Adaptive algorithm of the suboptimal Kalman filter
Ekaterina KLIMOVA
Prosp. Akad. Lavrentyeva 6, Novosibirsk 630090
Russia
klimova@ict.nsc.ru
The implementation of the Kalman filter theory in meteorological data
assimilation presents a severe problem, since the Kalman algorithm is
computationally expensive here and requires a great body of information. One of
the ways to solve this problem is to use simplified models in the Kalman filter for
the calculation of prediction error covariances (the suboptimal Kalman filter).
A suboptimal Kalman filter algorithm with a regional baroclinic model is
proposed. Some simplifications are made, which are based on splitting up into
physical processes and quasi-geostrophic theory.
In case when the covariance matrix of the model noise is assumed to be zero,
there occurs a fast decrease in the theoretical error of the Kalman filter algorithm
and, as a consequence, the observations at the analysis step have smaller factors.
This effect is named “divergence of the Kalman filter algorithm”. In this paper, an
adaptive Kalman filter algorithm for the estimation of model error covariances is
considered. The algorithm allows one to correct the prediction error covariance
matrix, which is also calculated with the help of a simplified operator. The
algorithm is based on the use of vectors of “residuals” (i.e. the difference between
the observed and predicted data). In the adaptive algorithm, the diagonal elements
of the prediction error covariance matrix are calculated by a method of successive
correction. In this method we use the residuals to obtain “observed” values of the
diagonal elements of the prediction error covariance matrix. The full matrix is
restored, assuming that the correlations are calculated precisely. Results of
numerical experiments of the modeled data assimilation are presented.
The problem of optimal filtration and suboptimal algorithms
Nowadays, the Kalman filter algorithm is one of the most popular
approaches to the problem of data assimilation in the atmosphere and ocean. The
calculation of the matrix of prediction error covariances in the Kalman filter
algorithm is most time-consuming due to a large dimensionality of this matrix. In
the suboptimal algorithms of the Kalman filter, at the step of calculation of this
matrix, the operator of a simplified model (of smaller dimensionality) is used
instead of the operator of the initial model. We proceed from the idea of splitting
of the model’s dynamic operator. We assume that the prognostic system under
consideration is linearized with respect to some background flow. The system of
prognostic equations is represented in the following form:
d
 A1  A2 ,
dt
where  is the vector of meteoelements being predicted ( A1 can be a finitedifference analog of the system of advection equations, and A2 can be a finitedifference analog of the corresponding operator of the system of adaptation
equations). Let the operator A2 be skew-symmetric. Then, on the basis of the
equation for P(t) in the discrete-continuous algorithm of the Kalman filter at a
small time interval t , the solution to this equation is as follows:
T
T
P(t0  t )  P(t0 )  t ( A1 P(t0 )  P(t0 ) A1 )  t ( A2 P(t0 )  P(t0 ) A2 )  O( t 2 ).
The second bracket in this expression can be omitted, if A2 P is small or if
the matrices A2 and P are commutative (that is, they have a common basis). This is
valid if we assume that the errors in the eigenvectors of the matrix A2 do not
correlate with each other. It has been shown in some papers that the eigenvectors
of the prediction error covariance matrix and the normal modes of the prognostic
model coincide in some cases. They coincide, in particular, if the operator of the
dynamic system is skew-symmetric. Thus, if the terms in the second bracket are
ignored, one can use a simplified model with the dynamic operator A1 to describe
the behavior of the covariance P(t).
Simplified models to calculate the prediction error covariance matrices
The simplified models to calculate the prediction error covariance matrices
considered here were obtained on the basis of a baroclinic adiabatic atmospheric
model for a region in the system of coordinates (x, y, p) [11]. An important feature
of the numerical algorithm for the realization of this model is the use of G. I.
