Motion estimation from satellite image sequences: validation
This report concerns the validation of surface velocity estimated from satellite images. The estimation is obtained with a dynamic model based on shallow-water equations. We first compare the stationary assumption to the shallow-water heuristics to justify our choice. Second, we quantify the qual...
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Морський гідрофізичний інститут НАН України
2011
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| Назва видання: | Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу |
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| Цитувати: | Motion estimation from satellite image sequences: validation / E. Huot, I. Herlin, N. Mercier, G. Korotaev, E. Plotnikov // Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу: Зб. наук. пр. — Севастополь, 2011. — Вип. 25, т. 2. — С. 79-90. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1126182017-01-25T03:02:24Z Motion estimation from satellite image sequences: validation Huot, E. Herlin, I. Mercier, N. Korotaev, G. Plotnikov, E. Моделирование процессов в Мировом океане This report concerns the validation of surface velocity estimated from satellite images. The estimation is obtained with a dynamic model based on shallow-water equations. We first compare the stationary assumption to the shallow-water heuristics to justify our choice. Second, we quantify the quality of the estimation by measuring the misfit between the model output and the altimetry measures. Experiments are achieved on Sea Surface Temperature data acquired by the NOAA/AVHRR satellites over the Black Sea. The altimetry measures are obtained by two radar sensors: Envisat and GFO. The good adequacy between the shallow-water output and the altimetry data validates our motion estimation approach. У статті описуються результати валідації полів швидкості поверхневих течій, розрахованих по серіях супутникових зображень. Розрахунок проводився за допомогою динамічної моделі, заснованої на рівняннях мілкої води. Спочатку демонструються переваги методу, що використовує рівняння дрібної води перед стаціонарним алгоритмом. Потім, оцінюється кількісне розбіжність між результатами моделювання і вимірювань, проведених за допомогою альтиметричніх даних. Експерименти засновані на картах температур акваторії Чорного моря, отриманих за допомогою сканера AVHRR NOAA. Альтиметричні дані отримані за допомогою сенсорів Envisat і GFO. Близькість результатів, розрахованих за цими джерелами даних, підтверджують високу якість моделювання. В статье описываются результаты валидации полей скорости поверхностных течений, рассчитанных по сериям спутниковых изображений. Расчет производился при помощи динамической модели, основанной на уравнениях мелкой воды. Сначала демонстрируются преимущества метода, использующего уравнения мелкой воды перед стационарным алгоритмом. Затем, оценивается количественное расхождение между результатами моделирования и измерений, произведенных при помощи альтиметрических данных. Эксперименты основаны на картах температур акватории Черного моря, полученных с помощью сканера AVHRR NOAA. Альтиметрические данные получены с помощью спутников Envisat и GFO. Близость результатов, рассчитанных по этим источникам данных, подтверждает высокое качество моделирования. 2011 Article Motion estimation from satellite image sequences: validation / E. Huot, I. Herlin, N. Mercier, G. Korotaev, E. Plotnikov // Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу: Зб. наук. пр. — Севастополь, 2011. — Вип. 25, т. 2. — С. 79-90. — Бібліогр.: 14 назв. — англ. 1726-9903 http://dspace.nbuv.gov.ua/handle/123456789/112618 551.46(262.5) en Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу Морський гідрофізичний інститут НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Моделирование процессов в Мировом океане Моделирование процессов в Мировом океане |
| spellingShingle |
Моделирование процессов в Мировом океане Моделирование процессов в Мировом океане Huot, E. Herlin, I. Mercier, N. Korotaev, G. Plotnikov, E. Motion estimation from satellite image sequences: validation Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу |
| description |
This report concerns the validation of surface velocity estimated from satellite
images. The estimation is obtained with a dynamic model based on shallow-water
equations. We first compare the stationary assumption to the shallow-water heuristics to
