Data assimilation and sensitivity of the Black Sea model to parameters
An adjoint based technique is applied to a Shallow Water Model in order to estimate influence of the model’s parameters on the solution. Among parameters the bottom topography, initial conditions, boundary conditions on rigid boundaries, viscosity coefficients and the amplitude of the wind stress...
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irk-123456789-1126642017-01-26T03:02:57Z Data assimilation and sensitivity of the Black Sea model to parameters Kazantsev, E. Моделирование процессов в морях и внутренних водоемах: Черное и Азовское моря An adjoint based technique is applied to a Shallow Water Model in order to estimate influence of the model’s parameters on the solution. Among parameters the bottom topography, initial conditions, boundary conditions on rigid boundaries, viscosity coefficients and the amplitude of the wind stress tension are considered. Their influence is analyzed from different points of view. Two configurations have been analyzed: an academic case of the model in a square box and a more realistic case simulating Black Sea currents. It is shown in both experiments that the boundary conditions near a rigid boundary influence the most the solution. This fact points out the necessity to identify optimal boundary approximation during a model development. Методика, заснована на сполучених рівняннях, була застосована до моделі «дрібної води» для оцінки чутливості її рішення до параметрів моделі. Розглядалися такі параметри як рельєф дна, початкові і граничні умови, коефіцієнти в'язкості і напруга тертя вітру на поверхні. Їх вплив аналізувалося з різних точок зору. Задача була розглянута в двох конфігураціях: академічний варіант моделі в квадратному басейні і більш реалістична версія, що моделює течії в Чорному морі. Було показано, що граничні умови на твердих кордонах мають найбільший вплив на рішення в обох випадках. цей факт вказує на необхідність пошуку оптимальних граничних умов при розробці моделі. Методика, основанная на сопряженных уравнениях, была применена к модели «мелкой воды» для оценки чувствительности ее решения к параметрам модели. Рассматривались такие параметры как рельеф дна, начальные и граничные условия, коэффициенты вязкости и напряжение трения ветра на поверхности. Их влияние анализировалось с различных точек зрения. Задача была рассмотрена в двух конфигурациях: академический вариант модели в квадратном бассейне и более реалистичная версия, моделирующая течения в Черном море. Было показано, что граничные условия на твердых границах оказывают наибольшее влияние на решение в обоих случаях. Этот факт указывает на необходимость поиска оптимальных граничных условий при разработке модели. 2011 Article Data assimilation and sensitivity of the Black Sea model to parameters / E. Kazantsev // Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу: Зб. наук. пр. — Севастополь, 2011. — Вип. 25, т. 2. — С. 161-181. — Бібліогр.: 21 назв. — англ. 1726-9903 http://dspace.nbuv.gov.ua/handle/123456789/112664 551.465 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 |
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DSpace DC |
| language |
English |
| topic |
Моделирование процессов в морях и внутренних водоемах: Черное и Азовское моря Моделирование процессов в морях и внутренних водоемах: Черное и Азовское моря |
| spellingShingle |
Моделирование процессов в морях и внутренних водоемах: Черное и Азовское моря Моделирование процессов в морях и внутренних водоемах: Черное и Азовское моря Kazantsev, E. Data assimilation and sensitivity of the Black Sea model to parameters Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу |
| description |
An adjoint based technique is applied to a Shallow Water Model in order to estimate
influence of the model’s parameters on the solution. Among parameters the bottom topography,
initial conditions, boundary conditions on rigid boundaries, viscosity coefficients
and the amplitude of the wind stress tension are considered. Their influence is analyzed
from different points of view.
Two configurations have been analyzed: an academic case of the model in a square
box and a more realistic case simulating Black Sea currents. It is shown in both experiments
that the boundary conditions near a rigid boundary influence the most the solution.
This fact points out the necessity to identify optimal boundary approximation during a
model development. |
| format |
Article |
| author |
Kazantsev, E. |
| author_facet |
Kazantsev, E. |
| author_sort |
Kazantsev, E. |
| title |
Data assimilation and sensitivity of the Black Sea model to parameters |
| title_short |
Data assimilation and sensitivity of the Black Sea model to parameters |
| title_full |
Data assimilation and sensitivity of the Black Sea model to parameters |
| title_fullStr |
Data assimilation and sensitivity of the Black Sea model to parameters |
| title_full_unstemmed |
Data assimilation and sensitivity of the Black Sea model to parameters |
| title_sort |
data assimilation and sensitivity of the black sea model to parameters |
| publisher |
Морський гідрофізичний інститут НАН України |
| publishDate |
2011 |
| topic_facet |
Моделирование процессов в морях и внутренних водоемах: Черное и Азовское моря |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112664 |
| citation_txt |
Data assimilation and sensitivity of the Black Sea model to parameters / E. Kazantsev // Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу: Зб. наук. пр. — Севастополь, 2011. — Вип. 25, т. 2. — С. 161-181. — Бібліогр.: 21 назв. — англ. |
| series |
Екологічна безпека прибережної та шельфової зон та комплексне використання ресурсів шельфу |
| work_keys_str_mv |
AT kazantseve dataassimilationandsensitivityoftheblackseamodeltoparameters |
| first_indexed |
2025-07-08T04:22:28Z |
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| _version_ |
1837051201748729856 |
| fulltext |
161
© E. Kazantsev, 2011
УДК 551 .465
E. Kazantsev
INRIA, MOISE team, Laboratoire Jean Kuntzmann. Grenoble
DATA ASSIMILATION AND SENSITIVITY
OF THE BLACK SEA MODEL TO PARAMETERS
An adjoint based technique is applied to a Shallow Water Model in order to estimate
influence of the model’s parameters on the solution. Among parameters the bottom topo-
graphy, initial conditions, boundary conditions on rigid boundaries, viscosity coefficients
and the amplitude of the wind stress tension are considered. Their influence is analyzed
from different points of view.
Two configurations have been analyzed: an academic case of the model in a square
box and a more realistic case simulating Black Sea currents. It is shown in both experi-
ments that the boundary conditions near a rigid boundary influence the most the solution.
This fact points out the necessity to identify optimal boundary approximation during a
model development.