Marchuk’s idea to solve the problem by the splitting up of the physical processes
into the step of transfer along trajectories and the step of adaptation of
meteorological fields [9]. Let us describe the models used in the numerical
experiments on assimilation of meteorological data to calculate the prediction error
covariance matrices.
model-0 is a model based on primitive equations. We assume that this model
describes a "real" state of the atmosphere.
model-1 is a model to calculate the prediction error covariances in the suboptimal
algorithm of the Kalman filter. It is based on the following assumptions:
 the state of the atmosphere in the Kalman filter algorithm is estimated for
vertical normal modes of the prognostic model; the vertical normal modes
are the eigenvectors of a finite-difference analog of the operator
 p2 
R 2T * ( a   0 )
2
Lp  
,m 
.
p m 2 p
gf 2
G.I. Marchuk has shown that the domain of influence of the vertical normal
modes decreases as their number increases; moreover, the horizontal scales
of different vertical modes differ greatly from one another. Therefore, it
makes sense to analyze the data of meteorological observations for the
coefficients of expansion in terms of the vertical normal modes
independently. Besides, for n beginning from some value n , the domain of
influence becomes smaller than the grid size and, hence, the analysis makes
sense only for the first n vertical modes;
 the calculation of covariances of prediction errors is based on the
assumption that the errors of vertical normal modes do not correlate with
each other; it is well-known that the eigenvectors of the vertical operator are
close to the natural orthogonal basis. Therefore, they can be assumed to be
statistically independent;
 the covariances of prediction errors are calculated only for the height field of
an isobaric surface, and the covariances of wind field errors are calculated
on the basis of geostrophic relations. Since our initial prognostic model is
based on the method of splitting-up into physical processes, only the
advection step is used to calculate the covariances. In this case, only the
temperature advection remains in the operator A1 .
 the wind velocity fields in the advection operator A1 do not depend on the
vertical coordinate p (that is, the background flow is close to a barotropic
one). Under this condition, one can calculate the covariances of prediction
errors for each vertical mode separately.
Let us consider the initial prognostic model (model-0) on a small time
interval (tn , tn 1 ) . Let the wind velocity components in the advection operator be
given at the time tn . Then the model is linear on this time interval. We write z, u,
and v in the form of series,
N
N
N
n 1
n 1
n 1
z   zn g n , u   un g n , v   vn g n ,
where g n are eigenvectors of a finite-difference analog of the operator Lp . Under
the above assumptions, we have the following model for the coefficients z n , un , vn
(model-1):
zn ~ zn ~ zn
U
V
 0,
t
x
y
g zn
g zn
un  
,v 
,
f 0 y
f 0 x
~
~
where U ( x, y ),V ( x, y ) are wind velocity fields at the time tn such that
~
~
U V

 0,
p p
f 0 is the average value of the Coriolis parameter. The assumption that the
eigenfunctions of the model’s vertical operator do not correlate with each other
makes it possible to decrease greatly the number of operations in the calculation of
the prediction error covariance matrix, because in this case the matrix becomes a
block diagonal one.
Now, let us consider a more general case, in which a simplified model is
obtained also from the assumption of geostrophicity, but the substance advection in
the operator A1 is not omitted. Then, under the condition of geostrophicity, we
have the following equation of advection of the quasi-geostrophic potential
vorticity:
d
 0,
dt
g
  Lp z  z  f ,
f0
d  ~  ~ 
 U V
,
dt t
x
y
~
~
where U ( x, y ),V ( x, y ) are wind velocity fields at the time tn such that
~
~
U V

 0,
p p
In this case, f is the Coriolis parameter, and f 0 is the average value of the
Coriolis parameter.
From the above, we obtain one more model, model-2, which is a quasi-linear
model described by the equation of vorticity transfer.
In practice, model-1 and model-2 can be used to calculate the prediction
error covariance matrices for the vertical modes of the prognostic model, because
in these models the errors in different vertical modes do not correlate with each
other. Under the assumption that the errors of vertical modes do not correlate, the
corresponding covariance matrix becomes a block diagonal one, which makes it
possible to decrease greatly the cost of computer storage and calculation.
