Inversion Study of Vertical Eddy Viscosity Coefficient Based on an Internal Tidal Model with the Adjoint Method

Abstract

Based on an isopycnic-coordinate internal tidal model with the adjoint method, the inversion of spatially varying vertical eddy viscosity coefficient (VEVC) is studied in two groups of numerical experiments. In Group One, the influences of independent point schemes (IPSs) exerting on parameter inversion are discussed. Results demonstrate that the VEVCs can be inverted successfully with IPSs and the model has the best performance with the optimal IPSs. Using the optimal IPSs obtained in Group One, the inversions of VEVCs on two different Gaussian bottom topographies are carried out in Group Two. In addition, performances of two optimization methods of which one is the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method and the other is a simplified gradient descent method (GDM-S) are also investigated. Results of the experiments indicate that this adjoint model is capable to invert the VEVC with spatially distribution, no matter which optimization method is taken. The L-BFGS method has a better performance in terms of the convergence rate and the inversion results. In general, the L-BFGS method is a more effective and efficient optimization method than the GDM-S.

Full text

Research Article
Inversion Study of Vertical Eddy Viscosity Coefficient Based on
an Internal Tidal Model with the Adjoint Method
Guangzhen Jin,1Qiang Liu,2and Xianqing Lv1
1Laboratory of Physical Oceanography, Ocean University of China, Qingdao 266100, China
2College of Engineering, Ocean University of China, Qingdao 266100, China
Correspondence should be addressed to Xianqing Lv; xqinglv@ouc.edu.cn
Received  March ; Revised  August ; Accepted  August 
Academic Editor: Fatih Yaman
Copyright ©  Guangzhen Jin et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Based on an isopycnic-coordinate internal tidal model with the adjoint method, the inversion of spatially varying vertical eddy
viscosity coecient (VEVC) is studied in two groups of numerical experiments. In Group One, the inuences of independent point
schemes (IPSs) exerting on parameter inversion are discussed. Results demonstrate that the VEVCs can be inverted successfully
with IPSs and the model has the best performance with the optimal IPSs. Using the optimal IPSs obtained in Group One, the
inversions of VEVCs on two dierent Gaussian bottom topographies are carried outin Group Two.In addition, performances of two
optimization methods of which one is the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method and the other is
a simplied gradient descent method (GDM-S) are also investigated. Results of the experiments indicate that this adjoint model is
capable to invert the VEVC with spatially distribution, no matter which optimization method is taken. e L-BFGS method has a
better performance in terms of the convergence rate and the inversion results. In general, the L-BFGS method is a more eective
and ecient optimization method than the GDM-S.
1. Introduction
Internal tide, which is the internal wave of tidal frequency,
is a ubiquitous phenomenon in the oceans. Rattray [],
Baines [], Bell [], Baines [], Craig [], Gerkema [], and
Llewellyn Smith and Young [] have developed theoretical
models and obtained some analytical solutions of internal
tide on ideal topographies. ese theoretical models helped
them investigate the generation and propagation of inter-
nal tide. Although great progress has been made on the
internal tide theory and some analytical works were carried
out, only a small amount of solutions can be provided,
due to the complexity of the problems. For this reason,
quantitative analysis with practical signicance still has to
rely on the combination of numerical simulation, theoretical
analysis, experiment, and observation. Numerical simulation
is an eective method in marine research and has been
widely used in the internal tide research. Kang et al. []
investigated the 2internal tide near Hawaii with a two-
dimensional, two-layered numerical model and conrmed
that the internal tide was generated by barotropic forcing
at the Hawaiian Ridge and propagated in north-northeast
and south-southwest directions. Based on a high-precision
three-dimensional Princeton Ocean Model (POM), Niwa and
Hibiya [] obtained the distribution of the 2internal tide in
the Pacic Ocean using the TOPEX/Poseidon (T/P) satellite
data. Cummins et al. [] simulated the generation and
propagation of internal tides near the Aleutian Ridge using
T/P altimeter data. e comparison between the altimeter
data and their model results showed good agreement for the
phase, which also provided evidence for wave fraction near
theAleutianRidge.Withathree-dimensionalPOM,Niwa
and Hibiya [] investigated the distribution and the energy
of the 2internal tide around the continental shelf edge
in the East China Sea. eir numerical experiment results
indicated that 2internal tides are eectively generated over
prominent topographies such as sea ridges, island chains,
and straits. Jan et al. []modiedthePOMtostudythe
generation of the 1internal tide and its inuence on
surfacetideintheSouthChinaSea.econversionfrom
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2015, Article ID 915793, 14 pages
http://dx.doi.org/10.1155/2015/915793