K EYWORDS: Variational Data Assimilation, Sensitivity to parameters, Boundary
conditions, Shallow water model.
1. Introduction
Thirty years ago model and data were considered as independent one on
another. Observational data were interpolatedon the model grid in order to provide
the model with the initial conditions, forcings and all the other necessary parame-
ters. However, since the pioneering work [1] of Edward Lorenz, we know that a
geophysical fluid is extremely sensitive to initial conditions. A perturbation of ini-
tial state may grow exponentially in time limiting the validity of the forecast. This
discovery leads to understanding that observational data can not be considered as
independent of the model. We must perform a joint analysis of the model and data
in order to choose the optimal initial point for the model.
This become possible by using variational data assimilation technique, first
proposed in [2, 3], which is based on the optimal control methods [4] and perturba-
tions theory [5]. This technique allows us to retrieve an optimal data for a given
model from heterogeneous observational fields ensuring a better forecast.
However, even now, all other forcings and parameters of the model are ob-
tained from data by more or less sophisticated interpolation and they can not be
considered as optimal for a given model. In the same time, we may suppose, that
their influence on the models solution is as strong as the influence of initial state. In
this case, we should also analyze the possibility and utility to apply the data assimi-
lation techniques to identify optimal values for all these parameters in order to im-
prove the forecast.
The purpose of this paper is to analyze the sensitivity of a Shallow-Water
model and, in particular, compare the influence of initial conditions with the
influence of other parameters. Among these parameters, we consider the boundary
conditions on the rigid boundaries, bottom topography, empirical coefficients like
reduced gravity, forcing amplitude and dissipation.
162
2. Sensitivity and control of the boundary conditions
We shall focus our attention on the boundary conditions because (as we shall
see later) they represent the most unusual control variable.
However, as it has been noted in [6], particular attention must be paid to the
discretization process which must respect several rules because it is the discreti-
zation of the model’s operators that takes into account the set of boundary condi-
tions and introduces them into the model. Consequently, instead of controlling
boundary conditions them-self, it may be more useful to identify optimal discreti-
zation of differential operators in points adjacent to boundaries because this is
more general case. Indeed, boundary conditions participate in discretized opera-
tors, but considering the discretization itself, we take into account additional pa-
rameters like the position of the boundary, lack of resolution of the grid, etc.
Boundary conditions are usually introduced into the model by a particular
discretization of operators near the boundary. For example, taking into account
the condition u0 = 0 we can calculate the derivative at the point x = h/2 as
h
u
x
u 1
2/1
=
∂
∂
.
In this paper, we shall write the approximation of the derivative in a general
form
h
u
x
u 110
2/1
αα +
=
∂
∂
.
Coefficients 0α and 1α will be used as controls. That means we shall let them
vary in the data assimilation procedure in order to find an optimal pair that realiz-
es the minimum of the cost function.
2.1. Example: one-dimensional wave equation
In order to understand what happens when the data are assimilated to control the
boundary conditions, we propose to take a look on a scholar example: one-
dimensional wave equation written for u = u(x,t) and p = p(x,t) in the following way:
0
0
=
∂
∂−
∂
∂
=
∂
∂−
∂
∂
x
u
t
p
x
p
t
u
(1)
This equation is defined on the interval 0 < x < 1 with boundary conditions
prescribed for u only:
u(0,t)= u(1,t) =0 (2)
Initial conditions are prescribed for both u and p
uxu =)0,( ; pxp =)0,( (3)
163
The equation is discretized on a regular grid that is somewhat similar to Ara-
kawa C grid [7] in two dimensions:
Discrete derivatives of u and p are defined as follows
h
pp
x
p pii
i
2/2/1)( −+ −
=
∂
∂
,
h
uu
x
u pii
i
2/2/1
2/1)( −+
+
−
=
∂
∂
(4)
in all internal points i.e. 2 ≤ i ≤ N – 2 for
ix
p
)(
∂
∂
and 1 ≤ i ≤ N – 2 for 2/1)( +∂
∂
ix
u
.
Near the boundary, at points 1/2, 1, N – 1, N – 1/2, we write the derivatives
in a general form, like
h
pp
x
p pp
2/312/10
1)(
αα −
=
∂
∂
,
h
u
x
u uu
110
2/1)(
αα +
=
∂
∂
(5)
considering α0 and α1as the control coefficients. Leap-frog scheme was used for
time stepping.
We introduce the simplest cost function that represents the distance between
the model solution and observation at time t:
L dxdttxptxptxutxu obs
T
obs 2
0
1
0
2 )),(),,(()),(),,(()( −+−= ∫ ∫ ααα (6)
and we calculate its gradient using the adjoint to the derivative of the solution
with respect to control coefficients α p,α u:
∫
∂
∂=∇
T
tptu
0
*
)(),(
2
α
L dt
txptxp
txutxu
obs
obs
−
−
),(),,(
),(),,(
α
α
(7)
Once we prescribe the initial conditions for the equation
u(x,0) = sin(kπx) p(x,0) = cos(kπx),
we can calculate its exact solution:
ue x ac t(x,t)= 2 sin(kπt – π /4)sin(kπx),
pe x ac t(x,t)= – 2 cos(kπt – π/4)cos(kπx).
The exact solution is used as artificial observational data in this example. We
perform the minimization of the cost function (6). The minimization procedure
used here was developed by Jean Charles Gilbert and Claude Lemarechal, INRIA
[8]. The procedure uses the limited memory quasi-Newton method.
The difference between the models solution and the exact one is shown in fig.1.
p1/2 p3/2 p5/2 pN-5./2 pN-3/2 pN-1/2
u0 u1 u2 u3 uN-3 uN-2 uN-1 uN
164
Fig. 1. Difference between the models solution and the exact one: classical
BC – dashed line ( ), optimal BC – solid line ( ).