If the deviation of the wind averaged over the vertical from its actual value is
great (for instance, in frontal zones), one can consider the transfer equation with
"generalized velocities".
We set
N
N
u~  u~ g , v~  v~ g

n 1
n

g
n 1
n
g
Then the transfer equation in the case of uncorrelated errors of vertical
normal modes has the following form:
zn ~ zn ~ zn
U
V
 0,
t
x
y
~ N
~ N
U    mnnu~m ,V    mnnv~m ,
m 1
m 1
 mnn  ( g m g l , g ).
where gn* is the n-th eigenvector L*p .
*
n
A model to calculate the covariances of homogeneous isotropic random fields
of prediction errors
Here, we consider a simplified model to calculate the covariances of homogeneous
isotropic random fields of prediction errors. By analogy with the derivation of
equations for the structural functions in turbulence theory [10], we derive an
analytical equation for the local prediction error covariances between two given
points. The differential equations are derived for the local prediction error
covariances at two given points.
Let the deviation of prognostic fields from “real” values be termed the
prediction error. In this case, it is assumed that the prediction errors depend only
on the errors in the initial data. We describe a "real" state of the atmosphere by
using a baroclinic adiabatic model of the atmosphere for a region in the system of
coordinates (x, y, p). We consider a system of prognostic equations on a time
interval (tn , tn 1 ) . Let u~  u(tn ), v~  v (tn ) . Since the prediction errors of the fields of
wind velocity components and temperature depend only on errors in the initial
data, the system of equations for prediction errors has the following form on this
time interval (we assume that the vertical velocity ~  0 ):
u ~ u ~ u
z
u
v
 fv  g
 0, (1)
t
x
y
x
v ~ v ~ v
z
u
v
 fv  g
 0, ( 2)
t
x
y
y
T ~ T ~ T
u
v
   0, (3)
t
x
y
u v 


 0, ( 4)
x
y
x
p z
( a   ) RT *
T   g
, 
.(5)
R p
gp
Here ( u, v,  , z, T ) are the prediction errors of the wind velocity, height,
and temperature fields, respectively. Let us derive equations for the covariances of
prediction errors of the wind velocity field component u between two points with
coordinates ( x1 , y1 , p1 ) and ( x2 , y2 , p2 ) . Let u1 ,u~1 be the values of the quantities
u, u~ at the point with coordinates ( x1 , y1 , p1 ) ; and u2 ,u~2 be the values of these
quantities at the point ( x2 , y2 , p2 ) . Let us write the first equation at points 1 and 2
(assuming that the value at point 1 does not depend on the value at point 2). We
multiply the first of these equations by u2 , and the second one by u1 , add them
up, and average the result obtained (here we mean the theoretical-probabilistic
averaging). We have the following equality:
u1u2 ~ u1u2 ~ u1u2
u1u2 ~ u1u2
 u1
 v1
 f v1u2  u~2
 v2

t
x1
y1
x2
y2
 f v2u1  g
z1u2
z2u1
g
 0.
x1
x2
Equations for the covariances of the other fields are derived in a similar way.
It is well-known from turbulence theory that the homogeneous isotropic solenoidal
vector field does not correlate with any scalar field. In the prognostic model under
consideration, the continuity equation is satisfied, that is, the field (u, v,  ) is
solenoidal. Hence, in case when we assume that the random fields of prediction
errors are homogeneous and isotropic, the terms with pressure gradient in the
model’s equations can be dropped. Moreover, the terms with  in the third
equation are dropped.
Since we comply with the idea of splitting up the model’s dynamic operator
into the stage of transfer along trajectories and the adaptation stage, to calculate the
covariances of homogeneous isotropic prediction errors at the adaptation stage, the
pressure gradient and vertical velocity terms in the corresponding equations must
be dropped. Hence, only the terms with the Coriolis force remain. In this case, the
system of adaptation equations has the following form:
u
 fv  0,
t
v
 fu  0.
t
It follows from these equations that allowance for the Coriolis forces means that
the wind velocity vector makes a turn. Since by definition all distribution densities
in a homogeneous and isotropic random field do not depend on any turns, one must
also drop the terms with the Coriolis force in the system of adaptation equations.