justify our choice. Second, we quantify the quality of the estimation by measuring the
misfit between the model output and the altimetry measures. Experiments are achieved on
Sea Surface Temperature data acquired by the NOAA/AVHRR satellites over the Black
Sea. The altimetry measures are obtained by two radar sensors: Envisat and GFO. The
good adequacy between the shallow-water output and the altimetry data validates our
motion estimation approach. |
| format |
Article |
| author |
Huot, E. Herlin, I. Mercier, N. Korotaev, G. Plotnikov, E. |
| author_facet |
Huot, E. Herlin, I. Mercier, N. Korotaev, G. Plotnikov, E. |
| author_sort |
Huot, E. |
| title |
Motion estimation from satellite image sequences: validation |
| title_short |
Motion estimation from satellite image sequences: validation |
| title_full |
Motion estimation from satellite image sequences: validation |
| title_fullStr |
Motion estimation from satellite image sequences: validation |
| title_full_unstemmed |
Motion estimation from satellite image sequences: validation |
| title_sort |
motion estimation from satellite image sequences: validation |
| publisher |
Морський гідрофізичний інститут НАН України |
| publishDate |
2011 |
| topic_facet |
Моделирование процессов в Мировом океане |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112618 |
| citation_txt |
Motion estimation from satellite image sequences: validation / E. Huot, I. Herlin, N. Mercier, G. Korotaev, E. Plotnikov // Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу: Зб. наук. пр. — Севастополь, 2011. — Вип. 25, т. 2. — С. 79-90. — Бібліогр.: 14 назв. — англ. |
| series |
Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу |
| work_keys_str_mv |
AT huote motionestimationfromsatelliteimagesequencesvalidation AT herlini motionestimationfromsatelliteimagesequencesvalidation AT merciern motionestimationfromsatelliteimagesequencesvalidation AT korotaevg motionestimationfromsatelliteimagesequencesvalidation AT plotnikove motionestimationfromsatelliteimagesequencesvalidation |
| first_indexed |
2025-07-08T04:18:04Z |
| last_indexed |
2025-07-08T04:18:04Z |
| _version_ |
1837050926609727488 |
| fulltext |
79
© E. Huot, I. Herlin, N. Mercier, G. Korotaev and E. Plotnikov, 2011
УДК 551 .46(262 .5 )
Etienne Huot 1,2,3, Isabelle Herlin 1,2, Nicolas Mercier 1,2,
Gennady Korotaev 4 and Evgeny Plotnikov 4
1 INRIA, Institut National de Recherche en Informatique et Automatique, France
2 CEREA, joint laboratory ENPC – EDF R&D, Université Paris-Est, France
3 Université de Versailles - Saint-Quentin-en-Yvelines, France
4 Marine Hydrophysical Institute, National Academy of Sciences, Sevastopol, Ukraine
MOTION ESTIMATION FROM
SATELLITE IMAGE SEQUENCES: VALIDATION
This report concerns the validation of surface velocity estimated from satellite
images. The estimation is obtained with a dynamic model based on shallow-water
equations. We first compare the stationary assumption to the shallow-water heuristics to
justify our choice. Second, we quantify the quality of the estimation by measuring the
misfit between the model output and the altimetry measures. Experiments are achieved on
Sea Surface Temperature data acquired by the NOAA/AVHRR satellites over the Black
Sea. The altimetry measures are obtained by two radar sensors: Envisat and GFO. The
good adequacy between the shallow-water output and the altimetry data validates our
motion estimation approach.
K EYWORDS: Data assimilation, Motion estimation, Shallow-water, image models,
Validation, Altimetry.
1. Introduction
The issue of surface velocity estimation from satellite images has been
extensively studied in the literature [1 – 7]. Data Assimilation (DA) techniques
have been applied in the last five years and gain importance in the scientific
community [8 – 11]. The key points of the DA approach are: availability of
heuristics on the dynamics of a satellite sequence, knowledge on links between
velocity and image data.
This paper proposes an analysis and a validation of the DA approach for
motion estimation from ocean satellite images. Two Image Models were proposed
in [12 – 14]. They express heuristics on the dynamics of the motion field. The
comparison of the estimation using the two models allows us to analyze the
impact of these heuristics. The main issue of the paper is then to validate the
estimation approach by evaluating the quality of the result, compared to real data.