We see that optimal discretization of derivatives near the boundary brings the
solution much closer to the exact solution, but the set of optimal coefficients α
does not approximate a derivative:
h
uu
x
u 01
2/1
048,1
−=
∂
∂
h
pp
x
p 2/12/3
1
828,2014,3 −=
∂
∂
(8)
Neither expression for
x
u
∂
∂
, nor for
x
p
∂
∂
has any reasonable order of approxi-
mation. The first one is of «0» order, the second is of «-1» order. Moreover, while
we get always the same formula for
x
u
∂
∂
, approximation of the derivative of p
varies in different assimilation experiments. Assimilations performed with
different assimilation windows, for example, result in different coefficients for
x
p
∂
∂
. In fact, any combination pp
10 ,αα in (5) may be found as the result of assimi-
lation under condition
107,0104,1 01 −−= pp αα . (9)
This linear relationship has been obtained experimentally performing assimila-
tions with all assimilation windows in range from 600 to 2400 time steps (with
the time step equal to 1/120 of the time unit).
0 50 100 150 200 250 300
Time
102
101
100
10-1
10-2
10-3
10-4
10-5
10-6
D
iff
er
en
ce
w
ith
th
e
ex
ac
t s
o
lu
tio
n
165
To explain this strange result, we analyze the numerical solution of the equa-
tion. It is well known, the principal numerical error of the scheme is a wrong
wave velocity. The wave speed, that must be equal to 1, is replaced by
)2/sin(2
)sin(
kh
kh
τ
τβ =
which depends on the time step τ and the grid step h. For the given parameters
(k = 3, h = 1/30 and τ =120), error in the wave velocity is equal to 3,09 × 10−3.
The data assimilation and control of the boundary derivatives can not modify
numerical wave velocity. The only way for this control to get a better solution
consists in modifying the length of the interval. A numerical wave with wrong
velocity will propagate on the interval with wrong length. But the length of the
interval is adapted by data assimilation in order to ensure the wave with numeri-
cal velocity propagates the modified interval in the same time that the exact wave
propagates the exact interval. So far, the control can not correct the error in the
wave velocity, it commits another error in length in order to compensate the first
one as it is illustrated in fig. 2.
p1/2 p3/2 p5/2
u0 u1 u2
p1/2 p3/2 p5/2
u0 u1 u2
Fig. 2. Modification of the intervals length.
Non uniqueness of optimal p1α and p
0α can be explained if we take into ac-
count that p has also a form of cosine of 3πx. Hence, at any time p1/2 = A(t)cos(3πh/2)
and p3/2= A(t)cos(9πh/2) with some A depending on time. Their linear combina-
tion 2/302/11 pp pp αα + can vanish if
3)2/(cos4 2
0
1 −
−=
hk
p
p
π
αα (10)
Consequently, all couples p1α , p
0α belonging to the line that passes by the
point p
0α = – 1,023, p
1α =1,023 with tangent =
−
−
3)2/(cos4 2
0
hk
p
π
α
– 1,108 pro-
duce the same derivative. This line coincides withing accuracy of computation
with the set (9) obtained numerically. Any point on this line gives coeffcients pα
that theoretically provide the same value of the derivative and the same value of
the cost function.
Of course, in this simple example we can avoid the ambiguity in the solution:
it is sufficient to control only one coefficient pα rather than two. But, in more
complex problems, it may be difficult to locate and avoid the presence of kernels.
166
Consequently we can say that the data assimilation allows to place the boundary
in the optimal position resulting in a solution closer to the exact one. Boundary con-
trol allows to compensate numerical errors committed in the interior of the domain,
but it may be difficult to understand the physical meaning of optimal coefficients α
and non-null kernels may exist leading to non unique result.
More details of this study can be found in [9]
2.2. Shallow Water Model
In this paper we consider a shallow-water model written in a conservative form:
−
∂
∂−−+
∂
∂−=
∂
∂
x
u
hHhghhu
x
fhu
t
hu µ)(2
,0 xhu
y
u
hhuv
y
ττσµ +−
∂
∂−
∂
∂−
−
∂
∂−
∂
∂−−=
∂
∂
x
u
hhuv
x
fhu
t
hv µ (11)
xhv
y
v
hHhghhv
y
ττσµµ 0
2 )( +−
∂
∂−−+
∂
∂− ,
y
hv
x
hu
t
h
∂
∂−
∂
∂−=
∂
∂
,
where hu(x,y,t) and hv(x,y,t) are two flux components that represent the product
of the velocity by the ocean depth, h(x,y,t), that corresponds to the distance from
the sea surface to the bottom of the ocean. The sea surface elevation is
represented by the difference h(x,y,t) – H(x,y), where H(x,y) is the bottom to-
pography. The model is driven by the surface wind stress with components
τx(x,y,t) and τy(x,y,t) normalized by τ0 and subjected to the bottom drag that is
parameterized by linear terms σhu and σhv. Horizontal eddy diffusion is
represented by harmonic operators div( uh∇µ ) and div( uh∇µ ) . Coriolis para-
meter is represented by the variable f(y) that is equal to f0+βy assuming β-plane
approximation. Parameter g is the reduced gravity. The system is defined in some
domain Ω with characteristic size L requiring that both hu and hv vanish on the
whole boundary of Ω. No boundary conditions is prescribed for h. Initial condi-
tions are defined for all variables: hu, hv and h.
As usual, initial conditions are considered as the control parameter of the
model in this paper. We study the sensitivity of the model to its initial point and
assimilate data to find its optimal value. However, in addition to initial condi-
tions, all other parameters of the model, and namely the discretization of opera-
tors near the boundary, its bottom topography H(x, y), scalar coefficients µ, σ, g
and τ0, are also considered as control variables. All of them are allowed to vary in
the data assimilation procedure in order to bring them to their optimal values.
We discretize all variables of this equation on the regular Arakawa’s C-grid
[10] with constant grid step δx =
N
L
in both x and y directions. Discretizing the
167
system (11), we replace the derivatives by their finite difference representations
Dx and Dy and introduce two interpolations in x and y coordinates Sx and Sy. Inter-
polations are necessary on the staggered grid to calculate the variable’s values in
nodes where other variables are defined. The discretized system (11) writes
+
−−++−
∂
∂
hS
hu
hDyxHhgh
h
huS
DhuSfS
t
hu
x
x
x
xyx µ)),((
)( 2
x
x
yyx
yx
yx
y hu
hS
hu
DhSS
hSS
huShuS
D ττσµ 0)(
)(
+−=
−+ ;
+
−++
∂
∂
hS
hv
DhSS
hSS
huShuS
DhuSfS
t
hu
y
yyx
yx
yx
xyx )(
)(
µ (12)
y
y
y
x
y hu
hS
hu
hDyxHhgh
h
huS
D ττσµ 0
2
)),((
)(
+−=
−−++ ;
hvDhuD
t
h
yx −−=
∂
∂
.