Thus, the dynamics of prediction error covariances in case when they are
homogeneous and isotropic is described by the prognostic model of substance
transfer along particle trajectories (on a small time interval).
Strictly speaking, the property that the vector solenoidal and scalar fields do
not correlate is valid only for an orthogonal system of coordinates. The system
(x,y,p) is not orthogonal. Since, however, the condition that the meteorological
fields are homogeneous and isotropic is valid at small distances, the local system
of coordinates for the points at which the covariance is estimated can be considered
orthogonal with high degree of accuracy.
For meteorological fields, the condition of homogeneity and isotropy is
usually considered only for horizontal directions. In this case, one can make
similar calculations, if it is assumed that the two-dimensional vector (u, v )T is
solenoidal. Let us consider a system of equations similar to (1)-(5) for prediction
errors and assume that the background wind velocity fields do not depend on p.
Then, if the prediction error fields are homogeneous and isotropic, to describe
them on a small time interval one can use the system of equations of transfer along
trajectories for the coefficients of expansion in terms of the vertical normal modes
(at the condition that the horizontal velocity vector is solenoidal). If the prediction
errors of height and velocity fields are related geostrophically, we have model-1,
which was described above.
However, the property of homogeneity and isotropy is not valid for
atmospheric motions of large scales. However, it is well-known that the large-scale
motions are described well with the help of quasi-geostrophic approximation.
Thus, since the domain of dependence of the coefficients of expansion in terms of
the vertical normal modes decreases with distance, let us consider a "hybrid"
model. In this model, to calculate the error covariances of the first, large-scale
modes, the quasi-geostrophic vorticity equation for the n-th coefficient of
expansion in terms of the vertical normal modes is used, and for the other modes
the system of equations of model-1 is used. The background flow is considered to
be close to a barotropic one, that is, the transfer is realized with a wind averaged
over the vertical. Let this model be called model-3.
Adaptive assimilation algorithm based on the Kalman filter
In the discrete algorithm of the generalized Kalman filter, the equation for
prediction error covariances has the following form:
Pk f  Ak 1 Pka1 AkT1  Qk 1
where Pk f is the prediction error covariance matrix at the k-th time step and Pka is
the analysis error covariance matrix at the (k-1) -th time step, Ak 1 is the operator
of the prognostic model (the case under consideration is that of a linearized
model), and Qk 1 is the matrix of model errors (“noise”). Here, if a suboptimal
~
assimilation algorithm is considered, the matrix Ak 1 is used instead of the
matrix Ak 1 .
If the “model noise” matrix is considered zero in the calculation of the error
covariance matrix, the norm of matrix Pk f decreases with time. This leads to
"divergence" of the Kalman filter algorithm. To specify the matrix Qk 1 is a
difficult problem, because the exact value of the matrix is unknown. In this section,
we consider a procedure of estimating Qk 1 for a suboptimal assimilation algorithm
based on the Kalman filter. The algorithm also makes it possible to correct the
matrix Pk f , which is calculated by using a simplified operator.
The properties of the adaptive algorithm proposed were investigated with the
help of numerical experiments on the assimilation of modeled data with the use of
a suboptimal filter. To calculate the prediction error covariances, model-3 was used
as the simplified one.
All algorithms of the adaptive Kalman filter now in use are based on an idea
described in the classical book [1]. The authors of this book propose an adaptive
algorithm of the Kalman filter in which the elements of the matrix Qk 1 (in case
when this matrix is diagonal) are estimated by using the following “residual”
vectors  k :
 k  yk0  M k xkf .
E k  0,
E ( k )( k )T  M k ( Ak 1 Pka1 AkT1  Qk 1 ) M k  Rk .