Mathematical Problems in Engineering
the barotropic energy to the baroclinic energy over topo-
graphicridgesintheLuzonStraitwasalsoestimated.
Determination of the vertical eddy viscosity coecient
(VEVC), which describes the vertical mixing in the ocean,
plays an important role in the study of energy exchange and
material transportation. e VEVC is regularly regarded as
a constant in numerical models. Schemes to determine the
VEVC mainly include the Prandtl mixing-length hypothesis
model, the -model, the Pacanowski-Philander mixing
model [], and some turbulent closure models that are
more complicated. Many studies have been carried out to
investigate the variation of the VEVC [–]. All these
mentioned studies indicate that due to dierent intensions
of the vertical mixing in sea water, the VEVC should
not be treated as a constant but a parameter with spatial
distribution.
Satelliteremotesensingtechnologyandotherrelated
technologiesprovideuswithalargenumberofdata.us,itis
one of the most important missions in physical oceanography
to make use of the data eciently and precisely as well
as to combine the observation data with present numerical
models. Indeed, data assimilation with the adjoint method
provides an eective access to these missions. e use of the
adjoint method in marine science can be traced back to s.
e adjoint model is capable of optimizing control param-
eters in numerical simulation. Bennett and McIntosh []
applied the weak constraint thought to solve the tidal problem
andthegeostrophic-owproblem.YuandO’Brien[]
assimilated both meterological and oceanographic data into
an oceanic Ekman layer model and deduced the unknown
boundary condition, the unknown vertical eddy viscosity,
and the current eld. Based on a tidal model with a two-level
leapfrog method, Lardner [] inverted the open boundary
conditions in three-test problems. Seiler [] used the adjoint
method to assimilate observations into a quasi-geostrophic
ocean model and estimated the lateral boundary values in
ideal experiments. Navon []wroteasummaryofthe
parameter estimation in meteorology and oceanography in
the view of applications with four- dimensional variational
data assimilation techniques. Using an automatic dieren-
tiation compiler, Ayoub [] constructed the adjoint model
oftheMassachusettsInstituteofTechnologyOceanGeneral
CirculationModelandinvertedtheopenboundarycondi-
tions in the North Atlantic. Zhang and Lu []developed
a three-dimensional nonlinear numerical tidal model with
the adjoint method and designed several numerical exper-
iments to estimate three kinds of parameters including the
open boundary conditions, the bottom friction coecients,
and the vertical eddy viscosity coecients. Zhang and Lu
[] employed a two-dimensional tidal model to study the
inversion of the bottom friction coecients in the Bohai Sea
and the Yellow Sea with the adjoint method. Chen et al. []
constructed a three-dimensional internal tidal model with
theadjointmethodandestimatedsixdierentkindsofopen
boundary conditions on fourteen types of topography. Based
on a tidal model, Zhang and Chen [] carried out several
semiidealized experiments to estimate the partly and fully
spatial varying open boundary conditions. Cao et al. []
investigated the inversion of open boundary conditions with
a three-dimensional internal tidal model and simulated the
2internal tide around Hawaii by assimilating T/P data.
ere are two main objectives of this paper. One is to
study the inversion of the VEVC with an internal tidal model
and the adjoint method. According to the introductions
above, a lot of studies have been carried out to investigate the
inversion of the control parameters of internal tide such as
the open boundary condition [,]andthebottomfriction
condition [,]. However, few works are found to study
the inversion of VEVC. Since VEVC is a decisive factor to
describe the vertical mixing in the ocean, it is necessary to pay
attention to the inversion of the VEVC. e other objective is
to make a computational investigation on the performance
of the gradient descent method and the limited-memory
Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method for
the inversion of VEVCs based on the model constructed by
Chen et al. []. Both of the methods do not require any
evaluations of the Hessian matrices but gradient vectors and,
thus, are computationally feasible. Chen et al. [] have made
a comparative study on several optimization methods but it
is on the inversion of the open boundary conditions which
is a one-dimension case. e feasibility of these optimization
methods for two-dimensional case such as the inversion of
the VEVC needs further studies.
Two groups of numerical experiments are carried out
to study the inversion of spatially varying VEVCs based
on an isopycnic-coordinate internal tidal model with the
adjoint method. In Group One, the inuences of independent
point schemes (IPSs) exerting on parameter inversion are
discussed. Group Two investigates the inversions of VEVCs
on two dierent Gaussian bottom topographies and the
performances of two optimization methods which are the
GDM-S and the L-BFGS methods.
is paper is organized as follows. Section  briey
introduces the adjoint tidal model and the methodology.
Two optimization methods, including the GDM-S and the L-
BFGS methods, are described in Section .Section  presents
design and process of the experiments in detail. Results of
the experiments are discussed in Section . Finally, we make
a summary and draw some conclusions in Section .
2. Numerical Model Introduction
An isopycnic-coordinate internal tidal model with adjoint
assimilation method is employed in this paper. ere are two
parts in the internal tide model. One is forward model with
the governing equations and the other is adjoint model with
the adjoint equations. e two models are used to simulate
the internal tide and to optimize the control variables,
respectively. Chen et al. [] had introduced the two parts in
great detail and tested the reasonability and feasibility of the
model. e formulation will not be presented in this paper.
e derivation of VEVC adjustment, introduction of the two
optimization method, test of the adjoint method, and the
independent point scheme (IPS) are described in details in
this part.