Discretized operators Dx, Dy and Sx, Sy are defined in a classical way at all internal
points oft he domain. For example, the second order derivative and the interpola-
tion operator of the variable hu defined at corresponding points write
dx
huhu
huD jiji
jix
2/1,12/1,
2/1,2/1)( −−−
−−
−
= for i = 2, … N – 1,
2
)( 2/1,12/1,
2/1,2/1
−−−
−−
−
= jiji
jix
huhu
huS for i = 2, … N – 1.
Discretization of operators in the directly adjacent to the boundary nodes are
different from (13) and represent the control variables in this study. In order to
obtain their optimal values assimilating external data, we suppose nothing about
derivatives and interpolations near the boundary and write them in a general form
dx
huhu
huD j
D
j
D
D
jx
hu
x
hu
xhu
x 2/1,122/1,01
02/1,2/1)( −−
−
−
+=
σσ
σ ;
2
)( 2/1,122/1,01
02/1,2/1
−−
−
−
+= j
SD
j
S
S
jx
huhu
huS
hu
x
hu
xhu
x
σσ
σ .
This formula represents a linear combination of values of hu at two points
adjacent to the boundary with coefficients α . The constant 0α maybe added in
some cases to simulate non-uniform boundary conditions like 0),0( 0 ≠= αyhu .
We distinguish α for different variables and different operators allowing
different controls of derivatives because of the different nature of these variables
(13)
(14)
168
and different boundary conditions prescribed for them. It is obvious, for example,
that the approximation of the derivative Dx in the first equation may differ from
the approximation of Dx in the third one. Although both operators represent a de-
rivative, boundary conditions for hu and h are different, these derivatives are
defined at different points, at different distance from the boundary. Consequently,
it is reasonable to let them be controlled separately and to assume that their op-
timal approximation may be different with distinct coefficients
hu
xDα and
h
xDα .
Time stepping of this model is performed by the leap-frog scheme. The first
time step is splitted into two Runge-Kutta stages in order to ensure the second
order approximation.
As well as before, the approximation of the derivative introduced by (13) and
(14) depends on variables α . These variables are added to the set of control va-
riables enumerated above. Operators are allowed to change their properties near
boundaries in order to find the best fit with requirements of the model and data.
To assign all control variables we shall perform data assimilation procedure and
find their optimal values. Variational data assimilation is usually performed by
minimization of the specially introduced cost function. The minimization is
achieved using the gradient of the cost function that is usually determined by the
run of the adjoint to the tangent linear model.
To define the cost function we introduce dimensionless state vector φ that is
composed of three variables of the model t
hhvhu hwhvwhuw },,{=φ weighted by
coeffcients w . These weights are used to normalize values of the flux components
by 1
00 )( −== gHHww hvhu and the Sea surface elevation by 1
0)( −= Hwh . The
distance between the model solution and observations is defined as the Euclidean
norm of the difference
∑ =−==
k
obs
kktp 222 )()),(( φφφξξ (15)
2
,
,,
22
,
,,
22
,
,,
2 )()()( ∑∑∑ −+−+−=
jt
obs
jijih
jt
obs
jijihu
jt
obs
jijihu hhwhvhvwhuhuw . (16)
In this expression, we emphasize implicit dependence of ξ on time and on the set
of the control parameters p that is composed of:
– the set of initial conditions of the model }|,|,|{ 0000 ==== ttt hhvhuφ ;
– the set of the coefficients α that controls the discretizations of operators
near the boundary;
– the bottom topography H(x, y);
– four scalar parameters σ, µ, g, τ0.
Taking into account the results obtained in [11], we define the cost function as
L ∫=
T
dttptp
0
2 )),(()( φξ
(17)
169
that gives higher importance to the difference 2ξ at the end of assimilation interval.
It should be noted here, that this cost function can only be used in the case of
assimilation of a perfect artificially generated data. When we assimilate some
kind of real data that contains errors of measurements and is defined on a
different grid, we should add some regularization term to the cost function (like
the distance from the initial guess) and use some more appropriate norm instead
of the Euclidean one (see, for example [12] for details).
The n-th component of the gradient of the cost function can be calculated as
the Gateaux derivative of an implicit function:
∇( L n) = =
∂
∂
∂
∂=
∂
∂=
∂
∂
∫ ∫ ∑ dt
p
tdt
p
t
p
T T
n
k
k knn 0 0
22 φ
φ
ξξL
dt
p
t
T
n
k
k
obs
kk∫ ∑
∂
∂
−=
0
)(2
φφφ
because the derivative
kφ
ξ
∂
∂ 2
can easily be calculated from (15): )(2
2
obs
kk
k
φφ
φ
ξ −=
∂
∂
.
The second term in (18),
n
k
p∂
∂φ
, represents the matrix of the tangent linear model
that relates the perturbation of the parameter pn and the perturbation of kth com-
ponent of the model state vector kφ . This relationship, of course, is assumed in
the linear approach, that means it is only valid for infinitesimal perturbations.
In the classical case, when initial conditions are considered as the only con-
trol variable, the derivative
0
)()(
φ
φφ
∂
∂=
∂
∂ t
p
t
is the classical tangent model that de-
scribes the temporal evolution of a small error in the initial model state. The ma-
trix is a square matrix that is widely studied in numerous sensitivity analyses. Its
singular values at infinite time limit are related to well known Lyapunov expo-
nents that determine the model behavior (chaotic or regular) and the dimension of
it’s attractor.
In our case, the matrix
0
)(
φ
φ
∂
∂ t
is rectangular. It describes the evolution of an
infinitesimal error in any parameter (including initial state). However, we can
study it’s properties in the similar way as we do with the classical tangent linear
model. Its structure and composition is described in [11] for the case of using
coefficients α as control parameters and in [13] for the case when the bottom to-
pography is used to control the model’s solution.