The motion estimation is performed on NOAA/AVHRR Sea Surface Temperature
(SST) data acquired over the Black Sea. The analysis is conducted by comparing the
stationary and the shallow-water heuristics. The validation is obtained by quantifying
the discrepancy of the water layer thickness, estimated with the shallow-water image
model, and the one computed from altimetry data. The altimetry measures, used in
this study, are acquired by the Envisat and GFO sensors.
The paper is organized as follows. Section 2 summarizes the principles of
variational data assimilation. The definition of the Stationary Image Model (SIM)
and Shallow Water Image Model (SWIM) is given in Section 2. Section 2
describes the application of DA to perform motion estimation. Section 5 describes
80
the SST images (5.1), displays and analyzes the estimated motion results (5.2),
describes the altimetry data (5.3), and validates the approach (5.4).
2. Variational Data Assimilation
2.1. Mathematical setting
Let X being the state vector depending on the spatial coordinate x ( ),( yx=x for
image data) and time t . X is defined on ],0[ τ×Ω=A , Ω being the bounded spatial
domain and ],0[ τ the temporal domain.
We assume X is evolving in time according to:
0),)((),( =+
∂
∂
tMt
t
xXx
X
(1)
M , named the evolution model, is supposed differentiable.
Observations ),( txY , for instance satellite image acquisitions, are available at
location x and date t and linked to the state vector through an observation equation:
0),(),)((),( 0 =+= tEtHt xxXxY (2)
In this paper, we assume that one component of X is a pseudo-image
directly comparable to Y . Consequently, H reduces to a projection operator.
The observation error 0E simultaneously represents the imperfection of the
observation operator H and the measurement errors.
We consider having some knowledge on the initial condition of the state
vector at 0=t :
)()()0,( xxXxX bb E+= (3)
bX is named background value of the initial condition and bE the background
error.
bE and oE are assumed to be Gaussian and fully characterized by their
covariance matrices B and R .
2.2. Variational formulation
In order to solve the system (1), (2), (3) with respect to X having a maximal
a posteriori probability given the observations, a functional )(XE is defined and
minimized:
+−−= −
∫ dtdtHtttHtE
A
T xxXxYxRxXxYX )]),)((),()[,()],)((),([)( 1
+−−+ −
Ω
∫ dxb
T
b )]()0,()[()]()0,([ 1 xXxYxBxXxX Reg.
In this formulation, we consider no correlation of the errors between two
space-time positions. Reg is a regularization term used to obtain a convex
function and allow the minimization process to converge to a global minimum.
The minimization of )(XE is carried out with an iterative method based on the
one described in [10] and summarized in the following.
(4)
81
At each iteration k , the analysis k
aX is obtained from the background kbX
by computing the increment Xδ at 0=t .
)()0,()0,( xXxXxX δ+= k
b
k
a (5)
1. Initialization
(a) 0=k
(b) Compute ),(0 tb xX from the initial condition )(xXb of the state
vector at 0=t in (3):
)()0,(0 xXxX bb = (6)
0),)((),( 0
0
=+
∂
∂
tMt
t b
b xXx
X
, for 0=t to τ (7)
(c) Initialize the analysis ),(0 ta xX :
],0[),(),( 00 τ∈∀= ttt ba xXxX (8)
2. Repeat
(a) Compute the adjoint variable λ from τ=t to 0=t :
0),( =τλ x (9)
])([)()( 1
*
k
a
T HtHt
M
t
t
XYR
X
−=
∂
∂+
∂
∂− −λλ
, for τ=t to 0 (10)
(b) Update the value of the background variable:
k
a
k
b XX =+1 (11)
(c) Compute the incremental variable Xδ at 0=t :
)0,()()( xxxX λδ B= (12)
(d) Update the value of the analysis variable:
)()0,()0,( 11 xXxXxX δ+= ++ k
b
k
a (13)
(e) Compute ),(1 tk
a xX + from the initial condition:
0),)((),( 1
1
=+
∂
∂ +
+
tMt
t
k
a
k
a xXx
X
, for 0=t to τ (14)
(f) 1+= kk
Until εδ ≤2
X
Final result is k
aX .
Equation (10) makes use of the adjoint model
*
∂
∂
X
M
. In our study, the discrete
adjoint model is automatically obtained by the Tapenade software1.