In these formulas, Е is the theoretical-probabilistic averaging. One can see from
these formulas that the vector covariance matrix  k contains information about the
matrix Pk f .
Let us assume that
diag ( E ( k )( k )T )  diag (( k )( k )T ).
In the general case,
T
diag((k )(k )T )  diag( E(k )(k )T )   k ,
where  k is a random vector with zero mean value.
Let a preliminary estimate of the matrix Pk f be obtained by using the operator of
the simplified model:
Pk f  Ak 1 Pka1 AkT1.
f
f
Let the “”real” matrix Pk differ from Pk by the matrix  Pk . The matrix  Pk f is
f
f
f
restored by using the data  k . We denote   diag ( Pk ) . It is assumed that
( k k )i0i0  ( kf   kf  diag (Qk 1 ))i0i0  ( Rk )i0i0 .
T
In this case, we have the problem to restore the vector  kf by using the data
( k k )i0i0  ( Rk )i0i0  hi0 and the estimate  kf . This problem is solved by the
method of step-by-step correction:
T
N
( kf   kf )l   wli0 (hi0  ( kf )i0 ),
(6)
i0 1
where l is the grid node number, and the weight coefficients are
wli0 
cli0
N
c p   cli
,
i 1
0,5( r / L )
, rli is the distance between the grid node and the
Here, cli  F (rli ), F (r )  e
observation.
After the variances are estimated, the entire matrix is restored:
2
Pk f  ( kf )1/ 2 corrkf (( kf )1/ 2 )T ,
corrkf  ( kf )1/ 2 Pk f (( kf )1/ 2 )T .
(7)
It should be noted that this algorithm makes it possible to specify the values
f
of the entire matrix Pk , which includes the prediction on the basis of the
simplified model and the matrix Qk 1 , which is, as a rule, unknown.
Numerical experiments
The properties of the algorithm proposed above were investigated in
numerical experiments with modeled data for a regional baroclinic model of the
atmosphere [11]. The horizontal grid size in the model was equal to 300 km, and
15 standard levels were used in the vertical. In the numerical experiments, the data
of an archive of objective analyses for the regional model of the atmosphere on
April 01-03 1991 were used. The data assimilation was made for the regional
f
model. In the calculation of the matrix Pk in time, it was assumed that at the
boundary of the domain the values of matrix elements do not change with time. At
the “analysis” stage, the covariance matrix values were recalculated for all points
of the domain. Otherwise, unbalanced values of the matrix elements were obtained
inside the domain and at the boundary, which led to a quality decrease in the
assimilation procedure. The numerical experiments were performed with a
covariance matrix P0 given in the form
P0  cn2 I ,
where I is the identity matrix, and c n is the root-mean-square error of prediction of
the n-th vertical mode. In all experiments, the wind field errors at each time step
were determined with the help of geostrophic relations. At the analysis step, the
obtained fields of "corrections" of the first approximation (prediction) were subject
to a procedure of variational adjustment. The following calculation variants for 48hour prediction, with assimilation every 12 hours, beginning with 0 o’clock data on
the geopotential field, were considered.
In the first experiment series, it was assumed that the prediction error
depends only on the random error in the initial fields (so-called "twin"-type
experiments). The 48-hour prediction on the basis of the initial fields was
considered a "real" state of the atmosphere. Then, a random error was added to the
initial height and velocity fields. The error was distributed by the normal law with
a zero mean value and a variance corresponding to the prediction error level. The
prediction errors of the height and velocity fields at the initial time were balanced
on the basis of geostrophic relations. The observation data were given by adding to
the "real" field a random error distributed by the normal law with a zero mean
value and a variance corresponding to the error level of aerologic observations. It
was assumed in the experiments that the observation errors do not correlate with
each other. 48-hour calculations, with assimilation of modeled data on the
geopotential field every 12 hours, were made. The observation data were given at
regular grid nodes at 143 points uniformly distributed in the integration domain.
The "model noise" matrix was assumed zero.