The product ( )∑ ∂
∂−
k n
kobs
kk p
φφφ in (18) represents an unusual vector-matrix
(18)
170
product. To calculate this product directly we would have to evaluate all the ele-
ments of the matrix. This would require as many tangent model runs as the size of
the state vector is. So, instead of the tangent model, we shall use the adjoint one
that allows us to get the result by one run of the model. Backward in time adjoint
model integration that starts from ( obsφφ − ) provides immediately the product
)(
*
obs
p
φφφ −
∂
∂
which is exactly equal to
p
obs
∂
∂− φφφ )( in (18).
Using these notations, we write
dttt
p
t
t obs
T
))()((
)(
2
*
0
φφφ −
∂
∂=∇ ∫L (19)
where the expression in the integral is the result of the adjoint model run from t to
0 starting from the vector ( )()( tt obsφφ − ).
Tangent and adjoint models have been automatically generated by the Tape-
nade software [14, 15] developed by the TROPICS team in INRIA. This software
analyzes the source code of the nonlinear model and produces codes of it’s deriv-
ative
p∂
∂φ
and of the adjoint
*
∂
∂
p
φ
.
This gradient is used in the minimization procedure that is implemented in
order to find the minimum of the cost function:
)(min)( pp LL = (20)
Coefficients p are considered as coefficients achieving an optimal parameters
for the model. As it has been already noted, the set of parameters p is composed
of the set of initial conditions of the model 0φ , the set of the coefficients α that
controls the discretization of operators near the boundary, the bottom topography
H(x, y) and four scalar parameters σ, µ, g, τ0. We shall minimize the cost function
controlling either the total set of available parameters p or any possible subset,
comparing the efficiency of the minimization.
We use the minimization procedure developed by Jean Charles Gilbert and
Claude Lemarechal, INRIA [8]. The procedure uses the limited memory quasi-
Newton method.
In addition to the data assimilation, we perform also the sensitivity study of
the model solution to parameters enumerated above. We are looking for a pertur-
bation in the model’s parameters pδ that, for a given small norm, maximizes the
norm of the perturbation of the solution at time t.
p
t
t
δ
δφ
λ
)(
max)( = (21)
p
p
171
We can note that we already have all the necessary software to estimate λ(t).
Tangent linear model
∂
∂
p
t)(φ
allows us to calculate p
p
t
t δφδφ
∂
∂= )(
)( . Using
the scalar product that corresponds to the norm in the definition of the distance ξ
(16), we can write >>
=
>><<
>>
∂
∂
∂
∂<<
=
>><<
>><<=
pp
p
p
t
p
p
t
pp
tt
t
δδ
δφδφ
δδ
δφδφλ
,
)(
,
)(
max
,
)(),(
max)(
>><<
>>
∂
∂
∂
∂<<
=
pp
pp
p
t
p
t
δδ
δδφφ
,
,
)(
,
)(
max
*
(22)
This expression is a well known Rayleigh-Ritz ratio which is equal to the largest
eigenvalue of the problem
ϑλϑφφ
)(
)()(
*
t
p
t
p
t =
∂
∂
∂
∂
(23)
So far, we need just the maximal eigenvalue and the matrix of the problem is a
self-adjoint positive definite matrix, we can solve the problem (23) by the power
method performing successive iterations
n
n
n
p
t
p
t
p
t
p
t
ϑφφ
ϑφφ
ϑ
∂
∂
∂
∂
∂
∂
∂
∂
=+
)()(
)()(
*
*
1 , 0ϑ = random vector.
In the limit, the denominator of the right-hand-side tends to the largest eigenvalue
and nϑ – to the corresponding eigenvector of the matrix. The principal advantage
of this method consists in the fact that we do not need to calculate the matrix
itself, we just need a matrix-vector product. So far, we have both codes for the
tangentand adjoint models, we can successively run these models and get the left-
hand side of (23).
We should note here that when the initial conditions of the model are used as
the control parameters (i.e. )0(δφδ =p ), the sensitivity characteristics λ(t) are all
close to one when 0→t . It is evident because the perturbation has no time to
betransformed by the model’s dynamics and we get pt δδφδφ ==→ )0(|)( 0t .
When any other model parameter is used as the control and the error growing
time is small, all λ(t) are vanishing. This is also clear: the model’s dynamics has
no time to transmit the perturbations from the parameters to the solution. The per-
turbation of the solution remains, consequently, close to zero as well as the value
of 0|)( 0t =→tλ .
<<
<<
<<
<<
<<
<< >>
>>
>>
>>
>>
>>
172
In order to make the behavior of the sensitivity characteristics uniform with
different parameters, we shall use 1)( →tλ every time when the initial model’s
state is considered as the control parameter.
3. Configurations
3.1. Model in a square box
We start from the data assimilation in frames of the very well studied «aca-
demic» configuration. Several experiments have been performed with the model
in a square box of side length L = 2000 km driven by a steady, zonal wind forcing
with a classical sinusoidal profile
L
Ly
x
)2/(2
cos0
−= πττ
that leads to the formation of a double gyre circulation [16]. The attractor of the
model and the bifurcation diagram in a similar configuration has been described
in [17]. Following their results, we intentionally chose the model’s parameters to
ensure chaotic behavior. The maximal wind tension on the surface is taken to be
2
0 /5,0 cmdyne=τ . The coefficient of Eckman dissipation and the lateral friction
coefficient are chosen as σ = 5 × 10−8 (ms)-1 and µ = 200 m2/s respectively.
As it has been already noted, the Coriolis parameter is a linear function in y
with f0 = 7 × 10-5 s-1 and β = 2 × 10-11(ms)-1. The reduced gravity and the depth
are respectively equal to g = 0,02 m/s2 , H0 = 1000 m.
The resolution of the model in this section is intentionally chosen to be too
coarse to resolve the Munk layer [18] that is characterized by the local equili-
brium between the β-effect and the lateral dissipation. Its characteristic width is
determined by the Munk parameter
3/1
2
=
β
µ
d which is equal to 42 km in the
present case. The model’s grid is composed of 30 nodes in each direction, that
means the grid-step is equal to 67 km, that is more than the Munk parameter.