1 See http://www-sop.inria.fr/tropics/
82
3. Image models
The two Image Models used in the paper are based on the assumption that a
pixel value is a passive tracer transported by the surface velocity field. The state
vector X includes the motion vector W and a pseudo-image q that is also
transported by the motion vector and can be directly compared to the image
observations. The evolution of q is given by the advection-diffusion equation:
qvq
t
q
q∆=∇⋅+
∂
∂
W (15)
with qv standing for the diffusion coefficient.
The Stationary Image Model (SIM) is based on the restrictive assumption
that, at each space position, the velocity is constant over time. The underlying
hypothesis is that the surface velocity field evolves much slower than the
temperature field. This heuristic is acceptable for a large range of marine
processes. If a vortex, whose spatial scale is more than 10 – 50 km, is transported
with a velocity less than 0,1 to 0,5 m/s, then the temporal scale of that
phenomenon will be more than one day. It means that the surface velocity field
can be considered as stationary during one day. Defining Tqvu ),,(=X , with u
and v the two components of the 2D motion vector ,W SIM is defined as:
0=
∂
∂
t
u
;
0=
∂
∂
t
v
; (16)
qv
y
q
v
x
q
u
t
q
q∆+
∂
∂−
∂
∂−=
∂
∂
;
However, the stationary hypothesis makes this image model only applicable
on a short temporal window.
The shallow-water equations, derived from the Navier-Stokes equations, link
the 2D velocity ),( vu of the layer to its thickness h and take into account the
gravity and Coriolis forces. The state vector X is Tqhvu ),,,( and the Shallow
Water Image Model (SWIM) is defined as:
uvfv
x
B
t
u ∆+++
∂
∂=
∂
∂
)( ξ ;
vvfv
y
B
t
v ∆+++
∂
∂=
∂
∂
)( ξ ;
y
hv
x
hu
t
h
∂
∂−
∂
∂−=
∂
∂
;
qv
y
q
v
x
q
u
t
q
q∆+
∂
∂−
∂
∂−=
∂
∂
(17)
;
83
with )22(
2
1
vughB ++= , g the reduced gravity, f the Coriolis parameter
(depending on the latitude), ξ the vorticity )(
y
u
x
v
∂
∂−
∂
∂=ξ .
4. Application of Data Assimilation
Data Assimilation is applied to perform motion estimation. The sequence of
SST images ),( tT x is assimilated in the two models SIM or SWIM, using the
incremental method described in Section 2.2.
As said in Section 1, the pixel value ),(tT x is directly comparable to the
pseudo-image ),( tq x of the state vector. The observation operator H reduces to
a projection operator, ),()),(( tqtH xxX = . The regularization term is based on
the 2L -norm of the motion gradient (to obtain a smooth vector field) and on the
motion divergency (incompressibility assumption). Its impact is analyzed in [14].
As we consider perfect models, the value of )(tX is obtained from the initial
conditions )0(X by integrating in time. Hence, the cost function (4) only
depends on the initial conditions and is rewritten as:
+−−= −
∫ dtdqTtqTE
A
T xxRX ))(,()())0(( 1
+−−+ −
Ω
∫ xxXXxBxXX db
T
b )]()0()[()]()0([ 1
(16)
xx dvdivdvu 222 )( ∫∫
ΩΩ
+∇+∇+ βα
The choice of the covariance matrix R is crucial for the quality of results.
As the satellite images are provided with meta-data information (see section 1),
the quality of the acquisitions is approximately known. ),(1 txR− is then given a
small value when the acquisition is noisy at ),(tx (because of cloud occlusion for
instance). The choice of the initial background conditions has also a strong
impact on the quality of the result. It has been discussed in [14] that the best
results are obtained with the first observation as background for q , null value for
W and a constant value mh forh , with mh being the thickness value at rest state.
As the background value of q is reliable, qB is given a small value.
5. Results
5.1. Image data
A huge amount of images are acquired over the ocean by space remote sensors.