Mathematical Problems in Engineering 
2.1. Test of the Adjoint Method. According to the equations
and derivations of Chen et al. [], the formula to invert
the VEVC can be derived. e rst derivative of Lagrangian
function with respect to VEVC is obtained as follows:

V𝑖,𝑗,𝑘 =0, ()
where V𝑖,𝑗,𝑘 is the value of VEVC at grid (,)in the th layer.
e gradient of cost functions with respect to the VEVC in the
grid (,,)can be deduced as follows:

V𝑖,𝑗,𝑘
=𝑘+𝑘+1
×
𝑛
𝑛
𝑖,𝑗,𝑘 −𝑛
𝑖,𝑗,𝑘+1𝑛
𝑎𝑖,𝑗,𝑘 −𝑛
𝑎𝑖,𝑗,𝑘+1
𝑖,𝑗,𝑘 +𝑖+1,𝑗,𝑘 +𝑖,𝑗,𝑘+1 +𝑖+1,𝑗,𝑘+1
+𝑛
𝑖−1,𝑗,𝑘 −𝑛
𝑖−1,𝑗,𝑘+1𝑛
𝑎𝑖−1,𝑗,𝑘 −𝑛
𝑎𝑖−1,𝑗,𝑘+1
𝑖−1,𝑗,𝑘 +𝑖,𝑗,𝑘 +𝑖−1,𝑗,𝑘+1 +𝑖,𝑗,𝑘+1 

+𝑘+𝑘+1
×
𝑛
V𝑛
𝑖,𝑗,𝑘 −V𝑛
𝑖,𝑗,𝑘+1V𝑛
𝑎𝑖,𝑗,𝑘 −V𝑛
𝑎𝑖,𝑗,𝑘+1
𝑖,𝑗,𝑘 +𝑖+1,𝑗,𝑘 +𝑖,𝑗,𝑘+1 +𝑖+1,𝑗,𝑘+1
+V𝑛
𝑖−1,𝑗,𝑘 −V𝑛
𝑖−1,𝑗,𝑘+1V𝑛
𝑎𝑖−1,𝑗,𝑘 −V𝑛
𝑎𝑖−1,𝑗,𝑘+1
𝑖−1,𝑗,𝑘 +𝑖,𝑗,𝑘 +𝑖−1,𝑗,𝑘+1 +𝑖,𝑗,𝑘+1 
,
()
where 𝑘is the potential density in the th layer, 𝑛
𝑖,𝑗,𝑘 and
V𝑛
𝑖,𝑗,𝑘 are horizontal velocities at the th time step, 𝑛
𝑎𝑖,𝑗,𝑘 and
V𝑛
𝑎𝑖,𝑗,𝑘 are the adjoint variables of 𝑛
𝑖,𝑗,𝑘 and V𝑛
𝑖,𝑗,𝑘,respectively,
and 𝑖,𝑗,𝑘 is the initial thickness of the th layer. e detailed
derivation of () is presented in the appendix.
Accurately programming the adjoint in such problems as
the present one is quite tricky and experience has shown that
it is essential to check the accuracy of the adjoint computation
beforeproceedingwiththeminimizationruns[]. e
correctness of the adjoint method is veried in this section.
Take the rst-order term of a Taylor expansion for the cost
function and we obtain the following equation:
(p+U)=(p)+U⋅(p)+2. ()
Here, pis a general point of the control variable, (p)=
∇(p)isthecomputedgradient,andUis an arbitrary unit
vector in the parameter space. Based on () a function of 
canbewrittenasfollows:
Φ()=(p+U)−(p)
U⋅(p),()
where is a small real number that is not equal to zero.
If the adjoint methodology is correct, it is supposed that
lim𝛼→0Φ() = 1according to (). In this paper, the VEVC
variable Vis treated as pand test of the adjoint method is
based on ().
In order to test the accuracy of the adjoint method, two
experiments are carried out in which two dierent types of U
are used. e dierent vector directions are 1=()/|()|
and 2=(1/,1/,...,1/),respectively.
Figure  indicates the trends of Φ()as approaches to .
It is clear that in both experiments when is less than −3,
values of Φ(solidlines)arebothverycloseto(dashedlines).
Equation lim𝛼→0Φ() = 1is proved and the correctness of
gradient computed in the adjoint model is veried.
2.2. Independent Point Scheme. e available observation
data may not be sucient and control parameters to be
determined may be excessive in practice. at may cause
ill-posedness of the inversion problem. Richardson and
Panchang [] rst noted that if an adjoint method is applied,
when there is a big error in data, the solution will be unstable
and not unique. Many researchers have made progress in
solving this problem. A lot of work [,,,]havebeen
done to prove the capability and feasibility of the independent
point scheme (IPS) in solving ill-posed problems of inversion.
In this paper, the IPS is used to optimize the control
parameter. e basic idea of IPS is as follows: some grids (e.g.,
(,)) are selected as the independent points; it is assumed
that 𝑖𝑖,𝑗𝑗 represents the value of VEVC in grid (,),so
values of VEVC in all grids 𝑖,𝑗 can be calculated from 𝑖𝑖,𝑗𝑗
with linear interpolation method. e computing formula is
given as follows:
𝑖,𝑗 =∑𝑖𝑖,𝑗𝑗 𝑖,𝑗,𝑖𝑖,𝑗𝑗𝑖𝑖,𝑗𝑗
∑𝑖𝑖,𝑗𝑗 𝑖,𝑗,𝑖𝑖,𝑗𝑗 ,()
where 𝑖,𝑗,𝑖𝑖,𝑗𝑗 is the weight coecient of the Cressman form
[]:
𝑖,𝑗,𝑖𝑖,𝑗𝑗 =2−2
𝑖,𝑗,𝑖𝑖,𝑗𝑗
2+2
𝑖,𝑗,𝑖𝑖,𝑗𝑗 ,()
where 𝑖,𝑗,𝑖𝑖,𝑗𝑗 is the center distance between (,)and (,)
and is the inuence radius. According to (), the gradient
of with respect to 𝑖,𝑗 canbewrittenas

