100, and 1000 iterations are shown in Fig. 5. The VAN analysis obtained after one iteration is already quite different from the background and has many features of the BLUE solution. The VAN analyses after 10, 100, and 1000 iterations could be considered the same using any subjective judgment, although minor differences, for instance close to the northern boundary, do exist.

### f. VAF solutions

Using the Gaussian filter (9) with different orders, a series of experiments using the VAF scheme are conducted. First, a 40th-order filter without the Lanczos window is used as a check of the coding, since the VAF scheme is equivalent to the VAN scheme when the filter parameter (0.5*I* + 1) × (0.5*J* + 1) is equal to the number of grid points of the analysis domain *N*_{λ} × *N*_{ϕ}, which are all 21 × 21. The VAF results (not shown) are identical to the VAN results as expected.

Reducing the filter order by a factor of 2 (*I* = *J* = 20), the solutions (Fig. 6) are practically the same as the VAN solutions (Fig. 5). The solutions after 10 iterations are actually very close to the BLUE solution using *L* = 200 km (Fig. 3a).

A further halving of the filter order (*I* = *J* = 10) leads to a VAF solution with a smaller spatial scale (Fig. 7). As expected from the characteristics of the filter, the solutions converge to the BLUE solution using *L* = 100 km (Fig. 3b).

### g. Selection of *L*_{f} for VAF

Ideally the background error covariance matrix, **B***b*_{ij} and, in this case, the length scale *L* is computed from statistics. In other words, *L* should not be a free parameter to be adjusted.

As discussed in sections 2c and 3b, the length scale *L*_{f} used in VAF schemes is not necessarily the same as *L* due to the filter truncation and window selection. In most cases, *L*_{f} should be longer than *L.*

Using the BLUE solution as the “true solution” and the VAN solution as a reference solution, numerical experiments have been performed varying *L*_{f}. The results are summarized in Fig. 8, where the rms difference between VAF and BLUE solutions are shown as a function of iterations for the selected experiments. Note that logarithmic scales are used. There are two groups of experiments, one for VAN using *L* = 200 km and the 20th-order filter (Fig. 8a), and the other for VAN using *L* = 100 km and the 10th-order filter (Fig. 8b).

The selection of *L*_{f} = 260 km for *L* = 200 km and *L*_{f} = 128 km for *L* = 100 km was based on the final solution of each experiment, which has the minimum in the rms difference. As can be seen from the figure, the VAN solution is best at the end of iterations. This should be expected, since VAF is just an approximation to VAN.

A closer inspection of Fig. 8 also reveals a few interesting aspects of the VAF solutions. First of all, within the first 10 iterations the VAF solutions are not sensitive to *L*_{f} for a large range of *L*_{f}. This means that in practical applications the precise value of *L*_{f} may not be crucial. Second, there is an iteration range between 10 and 100 in which VAF solutions could be better than the VAN solutions. In practical implementations, there is often a predetermined maximum iteration number that could be in this range. In that case, it may be possible to improve the analysis by taking a *L*_{f} that is somewhat shorter than that selected based on the final solutions. Third, the distance between some VAF solutions and the “true solution” decreases through the early iterations and reaches the minimum at, for example, around 100 iterations. Further iterations would lead to an increase in the distance. Finally, it should be stressed that the rms difference between different solutions is plotted on logarithmic scales in Fig. 8 and is much smaller than the assumed errors in both background and observations.

## 4. A more realistic example

To give a more realistic picture, some of the experiments described in section 3 have been repeated with the following changes:

An increase of analysis grid points from 21 × 21 to 101 × 101.

An increase of observations from 38 to 1062.

*λ,*Δ

*ϕ,*

*σ*

_{o}, and

*σ*

_{b}, remain unchanged except that

*α*= 0.0005. The dimension of

**B**

^{8}, which makes the VAN scheme impossible to run even on some supercomputers. Therefore only a VAF solution is shown. The dimension of (

**HBH**

^{T}+

**R**

^{6}, which makes the matrix inversion of the BLUE scheme nontrivial to handle.

In this example, 2-m temperature observations at 0000 UTC 5 May 1999 within 35°–65°N and 10°W–50°E are selected from the Danish Meteorological Institute operational database. The locations and values of the observations are given by Fig. 9. The full analysis domain is shown in the figure. As in the previous section, the background field has the reasonable, large-scale structure of the observations (Fig. 9a).

Both the BLUE solution (with *L* = 200 km; Fig. 9b) and the VAF solution (with *L*_{f} = 260 km at the 1000th iteration; Fig. 9c) add smaller-scale information from observations to the background. Comparing the two analyses subjectively, it is obvious that the VAF solution is a good approximation of the BLUE solution. To give a quantitative comparison, the rms difference between the VAF and BLUE solutions as a function of iterations is plotted in Fig. 10. Based on this figure and the results from the previous section, it could also be argued that 10 iterations may be enough for the VAF scheme to produce an analysis of approximately the same quality as the BLUE solution.

## 5. Conclusions

In this paper, we have addressed two issues related to the variational data assimilation, VAR, as formulated by Lorenc (1986). The first is how to avoid the inversion of the large dimensioned background covariance matrix **B****B**

Following Lorenc (1988) and Derber and Rosati (1989), we have shown that a simple transform of the control variable leads to a modified scheme, named VAN—variational analysis with no inversion of the background error covariance matrix. The remaining storage and computation difficulties in VAN are mainly due to the presence of the matrix itself. Although a full-sized implementation of the VAN scheme is still difficult, the VAN scheme is considered as a step forward from VAR. The problems related to the inversion of a very large and often ill-conditioned matrix are avoided.

A limitation of the VAN scheme is the lack of preconditioning. If some preconditioning, for example, using the inverse of the Hessian matrix, is needed, the inversion of the background error covariance matrix may enter the preconditioning and approximations would be needed to avoid inversions.