It should be noted that the specification of a uniform observation data
network is a simplification of the initial problem. Numerous factors contribute to
the total prediction error of the assimilation procedure. These factors include the
interpolation procedures from observation points to regular grid nodes as well as
the procedures of initial processing and control of observation data. The simplified
specification of the observation data distribution was made to eliminate the
influence of the above factors.
In the first experiment, the modeled data were assimilated every 12 hours
during a 48-hour prediction by using the suboptimal algorithm based on the
Kalman filter. The simplified model described above was used to calculate the
covariance matrix.
In the second experiment, in addition to the prediction procedure of the
matrix Pk f , the matrix was corrected by using formulas (6)-(7). The scale L of the
Gaussian function in the correction procedure was taken equal to 1000 km. The
"residual” vectors  i were subject to a procedure of control.
The results of these experiments are presented in Fig. 1. Figure 1а shows the
behavior of root-mean-square deviations of the calculated geopotential fields from
the "real" ones for the surface of 500 millibar. In this figure, the prediction error
without assimilation is denoted by s0, and the prediction error with the use of the
suboptimal assimilation algorithm in the first and second experiments,
respectively, by s1 and s2. Figure 1b shows the behavior of the theoretical error of
0
the assimilation algorithm based on the Kalman filter, i.e. the trace of the
prediction error covariance matrix (for the first vertical mode) for the first and
second experiments (tr1 and tr2), respectively. One can see form this figure that the
f
correction procedure for the matrix Pk with the help of the adaptive algorithm
makes it possible to improve the quality of prediction with assimilation and avoid
f
the erroneous decrease in the elements of the matrix Pk .
In the second series of experiments, both random errors in the initial data
and observations and “model noise” were modeled (so-called experiments of the
"cousin" type). The "real" state of the atmosphere was modeled with the help of
prediction with a perturbed initial field. In this case, a random error distributed by
the normal law with a variance corresponding to root-mean-square error bn  0,3cn
was added at each prediction step to the wind velocity and height fields. The
observation data were modeled by adding a random error (the same as in the first
series of experiments) to the "real" field. The prediction errors for the wind
velocity and height fields were balanced with the help of geostrophic relations.
In the first experiment, the data were assimilated with the help of the suboptimal
algorithm based on the Kalman filter. The simplified model to calculate Pk f was
taken the same as in the experiments of the "twin" type. The matrix Qk 1 was
assumed zero. In this case, the values of the matrix Pk f were corrected by using
the adaptive algorithm.
f
In the second experiment, the matrix Pk was calculated with allowance for the
matrix Qk 1 , for the n-th vertical mode Qk 1  bn2 I . In this case, no correction of the
f
matrix Pk by the algorithm was made.
In the third experiment, the data were assimilated in such a way that the
f
matrix Pk was calculated without correction and under the condition Qk 1  0 .
The results of these experiments are presented in Figs. 2 and 3.
Figure 2 shows the root-mean-square deviations of the calculated geopotential
fields from the "real" ones for the surface of 500 millibar for prediction with
assimilation for the first, second, and third experiments (s1, s2, s3), respectively.
One can see from the figure that the prediction estimates obtained in the second
and third experiments are close. This means that the procedure of adaptive
correction of the matrix Pk f makes it possible to compensate for the specification
of the matrix Qk 1 , whose value is, as a rule, unknown. It follows from the
formulas for the analysis step of the assimilation procedure that the weight
coefficients for observation data depend on the elements of the prediction error
covariance matrix. Figure 3 shows the weight coefficients wi versus the grid node
number i. The data are assumed to be specified at the central point of the domain
(at a grid node). Then the weight coefficients at the assimilation of one observation
are calculated by using the formula
wi 
P(i, io )
,
( P(i0 , io )  r02 )
where i0 is the central node number, r0 is the root-mean-square error of
observations, and P is the prediction error covariance matrix (in the experiments,
the matrix for the first vertical mode is considered). The figure shows wi versus
the grid number for the grid nodes that are closest to i0 in the x - coordinate. The
central node in the figure has number "7" (the weight coefficients for 6 nearest
nodes are considered: this corresponds to a distance of 1800 km). In Fig. 3а, w1
and w2 denote the weight coefficients obtained for the second and third
f
experiments, respectively (the calculation of Pk was made for 12 hours). In Fig.