𝑖𝑖,𝑗𝑗 =
𝑖,𝑗 𝑖,𝑗,𝑖𝑖,𝑗𝑗 
𝑖,𝑗 
𝑖,𝑗 𝑖,𝑗,𝑖𝑖,𝑗𝑗,()
where /𝑖,𝑗 is the gradient of with respect to the VEVC
at the grid (,)and can be calculated using formula ().
According to Section ., the correctness of the gra-
dient with respect to the independent points should be
tested. Based on (), the perturbations ()applied to the
independent points have a linear impact on those applied
to the nonindependent points. e convergences for the
nonindependent points will remain the same aer linear
transformation for independent points.
e values of VEVC at the independent grids can be
calculated inside the model and values at other grids are

Mathematical Problems in Engineering
−5
−4−3−2−10
−0.5
0
0.5
1
1.5
2
2.5
log10Φ(𝛼 )
log10𝛼
U1
(a)
1
0.8
0.6
0.4
0.2
0
−0.2
U2
−5
−4−3−2−10
log10Φ(𝛼 )
log10𝛼
(b)
F : Variation of Φ()with respect to .
gained through interpolation using (). en the spatial
distribution of VEVC in the entire area is obtained.
3. Optimization Algorithms
ere have been many large-scale optimization methods to
solve the minimization problem []. Four main methods
arelinesearchmethod(e.g.,WolfeandGoldstein),trust-
region method, conjugate gradient method (e.g., Fletcher-
Reeves), and quasi-newton method (e.g., BFGS and L-BFGS).
However, the number of studies discussing the performances
of various optimization methods in the meteorological and
oceanographic application is still relatively small []. Line
search method requires repeated computations for the cost
function and the gradient. erefore it spends too much com-
putation resource during numerical simulations especially for
physical oceanography. Besides, line search methods may fail
in some cases []. e L-BFGS method is commonly used to
solve large-scale problems in oceanography and meteorology
[,,]. Chen et al. [] have made a computational
investigation on the performances of the L-BFGS method and
two versions of gradient descent method with an internal tide
model. In their work, a simplied gradient descent method
is applied which is able to avoid too much computation.
e step length is chosen according to the experience of the
modeler. Compared with other methods, there are two main
advantages of this plan. One is the less computation resource
usage and the other is the more controllable optimization
process. Many research papers have proved the feasibility of
this method [–,,].
Generally speaking, numerical methods to solve the
minimization problems have the similar iterative formula as
follows:
𝑛+1
V=𝑛
V+𝑛𝑛,()
where 𝑛
Vand 𝑛+1
Vare the priori and adjusted values of
VEVC in the th iteration, respectively and 𝑛and 𝑛
represent the iteration step length and the search direction,
respectively. ere are many methods to determine the search
direction 𝑛. Two dierent optimization methods employed
in this paper are the GDM-S and the L-BFGS methods.
3.1. Gradient Descent Method (GDM). e GDM is a simple
and feasible method to dene the search direction as follows:
𝑛=−𝑛=−/V𝑖,𝑗,𝑘𝑛




/V𝑖,𝑗,𝑘𝑛



,()
where ,,andare the zonal index, the meridional index,
and the layer index of the calculation grid, respectively; 
represents the step of iteration. (/V𝑖,𝑗,𝑘)𝑛is the 2norm
of the gradient of the cost function with respect to the VEVC
in the th iteration.
In the GDM-S, the step length is chosen to be a constant
according to the experience of the modeler. We have surveyed
the performances of dierent values of andtake.asthe
best choice. e optimized value of VEVC is obtained aer
the VEVC in every grid (,,) is adjusted according to ()
and ().
3.2. L-BFGS Method. L-BFGS is an optimization algorithm
in the family of quasi-Newton methods that approximates the
BFGS algorithm using a limited amount of computer mem-
ory. is method is rst described in the work of Nocedal
[],whereitiscalledtheSQNmethod.Itisapopular
algorithm for parameter estimation in machine learning [,
]. Due to its resulting linear memory requirement, the
L-BFGS method is particularly well suited for optimization
problems with a large number of variables.
It requires the search direction to be
𝑘=−𝑘𝑘,()