Those obtained by optical instruments, such as Sea Surface Temperature (SST) data,
display a strong space-time coherence. The images, used in the paper, are acquired
84
on-board NOAA-AVHRR satellites. Their spatial resolution is 1.1 km2 at nadir and
the temporal revisit is at best one day. However, several acquisitions over the
same area are usually acquired on the same day by different satellites. Some of
these data are contaminated by clouds or corrupted by noise. Fig. 1 displays a SST
image acquired over the Black Sea in October 2005, with the flat grey color
corresponding to clouds or noise.
HRPT. SST map (°C). Date 2005.10.23. Time 10 (h) 19 (m)
Fig. 1. Flat grey area corresponds to clouds or noise.
5.2. Analysis
In this paper, motion estimation is tested on a sequence of four images,
displayed on fig. 2. The flat grey areas, on the third and fourth frames, correspond
to missing data.
The two Image Models are used to estimate the surface velocity on these
data. Fig. 3 compares the motion fields estimated with SIM and SWIM, at 0=t .
The results obtained with SWIM visualize a cyclonic vortex on the western part of
the Black Sea. SWIM, due to its physical assumptions on the dynamic, permits a
more realistic motion estimation and characterizes structures occurring on the sea
surface. In comparison, the potential of SIM highly depends on the size of the
temporal window compared to the dynamics involved during that period. That
makes SIM no more relevant for data such as those displayed in fig. 2.
In conclusion, the DA approach for motion estimation permits to retrieve the
major currents of the Black Sea basin. Moreover, the high resolution of
NOAA/AVHRR images allows to better evaluate the size of some well known
mesoscale structures [12].
48°
47°
46°
45°
44°
43°
42°
41°
40°
L
a
ti
tu
d
e
(
N
o
rt
h
)
22
21
20
19
18
17
16
15
14
13
12
28° 30° 32° 34° 36° 38° 40° 42°
Longitude (East)
85
Fig. 2. SST data acquired from
October 23th to October 24th, 2005.
Fig. 3. Motion estimation: a – SIM; b – SWIM.
a) b)
c) d)
a) b)
86
5.3. Altimetry data
Satellite altimeters provide an accurate measure of the Sea Level Anomaly
(SLA) that corresponds to the sea surface deviation from its rest state (see the
black curve on fig. 4).
Fig 4 . Sea Level Anomaly.
The altimeters are nadir-pointing instruments providing an along-track
acquisition. The coverage of Envisat1 over the Black Sea is for instance displayed
on fig. 5. In this paper, we use altimetry measures provided by Envisat2 with a 35
days cycle and by GFO 3 with a 17 days cycle.
5.4. Validation
The outputs of SWIM are W , the surface velocity, and h the thickness of the
surface layer. The thickness anomaly, denoted SWIMh , is estimated from h as its
deviation from the value at rest. On another hand, the altimeters are 1-dimen-
sional instruments measuring the Sea Level Anomaly, denoted alth , along their
tracks. We then compare SWIMh and alth at the same positions. The physical
formula linking these two quantities is:
2 See – http://envisat.esa.int;
3 See – http://ilrs.gsfc.nasa.gov/satellite_missions/list_of_satellites/gfo1_general.html
S
L
S
LA
R
a
ng
e
O
rb
it
R
e
fr
e
n
ce
e
lli
p
so
id
Ocean Surface
Ocean Bottom
87
SWIMalt hh ×∆=× ρρ (17)
with ρ being the density of the upper layer, ρ∆ the difference of density
between the upper and the lower layer, alth the sea level anomaly measured by
the satellite, SWIMh the thickness anomaly estimated by SWIM.
Fig. 5. 35 days cycle of Envisat-1.
Fig. 6 displays the value of SWIMh , averaged in time. The two straight lines rep-
resent altimeter tracks. The green line comes from Envisat and the pink one from GFO.
The number of altimetry measures available on the same space-time period
than the SST data is rather small. However, we apply the conversion given in (17)
and perform a quantitative comparison of alth and SWIMh along a track. Fig. 7
displays these curves for the two tracks displayed on Fig. 6: on the left for
Envisat and on the right for GFO. Black crosses locate the altimeter measures.