Thus, there is only one grid node in the layer and the solution exhibits spurious
oscillations near the western boundary due to unresolved boundary layer.
Artificial «observational» data are generated by the same model with all the
same parameters but with 9 times finer resolution (7,6 km grid step). The fine
resolution model, having 7 nodes in the Munk layer, resolves explicitly the layer
and must have no spurious oscillations. All nodes of the coarse grid belong to the
fine grid, consequently, we do not need to interpolate «observational» data to the
coarse grid. We just take values in nodes of the high resolution grid that corres-
pond to nodes on the coarse grid.
The model on the fine grid has been spun up from the rest state during 3
years. The end of spin up was used as the initial state for the further integration of
the model. From the result of this integration we have extracted values of all three
variables at all grid points that belong to the coarse grid (as it has been noted, the
grids have been chosen so, that all grid points of the coarse grid belong to the fine
grid). This set is used as artificial observations in the following experiments.
So far the model is nonlinear with intrinsicly instable solution, there is no
hope to obtain close solutions in long time model runs because any difference
173
(even infinitesimal) between two models grows exponentially in time. Conse-
quently, we have to confine our study to the analysis of a short time evolution of
the model’s solution simulating the forecasting properties of the model.
As the initial guess for the initial conditions we use the state vector of the
high resolution model reduced on the coarse grid. This state is also used as the
initial conditions in all other assimilation experiments with other control parame-
ters. Noted above values of the model’s parameters (flat bottom topography, li-
near in y Coriolis parameter and scalar parameters (µ, σ, τ0, g) are used as the ini-
tial guess in the experiments that control these parameter, otherwise we simply
use these parameters in the model.
3.2. Model of the Black Sea
In this section we use the same model, but all the parameters are defined to
describe the upper layers circulation of the Black sea. Configuration of the model
and observational data have been kindly provided by Gennady Korotaev from the
Marine Hydrophysical Institute, National Academy of Sciences of Ukraine, Se-
vastopol, Ukraine. This configuration is described in [19].
The model grid counts 141 × 88 nodes that corresponds to the grid box of
dimension 7860 m and 6950 m in x an y directions respectively. 15 minutes
time step is used for integration of the model. The Coriolis parameter is equal to
f0 = 10-4s-1and β = 2 × 10-11(ms)-1. Horizontal viscosity is taken as µ = 50 m2s-1.
Using a typical density difference between upper and underlying layers of
3,1 kg/m3, and unperturbed layer thickness of H0 = 150 m, the Rossby radius of
deformation is estimated at about 22 km and the reduced gravity value
g = 0,031 m/s2. The grid therefore resolves the mesoscale processes reasonably well.
The model has been forced by the ECMWF wind stress data, available as dai-
ly averages for the years 1988 through 1999. Dynamical sea level reconstructed
in [20] was used as observational data in this section. These data have been col-
lected in ERS-1 and TOPEX/Poseidon missions and preprocessed by the NASA
Ocean Altimeter Pathfinder Project, Goddard Space Flight Center. Observational
data are available from the 1st May 1992 until 1999. These data have been linear-
ly interpolated to the model grid.
So far the sea surface elevation is the only observational variable available in
this experiment, we put 0== hvhu ww in (16). Consequently, the difference be-
tween the model’s solution and observations is calculated taking into account the
variable h only.
As it has been already noted, absence of observational data for the velocity fields
brings us to modify the cost function. We have to add the background term in the cost
function in order to require the velocity field to be sufficiently smooth. Otherwise,
lack of information about velocity components in observational data would result in a
spuriously irregular fields obtained in assimilation. To ensure necessary regularity of
hu and hv we add the distance from the initial guess to the cost function (17). In order
to emphasize the requirement of smoothness, this distance is measured as an enstro-
phy of the difference between the initial guess and current state:
174
2
,
0
,,
0
,, )()(
∑
∂
−∂
−
∂
−∂
=
ji
jijijiji
smooth y
huhu
x
hvhv
L (24)
where hu0, hv0 denote flux components of the initial guess of the minimization-
procedure.
Moreover, using real observational data requires to add at least one another
term to the cost function. One can see in the fig. 2 in [20], spatially averaged sea
surface elevation of the Black Sea exhibits a well distinguished seasonal cycle.
That means the mass is not constant during a year, it decreases in autumn and
increases in spring. Consequently, if we assimilate data during a short time (a
season or less), we assimilate also the information about the mass flux specific for
this season. This flux can not be corrected later by the model because the discreti-
zation of operators near the boundary (that controls the mass evolution) is obtained
once for all seasons. The mass variation of the Black sea reaches 25 centimeters of
the sea surface elevation. Assimilating data within one season may, consequently,
result in a persisting increasing or decreasing of the seal level of order of 50 cm per
year. To avoid this spurious change of the total mass, we must either take the assi-
milation window of at least one year, or prescribe the mass conservation to the
model’s scheme. One year assimilation window is computationally expensive and
is not justified by the model’s physics. On the other hand, prescribed mass con-
servation removes just the sinusoidal seasonal variation, allowing us to keep all
other processes and to choose any assimilation window we need.
To correct the mass flux of the model, we add the following term to the cost
function
dthth
T
ji
jijimass
2
0 ,
,, ))0()((∫ ∑
−=L . (25)
Similarly to (24), this term also ensures the regularity of the solution. It can
be noted here that other terms may be added to the cost function in order to make
a numerical scheme energy and/or enstrophy conserving, but we do not use them
in this paper.
The total cost function in this section is composed of three parts: (17), (24)
and (25)
maassmoothtotal LLLL 21 γγ ++= (26)
Coefficients γ are introduced to weight the information that comes from observa-
tional data (with L) and an a priori knowledge about mass conservation and regu-
larity of the solution.