Mathematical Problems in Engineering 
where
𝑘+1 =
𝑇
𝑘−1 ⋅⋅⋅𝑇
𝑘−𝑚0𝑘−𝑚 ⋅⋅⋅𝑘−1
+𝑘−𝑚 𝑇
𝑘⋅⋅⋅𝑇
𝑘−𝑚+1𝑘−𝑚𝑇
𝑘−𝑚 𝑘−𝑚+1 ⋅⋅⋅𝑘
+𝑘−𝑚 𝑇
𝑘⋅⋅⋅𝑇
𝑘−𝑚+1𝑘−𝑚𝑇
𝑘−𝑚 𝑘−𝑚+1 ⋅⋅⋅𝑘
+𝑘−𝑚 𝑇
𝑘⋅⋅⋅𝑇
𝑘−𝑚+1𝑘−𝑚𝑇
𝑘−𝑚 𝑘−𝑚+1 ⋅⋅⋅𝑘
...
+𝑘𝑘𝑇
𝑘.()
Note that 𝑘is the simplication of ∇(𝑘)for conve-
nience. Here, 𝑘=1/𝑘𝑇
𝑘,𝑘=1−𝑘𝑘𝑇
𝑘,𝑘=𝑘+1 −𝑘=
𝑘𝑘,and𝑘=
𝑘+1 −
𝑘. Many studies have shown that
typically 3≤≤7,where>7does not improve the
performanceofL-BFGS[]. So the number of corrections
m used in the L-BFGS update of this paper is taken as  [].
e version of L-BFGS used in this paper is described in Liu
and Nocedal [] and the Fortran codes are authorized by
Nocedal [].
4. Design of Experiments
All the experiments in this paper are implemented in an ideal
regional area from ∘Eto.
∘Eandfrom
∘N to .∘N
with in mind the practical sea area located around the Luzon
strait. e horizontal resolution is 10󸀠×10󸀠and there are
totally 49×31grids in the area. e maximum depth is set
to be  meters. e horizontal eddy coecient is chosen
to be ℎ=1000and the bottom friction coecient is taken
as  = 0.003. e Coriolis coecient is taken as the local
value. Only 2tide is considered and its angular frequency is
1.41×10−4 s−1. e whole-time step is . s which is /
of the period of 2tide. All the four boundaries are open
boundaries and boundary conditions are set to be the local
water levels in the Flather form.
Eastern and western boundaries:
−
󸀠=±1− 2
2
−
, ()
positive in eastern boundary and negative in western bound-
ary.
North and south boundaries:
−
󸀠=±1− 2
2
−
, ()
positive in north boundary and negative in south boundary.
is the surface elevation above the undisturbed sea level.
and are the zonal current velocity and the meridional
current velocity, respectively. 
󸀠,
,and
are the surface
elevation and the zonal and meridional current velocity
relating to the boundary barotropic tidal force, respectively.
is the Coriolis coecient and is the tidal frequency of 2
tide. is the local water depth and is the acceleration of
gravity.
Similar as Chen et al. [], the 2tidal force at the th
time step is subject to

󸀠2 =𝜁cos ()+𝜁sin (),()
the Fourier coecients 𝜁at four open boundaries are set to
be  and 𝜁are set to be  (−) at the north and west (south
and east) boundaries.
e T/P altimeter data is widely spread throughout the
oceananditcanbeusedtoinvertVEVC.Inthiswork,wepick
 calculating points based on the distribution features of T/P
altimeter observation as the observation points (Figure ).
Two kinds of topographies are tested in this paper and
they are generated based on the two formulas in (),
respectively (Figure ). e sea water in the computing area
is divided into two layers. e thicknesses of each layers are,
respectively, 1= 200mand2= 800m. e potential
densities of corresponding layer are 1=1021(kg/m−3)and
2=1024(kg/m−3), respectively. Consider
𝐴=0exp −−02+−02
22
−MaxDepth,
𝐵=01−exp −−02+−02
22
−MaxDepth,
()
where 0istheheightofthetopographyandMaxDepthisthe
depth of the water.
For each experiment, the optimization of the VEVC can
be implemented with the following steps.
Step 1. e prescribed VEVC is given and the forward model
is run. e whole simulation time is  period of the 2
tide in order to obtain a stable simulation result. e water
elevations in the observation positions are treated as the
“pseudoobservations.”
Step 2. Initial value of the control parameter (VEVC) is given
and forward model is run to get the simulated results of all the
state variables such as current velocity and water elevation.
e value of cost function is calculated.
Step 3. Dierence between the simulated elevation and the
“pseudoobservation” plays as the external force of the adjoint
model. Via backward integrating the adjoint equations in
aperiodofthe2tide values of the adjoint variables are
obtained.
Step 4. Using formula (), along with the state variables and
the adjoint variables obtained in Steps and ,thegradient
of cost function with respect to VEVC is calculated.