The shapes and values of alth and SWIMh curves are very similar. There is no
error in the slope directions. The extrema are well localized. It is almost perfect in the
case of Envisat. As the velocity field is strongly related to the shape of the thickness
46°
45°
44°
43°
42°
41°
L
at
itu
d
e
(N
o
rt
h
)
28° 30° 32° 34° 36° 38° 40° 42°
Longitude (East)
P
ix
el
n
u
m
b
er
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140 160 180 200
Pixel number
cm
60
50
40
30
20
10
0
Track1
Track2
altimeter asquisions +
Fig. 6. Two altimeter
tracks displayed over
the average of
hSWIM .
+
+
+
+
+
+
+
+
+
+
+
+
+
+
88
image, these promising results on thickness estimation validate the estimation of the
motion. Fig. 8 illustrates the link between velocity and thickness: a bump correspond
to an anticyclonic velocity field and a bowl to a cyclonic one.
Fig. 7. Sea Level Anomaly, given by the altimeters, compared with the estimation
with SWIM
6. Conclusion
In this paper, we propose an analysis and validation of the data assimilation
approach for motion estimation from satellite image sequences. We compared
two dynamic assumptions, i.e. we assimilated the same data in two image models,
SIM and SWIM, and analyzed motion results. Moreover, we used altimetry data to
quantify the quality of the estimation. The comparison between the surface
anomaly estimated by SWIM and measured by altimeters validates our approach.
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interpretation / CVPR. – 2000. – P. 2487-2492.
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Pattern Analysis and Machine Intelligence. – 2002. – vol. 24, № 3 (March). – P. 365-380.
7. Isambert T., Berroir J.P. and Herlin I. A multiscale vector spline method for estimating
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of International Conference on Image Processing, Atlanta, USA, October 2006.
1,0
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Fig. 8. From left to right: 1 – 3D water layer thickness; 2 – Its 2D projection.
The magenta line figures the track of an altimeter; 3 – SLA along this track; 4 – Velocity field.
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9. Papadakis N, Héas P. and Mémin E. Image assimilation for motion estimation of
atmospheric layers with shallow-water model / Proceedings of Asian Conference on
Computer Vision, Tokyo, Japan, November 2007. – P. 864-874.
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numerical models // Tellus A. – 2010, vol. 62. – P. 30-47.
12. Korotaev G., Huot E., Le Dimet F.-X., Herlin I., Stanichny S.V., Solovyev D.M. and Wu L.
Retrieving ocean surface current by 4-D variational assimilation of sea surface
temperature images // Remote Sensing of Environment. – 2008. – vol. 112, № 4 (Apriel).
– P. 1464-1475 (Special issue on data assimilation).
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Estimation of Ocean Circulation Velocity / Proceedings of IEEE International
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Rec eived 02 .11 .2011 ;
Материал поступил в редакцию 02 .11 .2011 г .
АНОТАЦ IЯ У статті описуються результати валідації полів швидкості поверхневих
течій, розрахованих по серіях супутникових зображень. Розрахунок проводився за до-
помогою динамічної моделі, заснованої на рівняннях мілкої води. Спочатку демонст-
руються переваги методу, що використовує рівняння дрібної води перед стаціонарним
алгоритмом. Потім, оцінюється кількісне розбіжність між результатами моделювання і
вимірювань, проведених за допомогою альтиметричніх даних. Експерименти засновані
на картах температур акваторії Чорного моря, отриманих за допомогою сканера AVHRR
NOAA. Альтиметричні дані отримані за допомогою сенсорів Envisat і GFO. Близькість
результатів, розрахованих за цими джерелами даних, підтверджують високу якість
моделювання.
АННОТАЦИЯ В статье описываются результаты валидации полей скорости
поверхностных течений, рассчитанных по сериям спутниковых изображений.
Расчет производился при помощи динамической модели, основанной на уравнениях
мелкой воды. Сначала демонстрируются преимущества метода, использующего
уравнения мелкой воды перед стационарным алгоритмом. Затем, оценивается
количественное расхождение между результатами моделирования и измерений,
произведенных при помощи альтиметрических данных. Эксперименты основаны на
картах температур акватории Черного моря, полученных с помощью сканера
AVHRR NOAA. Альтиметрические данные получены с помощью спутников Envisat
и GFO. Близость результатов, рассчитанных по этим источникам данных,
подтверждает высокое качество моделирования.
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