Based on the VAN scheme, we have formulated a new scheme, named VAF—variational analysis using a filter, which uses a spatial filter to model the background error covariance matrix and does not require the matrix explicitly. Using a truncated filter the spatial scale of the analysis is changed slightly. A redefinition of the length scale in the filter is necessary to adjust the effective scale of the analysis. The storage and computation required by the VAF scheme could be orders of magnitude smaller than the VAR and VAN schemes. A full-sized implementation of the VAF scheme is possible.

To illustrate how these schemes work, a small-sized and a more realistic two-dimensional univariate analysis problem are considered using real 2-m temperature observations. As the observation operator involved is linear, the best linear unbiased estimates are used as reference. It is shown that both VAN and VAF work satisfactorily.

In this paper, a Gaussian function is used to model **B****B****B****B**

In the construction of the spatial filter, a Lanczos window is introduced to reduce the truncation-related Gibbs oscillations. During the development of the VAF scheme, many problems caused by noise have occurred when the truncation was severe and the window was inactive. The Lanczos window is our first attempt and seems to be quite effective in removing noise. Although other windows may be used, a proper comparison method needs to be established to select a better window.

One of the advantages of using VAR, VAN, or VAF is the ability of assimilating indirect data. We plan to use the VAF scheme to include the ground-based Global Positioning System data in the type of analyses shown here. A more sophisticated minimization algorithm and a better preconditioning algorithm, for example as discussed by Dévényi and Benjamin (1998), will have high priority in the future development. A nonuniform grid version of VAF needs to be derived, as shown by Lorenc (1992), for more realistic implementations. A full-sized three-dimensional multivariant problem should be considered. In order to make efficient use of asynoptic data and to obtain a coherent four-dimensional analysis, a time-dependence and a numerical weather prediction model should also be included by the VAN and VAF schemes. These will be research subjects in the future.

## Acknowledgments

The author would like to thank Dezsö Dévényi, Nils Gustafsson, Erland Källén, Leif Laursen, Andrew Lorenc, Peter Lynch, and Henrik Vedel for comments on an earlier version of the manuscript. Comments from anonymous reviewers are also appreciated. This work has partly been supported by the Danish Research Councils through Grant 9702667.

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The background error covariance [*b*(*r*)] and the Gaussian filter coefficients [*g*(*r*)], as functions of distance (*r*).

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The background error covariance [*b*(*r*)] and the Gaussian filter coefficients [*g*(*r*)], as functions of distance (*r*).

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The background error covariance [*b*(*r*)] and the Gaussian filter coefficients [*g*(*r*)], as functions of distance (*r*).

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

BLUE analyses using (a) *L* = 200 km and (b) *L* = 100 km.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

BLUE analyses using (a) *L* = 200 km and (b) *L* = 100 km.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

BLUE analyses using (a) *L* = 200 km and (b) *L* = 100 km.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

(a) The cost functions *J*_{o}, *J*_{b}, and *J* at each iteration. (b) The rms difference in °C between VAN solution at each iteration and the BLUE solution. Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

(a) The cost functions *J*_{o}, *J*_{b}, and *J* at each iteration. (b) The rms difference in °C between VAN solution at each iteration and the BLUE solution. Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

(a) The cost functions *J*_{o}, *J*_{b}, and *J* at each iteration. (b) The rms difference in °C between VAN solution at each iteration and the BLUE solution. Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAN analyses obtained after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAN analyses obtained after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAN analyses obtained after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAF analyses using a 20th-order Gaussian filter after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAF analyses using a 20th-order Gaussian filter after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAF analyses using a 20th-order Gaussian filter after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAF analyses using a 10th-order Gaussian filter after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAF analyses using a 10th-order Gaussian filter after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

VAF analyses using a 10th-order Gaussian filter after (a) 1, (b) 10, (c) 100, and (d) 1000 iterations.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The rms difference in °C between VAN/VAF and BLUE solutions at each iteration for (a) VAF200 (*L* = 200 km) and VAF schemes with different *L*_{f} (from 220 km to 300 km) and (b) VAF100 (*L* = 100 km) and VAF schemes with different *L*_{f} (from 110 km to 150 km). Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The rms difference in °C between VAN/VAF and BLUE solutions at each iteration for (a) VAF200 (*L* = 200 km) and VAF schemes with different *L*_{f} (from 220 km to 300 km) and (b) VAF100 (*L* = 100 km) and VAF schemes with different *L*_{f} (from 110 km to 150 km). Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The rms difference in °C between VAN/VAF and BLUE solutions at each iteration for (a) VAF200 (*L* = 200 km) and VAF schemes with different *L*_{f} (from 220 km to 300 km) and (b) VAF100 (*L* = 100 km) and VAF schemes with different *L*_{f} (from 110 km to 150 km). Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The 2-m temperature observations (locations are indicated by stars and values in °C) at 0000 UTC 5 May 1999 and the (a) background field, (b) BLUE analysis, and (c) VAF analysis.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The 2-m temperature observations (locations are indicated by stars and values in °C) at 0000 UTC 5 May 1999 and the (a) background field, (b) BLUE analysis, and (c) VAF analysis.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The 2-m temperature observations (locations are indicated by stars and values in °C) at 0000 UTC 5 May 1999 and the (a) background field, (b) BLUE analysis, and (c) VAF analysis.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The rms difference in °C between VAF and BLUE solutions at each iteration. Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The rms difference in °C between VAF and BLUE solutions at each iteration. Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2

The rms difference in °C between VAF and BLUE solutions at each iteration. Logarithmic scales are used.

Citation: Monthly Weather Review 128, 7; 10.1175/1520-0493(2000)128<2588:VAUSF>2.0.CO;2