Mathematical Problems in Engineering
116.5 117 117.5 118 118.5 119 119.5 120 120.5
18.5
19
19.5
20
20.5
21
−1000
−950
−900
−850
−800
−750
−700
−650
−600
−550
−500
Latitude (∘N)
Longitude (∘E)
F : Planform of topography (e.g., topography A) and loca-
tions of the observations (white dots).
Step 5. Update the unknown control variables with a certain
optimization method.
Step 6. If the stopping criterion of iteration is reached, bring
the iteration to an end and return the optimized parameter.
Otherwise, update all the parameters and go back to Step .
In the experiments of this paper, all initial values of VEVC
are set to . and the total number of iterations is allowed
to be  at most. e chosen convergence criterion is that
the last two values of the cost function are suciently close,
which is dened by end −end−1<10−9,()
where end and end−1 are the last and the second last values of
the cost function, respectively.
Two groups of numerical experiments are carried out:
theinuenceofIPSsontheinversionofVEVCisstudied
inGroupOne;inGroupTwotheabilityofthisinternal
tide model to invert dierent kinds of VEVC with spatial
distribution is examined. Two kinds of spatial distribution
ofVEVCareprescribedandgiveninFigure .Forboth
distributions, the VEVC value ranges from 3×10−3 m2/s to
7×10−3 m2/s.
In Group One, nine experiments are carried out to discuss
the inuence of IPS on the inversion of VEVC. e distance
between independent points (IP) ranges from 󸀠(length of
grids)to
󸀠(length of  grids) and details are listed
in Table . Topography A is applied and distribution (a) is
chosen to be the prescribed spatial distribution of VEVC.
In Group Two, four numerical experiments (NEs) are
carriedoutwhicharenumberedasNE∼NE, respectively.
Each experiment is implemented with GDM-S and the L-
BFGS methods. Information of topographies and prescribed
VEVCs for all NEs is listed in Table .eIPSsarethe
optimal schemes obtained in Group One. Other parameters
are exactly the same as Group One.
All the experiments in Group One and Group Two are
carried out following Steps to described in Section .
VersionofL-BFGSmethodusedinthispaperisfromthe
work of Liu and Nocedal [].
T : Settings of independent point schemes in Group One.
IPS         
NumberofIPs  
Distance between IPs (󸀠)  
T : Topographies and distributions of VEVC in Group Two.
Experiment Topography Distribution
NE A a
NE A b
NE B a
NE B b
5. Experiment Results and Discussions
5.1. Results and Discussions of Group One. Figure  illustrates
therelationshipsbetweenmeanabsoluteerrors(MAE,which
reects the error between the inversion result and the given
VEVC) and the distance between independent points in
Group One.
As is shown in Figure , MAEs with dierent optimiza-
tion methods vary as the IPSs change. e minimum MAEs
with the two methods are not the same (dashed lines). e
minimum MAE with the GDM-S is 2.66× 10−4 m2/s while
that with the L-BFGS method reaches 9.14×10−5 m2/s. e
corresponding IP distances are 󸀠(length of  grids, GDM-
S) and 󸀠(lengthofgrids,L-BFGS),respectively.According
to the MAEs illustrated in Figure , the optimal IPSs of the
two methods are selected as IPS (GDM-S) and IPS  (L-
BFGS).
5.2. Results and Discussions of Group Two. With the respec-
tive optimal IPSs and the iteration process in Section ,the
VEVCs are inverted with GDM-S method (I) and L-BFGS
method (II), respectively, and the inversion results are shown
in Figure .
Comparison of the inversion results with the prescribed
VEVCs indicates that all the given spatial distributions of
VEVC are successfully inverted aer  iteration steps. e
main features of all distributions can be recovered very well.
Surfaces of the inverted VEVC with the L-BFGS are much
smoother than those with the GDM-S. Compared against
the inversion results with the GDM-S (le panels), patterns
with the L-BFGS method (right panels) are closer to the
prescribed VEVC. More statistic data will be presented in the
next paragraphs.
MAEs of the four numerical experiments aer assimila-
tion are calculated and listed in Table .einitialvaluesof
MAE in all NEs are 1×10−3 m2/s. Based on the results shown
in Table , all the MAEs aer assimilation are more than one
order of magnitude lower than the initial values, which means
the success for both of the two optimization methods. Note
that the MAEs with the L-BFGS method are quite close (less
than 1×10−8 m2/s and cannot be distinguished in the table)
and are less than half of those with the GDM-S. is result
demonstrates the ability of the L-BFGS method to deduce the
overall errors.

Mathematical Problems in Engineering 
10
20 30
40
10
20
30
−1000
−800
−600
−400
−200
0
Zonal index
Meridional index
A
(a)
10
B
20
30 40
10
20
30
−1000
−800
−600
−400
−200
0
Zonal index
Meridional index
10
2
0
30
40
10
20
Zon
al
index
idi
on
al
(b)
F : Topographies A and B. Note that the abscissa and ordinate axes are labeled with zonal index and meridional index, respectively.
116.5 117 117.5 118 118.5 119 119.5 120 120.5
18.5
19
19.5
20
20.5
21
3
3.5
4
4.5
5
5.5
6
6.5
7
×10−3
Latitude (∘N)
Longitude (∘E)
(a)
116.5 117 117.5 118 118.5 119 119.5 120 120.5
18.5
19
19.5
20
20.5
21
3
3.5
4
4.5
5
5.5
6
6.5
7
×10−3
Latitude (∘N)
Longitude (∘E)
(b)
F : Planform of two prescribed spatial distributions of VEVC.
T : Inversion errors of VEVC in Group Two (unit: m/s).
Method Experiment
NE NE NE NE
GDM-S(I) 2.42−04 2.43−04 2.23−04 2.24−04
L-BFGS(II) 9.14−05 9.14−05 9.14−05 9.14−05
To compare the eectiveness of the two methods to invert
the VEVC, we make statistics on the percentages of the grids
20 30 40 50 60 70 80 90 100
0
0.5
1
1.5
2
2.5
3
×10−3
MAE (m2/s)
IP distance
GDM-S
L-BFGS
F : MAEs versus IP distance in Group One. e abscissas
indicates distance between adjacent IPs (unit: 󸀠) while the ordinate
indicates MAE of inversion results. e solid lines are values of
dierent experiments and the dashed lines indicate the minimum
values of two solid lines, respectively.
at which the MAEs are less than 1×10
−4 m2/s, which is
listed in Tabl e . With the GDM-S, the inversion errors are
deduced by one order of magnitude at about % of the
total grids. By contrast, ratios of all NEs with the L-BFGS
method are .%, without dierences between NEs. is
phenomenon indicates that the L-BFGS method is eective at
more computation grids than the GDM-S. Furthermore, the
L-BFGS method maintains its eectiveness no matter which
topography is applied.
Combining the inversion patterns, the inversion errors of
the VEVC, and the eectiveness analyses, conclusions can be
drawn that the L-BFGS has a better performance in reducing
the inversion errors.
Finally we come to the optimization history for all the
experiments carried out in Group Two. e variations of the
cost function normalized by its initial value, that of the 2
norm of the gradient of the cost function with respect to the