This modification of the cost function results, of course, in additional terms
in the gradient:
( ) ( )∑ −+−+−+∇=∇
ji
jijixyytotal hvhvDDhuhuDD
,
,,20
*
0
*
1 .)0(2)()(2 ηηγγLL (27)
The model is spun up from the beginning of 1988 to May 1992 using the
wind tension data on the surface. The state corresponding to the 1st of May 1992
175
12h GMT is used as the initial guess in the data assimilation procedure controlling
initial conditions of the model. The assimilation controls the initial conditions
0φ only with the assimilation window T = 1 day and the regularization parameter
γ1 = 0,04. Such a short window was chosen in order to get almost instantaneous
state of the model to be used in further experiment as an initial state.
The behavior of the model solution is not chaotic in this configuration. Varia-
bility of the model is generated directly by the variability of the wind stress on the
surface. Consequently, we can compare particular trajectories of the model on any
time interval because their evolution is stable without exponential divergence.
Thus, we can hope that assimilating data in a relatively short window allows us to
bring the model’s solution closer to observation for a long integration period.
The minimization of the cost function has been accompanied by the mass
preserving correction (25) with γ2 = 0,01.
4. Sensitivity analysis
The flexibility of the model is illustrated in fig. 3. We perform the data assi-
milation experiment in two configurations using parameters described above as
initial guess. Due to high CPU time of the data assimilation, we limit the number
of iterations of the minimization procedure by 20. Thus, we have similar and rea-
sonable computational cost in each experiment.
In both configurations we examine the evolution of the distance «model–
observations» ξ(t) during assimilation and after the end of assimilation. Assimila-
tion window has been chosen as 5 days in the square box configuration and
T = 30 days for the Black Sea model. The distance is examined over longer inter-
vals: 20 days in the first case and 1 year in the second one.
Square box Black Sea
0 5 10 15 20
Time (days)
0 100 200 300 400
Time (days)
D
is
ta
n
ce
2,80
2,40
2,00
1,60
1,20
0,80
0,40
0,00
D
is
ta
n
ce
×
1
0-3
400
360
320
280
240
200
160
120
80
40
0
1
2
3
4
1
2
3
4
Fig. 3 : Distance between the model solution and observations for the model in the
square box (a) and the model of the upper layer of Black Sea (b). 1 – No Assimila-
tion; 2 – Optimal Init Cond.; 3 – Optimal Boundary; 4 – All Optimal.
a) b)
176
Analyzing the difference between the model solution and observations
shown in fig. 3, we see that in the assimilation window the model is almost equal-
ly flexible with respect to both initial and boundary conditions. Data assimilation
allows us to reduce the distance between the model solution and observations at
the end of the window approximately twice in both configurations. The only dif-
ference that can be seen in the assimilation window is that non-optimal initial
point leads to the spurious oscillations of the solution. These oscillations occur in
both configurations and show us the necessity to identify the optimal initial point.
However, the influence of parameters is significantly different beyond the
window. While the solution with optimal initial point tends towards the solution
obtained without any data assimilation, optimal set of boundary conditions en-
sures a new solution that is much closer to observational data. That means the
control of boundary conditions allows us to improve a long-range forecasting
quality of the model.
The third way of the sensitivity analysis consists in solving of the eigenvalue
problem (23) and analyzing λ(t) on different scales of error growing time from
about 10 minutes (10-3 day) to approximately one year. As it has been already
noted, λ(t)−1 is plotted in the case when initial conditions are considered as the
parameter.
Analyzing the figure (see fig. 4), we can see that three time scales can be
clearlydistinguished for the sensitivity characteristics of the model in both
configurations. The first, short time scales, approximately from 0 up to 2-3 hours
is characterized by the linear growth of λ(t). Indeed, the model behaves as a linear
model on this scales, the model’s solution can be well approximated by just one
step of the numerical time scheme.
The second time scale that can be distinguished in the figure (see fig. 4) cor-
responds to error growing times from 2-3 hours to 10 days. On these time scales
we see slower growth of the sensitivity characteristics λ(t) and, sometimes, no
growth at all. These time scales are characterized by the modification of the sta-
ble-instable subspaces of the model. Instable space on short time scale is not the
same as for long time evolution. Short time instabilities are usually localized in
space, while long time eigenvectors of (23) possesses a global structure.
The third time scale corresponds to the error growing times more than 100 days.
On these scales the model exhibits either non-linear chaotic behavior with exponen-
tial growth of all λ(t) (as it is the case in the square box), or stable behavior when
a perturbation of initial state decreases with time (as it is the case in the Black Sea
model).
In order to zoom these time scales, we plot the same data in the Log-Log and
Log-Linear coordinates in fig.4 on the left and on the right respectively. One can
see the error growth in the square box on this time scale is purely exponential
with the same exponent λ(t) = A·exp(0,027t). The multiplier A is particular for
each parameter, but the exponent is always the same. This confirms the remark
made in [13], [11]: no matter how the perturbation was introduced into the model,
it’s long-time growth is determined by the model’s dynamics.
Square box Black Sea
a) 103
102
101
100
5
2
5
2
5
2
5 3
4
b)
101
100
2
5
2
5
2
1
3
4
5
S
in
g
u
la
r
va
lu
es
S
in
g
u
la
r
va
lu
es
177
Comparing the evolution of the sensitivity of the model to different parame-
ters,we see that on small scales it is the bottom topography that the model is the
most sensitive to (thin solid line in fig. 4). An error in the topography produces 13
times bigger perturbation in the model state than a similar error in the model’s
initial conditions (thick solid line in fig. 4). However, λ(t) does not grow at all on
medium scales due to significant changes in the eigenvectors pattern. This leads
to the fact that on long scales, the sensitivity of the model to the bottom topogra-
178
phy is about 2 times lower than the sensitivity to initial conditions.
On the other hand, the sensitivity of the model to the discretization of operators
near the boundary exhibits the opposite behavior. On short scales, corresponding λ is
2 times lower than λ obtained for perturbations of 0φ , but there is no stagnation of the
growth on the middle scales. As a result, we see that the model is 4 times more sensi-
tive to α than to 0φ for long error growing times. Moreover, small perturbation of
initial conditions decreases in the Black sea configuration, while the perturbation of
α results in an increasing perturbation of the solution.