3b, w1 and w2 denote the weight coefficients obtained for the second and third
f
experiments, respectively (the calculation of Pk was made for 24 hours).
One can see from these figures that when a matrix Qk 1 different from 0 is
specified, the "weight" of the observation data in the assimilation procedure
increases, and in this case the prediction with assimilation becomes closer to the
"real" state of the atmosphere.
A more detailed description of the simplified models to calculate the
prediction error covariance matrices and numerical experiments to estimate their
properties can be found in [2-8].
References
1. Jazwinski A.H. (1970) Stochastic processes and filtering theory. -Academic
Press, New York.
2. Klimova E.G. (1997) Algorithm of data assimilation of meteorological
observations based on the extended suboptimal Kalman filter, Russian
Meteorology and Hydrology, 11, 40-47.
3. Klimova E.G. (1999) Asymptotic behavior of a data assimilation scheme
based on an algorithm of the Kalman filter, Russian Meteorology and
Hydrology, 8, 44-52.
4. E.G.Klimova (2000) Simplified models for the calculation of covariance
matrices in the Kalman filter algorithm. - Meteorologiya i Hydrologiya, No.
6. (Translated into English: Russian Meteorology and Hydrology), 18-30.
5. E.G.Klimova (2001) A model to calculate the covariances of homogeneous
isotropic stochastic fields of prediction errors. - Meteorologiya i
Hydrologiya, No. 10, 24-33 (In Russian).
6. E.G.Klimova (2001) Model for the calculation of the prediction error
covariances in a Kalman filter algorithm based on primitive equations. Meteorologiya i Hydrologiya, No. 11, 11-21 (In Russian).
7. E.G.Klimova (2003) Numerical experiments on meteorological data
assimilation with the help of the suboptimal Kalman filter. - Meteorologiya i
Hydrologiya, No. 10, 54-67 (In Russian).
8. E.G.Klimova (2005) Data assimilation algorithm based on the adaptive
suboptimal Kalman filter. - Meteorologiya i Hydrologiya, No. 2 (In
Russian).
9. Marchuk G.I. (1967) Numerical Methods in Weather Prediction,
Gidrometeoizdat, Leningrad.
10.Monin A.S., Yaglom A.M. (1971) Statistical Fluid Mechanics, vol.1, MIT
Press, Cambridge (Mass., USA).
11.Rivin G.S. (1996) Numerical modeling of background atmospheric
processes and the problem of aerosol transport in Siberian region,
Atmospheric and Oceanic Optics, 9, 780-785.
30
25
20
s0
15
s1
10
s2
5
0
0
10
20
30
40
50
60
Time (hour)
Fig.1a Root-mean-square forecast error (Q=0)
600000
500000
400000
ptr_adap
300000
ptr_assim
200000
100000
0
0
10
20
30
40
50
60
Время (час.)
Fig.1b Trace of the forecast error covariance
Fig.2 Root-mean-square forecast error
25
20
15
s_adapt
s_Q
s_assim
10
5
49
45
41
37
33
29
25
21
17
13
9
5
1
0
Время (час.)
Fig.3a Dependence of weight factors of the analysis on the mesh point number
(the forecast on 12 hours)
1
0,8
0,6
Счет P с ненулевой
Q
0,4
Счет P с Q=0
0,2
0
-0,2
0
2
4
6
8
10
12
14
Fig.3b Dependence of weight factors of the analysis on the mesh point number
(the forecast on 24 hours)
1
0,8
0,6
0,4
Счет P с ненулевой
Q
0,2
Счет P с Q=0
0
-0,2
0
2
4
6
8
10
12
14