Mathematical Problems in Engineering
117 118 119 120
19
20
21
Latitude (∘N)
Longitude (∘E)
(a) NE(I)
117 118 119 120
19
20
21
Latitude (∘N)
Longitude (∘E)
(b) NE(II)
117 118 119 120
19
20
21
Latitude (∘N)
Longitude (∘E)
(c) NE(I)
117 118 119 120
19
20
21
Latitude (∘N)
Longitude (∘E)
(d) NE(II)
117 118 119 120
19
20
21
Latitude (∘N)
Longitude (∘E)
(e) NE(I)
117 118 119 120
19
20
21
Latitude (∘N)
Longitude (∘E)
(f) NE(II)
117 118 119 120
19
20
21
Latitude (∘N)
Longitude (∘E)
(g) NE(I)
117 118 119 120
19
20
21
Latitude (∘N)
Longitude (∘E)
(h) NE(II)
F : Planform of inversion results in Group Two.

Mathematical Problems in Engineering 
T : Eectiveness analyses of inversions in Group Two.
Method Experiment
NE NE NE NE
GDM-S(I) .% .% .% .%
L-BFGS(II) .% .% .% .%
VEVCs and that of the inversion error, are plotted in Figures
(a),(b),and(c), respectively, as a function of the iteration
step.
Note that all experiments with the L-BFGS method reach
the convergence criterion and stop aer  iterations, which
indicates that the computation time for the L-BFGS method is
one twenty-h of that for the GDM-S. Figure (a) indicates
that all the cost functions are in downward trends throughout
the iteration process and decrease by more than  () orders of
magnitude for the GDM-S (L-BFGS method), which means
that the nal dierences between simulation value and the
observation of these two methods are less than one-tenth
and one-thousandth of their initial values, respectively. As is
shown in Figure (b),the2norms of gradient of the cost
function with respect to the VEVC decrease by more than
 order of magnitude (GDM-S) and  orders of magnitude
(L-BFGS), compared with their respective initial values. is
indicates that the inversion result is becoming increasingly
closer to the given VEVC during the iteration. As shown
in Figure (c), the inversion errors with the two methods
keep declining throughout the iterations until the stopping
criterions are satised. In general, the cost functions, norms
of gradient, and the inversion errors of the VEVC have
steady descent, which demonstrates that this adjoint model
is capable to invert the VEVC. What is more, both the
GDM-SandtheL-BFGSmethodsareeectiveinterms
of the inversion of the control parameters with spatially
distributions of internal tide.
It is also clear in Figure  that the convergence rate for the
cost functions, norms of gradient, and the inversion errors
of the VEVC is much faster with the L-BFGS method than
those with the GDM-S, which is consistent with the classic
theories about the convergence rate of the quasi-Newton
method and the GDM []. is trend is also consistent
with the results of numerical experiments to invert the open
boundary conditions in Chen et al. []. With no doubt, the
L-BFGS method is a more eective and ecient optimization
method to invert the spatially varying VEVC. However, the
GDM-S is easier to understand and to implement in the
model. Moreover, the step length and the search direction
in the process of GDM-S can be freely controlled by the
modelers, which is very convenient in practice. erefore,
for the inversion of the VEVC, the GDM-S should also be
regarded as a choice.
6. Summary and Conclusions
Based on an isopycnic-coordinate internal tidal model, the
inversion of VEVC is studied in this paper. A series of
numerical experiments are carried out to examine the inu-
ence factors on the inversion of VEVCs in four aspects:
independent point schemes (IPS), topography, the spatial
distribution of VEVC, and the optimization methods. For
each experiment, the cost function, the 2norm of gradient
of cost function with respect to the VEVC, and the inversion
errorarecalculatedandanalyzedindetails.
e IPS is introduced and discussed in Group One. All
the VEVCs can be inverted successfully with IPS. MAE is
regarded as the comparison criterion of the result. Aer
comparing the  experiments, the correctness of the IPS is
conrmed and the optimal IPSs are selected for the GDM-S
and the L-BFGS methods, respectively.
BasedontheoptimalIPSsinGroupOne,twokindsof
VEVC distributions are successfully inverted with this adjoint
model on two kinds of topography in Group Two. MAEs aer
optimization are at the level of −4 (−5)fortheGDM-S
(L-BFGS), which is one (two) order(s) of magnitude lower
than the initial value. All the cost functions and their gradient
norms with respect to the VEV lead satisfactory declines no
matter which optimization method is taken. Compared with
the GDM-S, the L-BFGS method has a remarkably better
performance, not only in terms of the convergence rate but
also in terms of the nal inversion results. e computation
time for the L-BFGS method is much shorter than that for
theGDM-S.Tosumup,theL-BFGSmethodisamore
eective and ecient method than the GDM-S in terms of
the inversion of the VEVC. Nevertheless, the GDM-S is more
convenientandcontrollablesoitshouldnotbeignoredand
should be taken seriously as a choice for the inversion of the
VEVC with spatially distribution.
e success of numerical experiments lays a solid founda-
tion for the practical experiments and encourages us to carry
out experiments in practical sea area with measured data and
the real T/P altimeter data.
Appendix
Derivation of ()
Let us start with the governing equations in Chen et al. [].
Layer  (surface layer)
1
 +1
cos 11
 +1
cos 1V1cos 
 =0,
(A.a)
1
 +1
cos 1
 +V1
1
 −1V1tan 
−V1−ℎ11
+1𝜆
1+
cos 
 𝑙