5. Modification of the boundary conditions
As it has been noted, it is useless to analyze the set of obtained coefficients α
to understand the modification of the boundary conditions. Instead of this, we
shall see the difference between velocity fields with classical and with optimal
coefficients α similarly to [21]. This difference has been averaged in time over
200 days time interval in order to reveal persistent modifications of the flow pro-
duced by the optimal discretization.
This average difference of the velocity together with the original velocity are
presented in fig. 5. We zoom the Southern part of the Black sea because it is in
this region the difference shows the biggest values reaching 5 cm/s while in the
middle of the sea it rarely exceeds 1 cm/s .
Fig. 5: Original velocity field (a) and its modification (b) near the boundary.
We can note several principal features of the flow that have been modified by
b)
a)
N
42,9°
42,6°
42,3°
42,0°
41,7°
41,4°
41,1°
N
42,9°
42,6°
42,3°
42,0°
41,7°
41,4°
41,1°
30° 31° 32° 33° 34° 35° 36° 37° E
5 cm/s
5 cm/s
179
boundary conditions. First, we can see a strong current on the boundary. The slip
condition (vanishing tangential velocity) has been replaced by a permanent cur-
rent along the boundary. Moreover, impermeability condition has also been
modified. The flow is now allowed to leave the domain ensuring, however, the
global mass balance. One can see a strong persistent vortex centered at 42,2°N,
32,8°E which southern part crosses the boundary resulting in not only tangential
but normal flux also. Similar vortices with lower amplitude can also be seen in
places where the boundary changes direction. Optimal discretization allows the
flow to cross the boundary in places where the direction change is not smooth.
Tangential velocity component is amplified in the direct vicinity of the boun-
dary. In these nodes we see a strong eastward flow that was forbidden by the
boundary conditions in the classical formulation of the model. On the other hand,
the eastward velocity is lower at nodes distanced by several grid cells from the
boundary. At these nodes we see westward flow in the difference of the optimally
discretized and classical models. That means the flow is moved towards the
boundary, allowing more optimal representation of a thin current on a coarse grid
that brings the model solution towards observations.
6. Conclusion
The comparative study presented in this paper shows the influence of
different model parameters on the solution. The study is confined to the analysis
of a low resolution model with a rather limited physics. Consequently we must
acknowledge the results may be valid only in the described case. Additional phys-
ical processes (baroclinic dynamics, variable density due to heat and salinity
fluxes, etc.) may modify results of this study revealing other parameters the mod-
el may be sensitive to.
The main conclusion we can made from this comparison is the important role
played by the boundary conditions on rigid boundaries. Almost all experiments
show that the model is the most flexible with respect to control of coefficients α ,
this control allows us to bring the model’s solution closer to the solution of the
high-resolution model or to the observed data.
Optimal α found in the assimilation window remain optimal long time after
the end of assimilation improving the forecasting ability of the model. We could
see that the fourth order model in the square box allows us to divide by two the
forecast error of the 20 days forecast. Optimal α obtained in one month assimila-
tion remains optimal even for a one year run of the Black sea model.
Finally, the long time sensitivity of the model’s solution to α exceeds the sensi-
tivity to almost all other parameters including the sensitivity to initial conditions. A
perturbation of α of a given small norm results in a bigger perturbation of the model’s
solution than a perturbation of some other parameter of an equal norm.
However, we could see that the influence of boundary conditions is only im-
portant on long time scales, i.e. time scales that exceeds the characteristic time of
the domain. In both experiments presented above the characteristic time was ap-
proximately equal to 5 days and in both experiments the sensitivity to α becomes
important on scale longer than 5 days. On the other hand, on short scales, it is the
bottom topography that influences the most the model’s solution. Both in the Black
sea and in the square box the sensitivity to topography is approximately 40 times
180
more important than the sensitivity to α .
In addition to that, we should note that usually prescribed boundary condi-
tions (impermeability and no-slip conditions have been used here as the initial
guess for the minimization of the cost function) seem not to be optimal for the
model. As we can see in fig. 3, modifying α we can bring the model much closer
to the high resolution model or to the observational data. But, the numerical
scheme is strongly modified in the assimilation process violating even impermea-
bility condition.
Taking into account an important influence of the numerical scheme that in-
troduces boundary conditions into the model, it is reasonable to think about iden-
tification of the optimal scheme by data assimilation process instead of prescrib-
ing classical boundary conditions.
Acknowledgments. Author thanks Gennady Korotaev from Marine Hydrophysical
Institute, National Academy of Sciences of Ukraine for providing the model parameters
and data for the upper layer model of the Black Sea.
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Rec eived 20 . 10 . 2011 ;
Материал поступил в редакцию 20 .10 .2011 г .
АНОТАЦ IЯ Методика, заснована на сполучених рівняннях, була застосована до
моделі «дрібної води» для оцінки чутливості її рішення до параметрів моделі. Розгля-
далися такі параметри як рельєф дна, початкові і граничні умови, коефіцієнти в'язкос-
ті і напруга тертя вітру на поверхні. Їх вплив аналізувалося з різних точок зору.
Задача була розглянута в двох конфігураціях: академічний варіант моделі в
квадратному басейні і більш реалістична версія, що моделює течії в Чорному морі.
Було показано, що граничні умови на твердих кордонах мають найбільший вплив
на рішення в обох випадках. цей факт вказує на необхідність пошуку оптимальних
граничних умов при розробці моделі.
АННОТАЦИЯ Методика, основанная на сопряженных уравнениях, была приме-
нена к модели «мелкой воды» для оценки чувствительности ее решения к парамет-
рам модели. Рассматривались такие параметры как рельеф дна, начальные и гра-
ничные условия, коэффициенты вязкости и напряжение трения ветра на поверхно-
сти. Их влияние анализировалось с различных точек зрения.
Задача была рассмотрена в двух конфигурациях: академический вариант моде-
ли в квадратном бассейне и более реалистичная версия, моделирующая течения в
Черном море. Было показано, что граничные условия на твердых границах оказы-
вают наибольшее влияние на решение в обоих случаях. Этот факт указывает на не-
обходимость поиска оптимальных граничных условий при разработке модели.
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