𝑚=1 𝑚
𝑚−𝑚+1
𝑘
=0,
(A.b)
V1
 +1
cos V1
 +V1
V1
 +2
1tan 
−1−ℎ1V1
+1𝜑
1+

 𝑙

𝑚=1 𝑚
𝑚−𝑚+1
𝑘
=0.
(A.c)

 Mathematical Problems in Engineering
−10
−5
0
log10(J/J0)
10 20 30 40 50 60 70 80 90 100
Iteration steps
(a)
10 20 30 40 50 60 70 80 90 100
Iteration steps
−4
−2
0
log10(G/G0)
(b)
NE1(I)
NE2(I)
NE3(I)
NE4(I)
NE1(II)
NE2(II)
NE3(II)
NE4(II)
10 20 30 40 50 60 70 80 90 100
Iteration steps
−4
−3.5
−3
log10(MAE)
(c)
F : Optimization history for experiments of Group Two, about (a) the cost function normalized by its initial value 0,(b)the2norm
of gradient of the cost function with respect to the VEVC, and (c) the MAEs between the inverted and prescribed VEVCs.
Layer (=2,...,−1)
𝑘
 +1
cos 𝑘𝑘
 +1
cos 𝑘V𝑘cos 
 =0,
(A.a)
𝑘
 +𝑘
cos 𝑘
 +V𝑘
𝑘
 −𝑘V𝑘tan 
−V𝑘−ℎ𝑘𝑘
−(𝑘−1)𝜆−𝑘𝜆
𝑘+
cos 
×𝑘−1

𝑚=1 1
𝑘−1
𝑚𝑚

+
 𝑙

𝑚=1 𝑚
𝑚−𝑚+1
𝑘
=0,
(A.b)
V𝑘
 +𝑘
cos V𝑘
 +V𝑘
V𝑘
 +2
𝑘tan 
+𝑘
−ℎ𝑘V𝑘−(𝑘−1)𝜑−𝑘𝜑
𝑘+

×𝑘−1

𝑚=1 1
𝑘−1
𝑚𝑚

+
 𝑙

𝑚=1 𝑚
𝑚−𝑚+1
𝑘
=0.
(A.c)
Layer (bottom layer)
𝑙
 +1
cos 𝑙𝑙
 +1
cos 𝑙V𝑙cos 
 =0, (A.a)
𝑙
 +𝑙
cos 𝑙
 +V𝑙
𝑙
 −𝑙V𝑙tan 
−V𝑙
−ℎ𝑙𝑙−(𝑙−1)𝜆−𝑏𝜆
𝑙+
cos 
×
 𝑙

𝑚=1 𝑚
𝑙−𝑚+1
𝑙
=0,
(A.b)
V𝑙
 +𝑙
cos V𝑙
 +V𝑙
V𝑙
 +2
𝑙tan 
+𝑙
−ℎ𝑙V𝑙−(𝑙−1)𝜑−𝑏𝜑
𝑙+

×
 𝑙

𝑚=1 𝑚
𝑙−𝑚+1
𝑙
=0.
(A.c)
e variables and background of the governing equations
have been introduced in Chen’s [] work in details. We will
not repeat them in this part. e interface and friction terms
are expressed by
𝑘𝜆,𝑘𝜑=V𝑘𝑘+1/2
𝑘+1/2 𝑘−𝑘+1,V𝑘−V𝑘+1,
=1,...,−1, (A.)
where Vis the vertical eddy viscosity coecient, 𝑘+1/2 =
(𝑘+𝑘+1)/2,and𝑘+1/2 =(𝑘+𝑘+1)/2.

Mathematical Problems in Engineering 
e cost function is dened as
,,V;p
=1
2

𝜍
𝑙

𝑘=1𝑙

𝑚=𝑘 𝑚
𝑚−𝑚
𝑘2
+𝑢
𝑙

𝑘=1𝑘−
𝑘2+V
𝑙

𝑘=1V𝑘−
V𝑘2

,
(A.)
which is exactly the same as that in []. en the Lagrangian
function is dened as
,,V;𝑎,𝑎,V𝑎;p
=,,V;p
+𝑎1 ⋅(A.1a)+𝑎11⋅(A.1b)+V𝑎11⋅(A.1c)
+⋅⋅⋅
+𝑎𝑘 ⋅(A.2a)+𝑎𝑘𝑘⋅(A.2b)+V𝑎𝑘𝑘⋅(A.2c)
+⋅⋅⋅
+𝑎𝑙 ⋅(A.3a)+𝑎𝑙𝑙⋅(A.3b)+V𝑎𝑙𝑙⋅(A.3c),
(A.)
where
(A.1a)
=−1
 +1
cos 11
 +1
cos 1V1cos 
 ,
(A.)
(A.1b)=−1
 +1
cos 1
 +V1
1
 −1V1tan 

−V1−ℎ11+1𝜆
1+
cos 
×
 𝑙

𝑚=1 𝑚
𝑚−𝑚+1
𝑘
,
(A.)
(A.1c)=−V1
 +1
cos V1
 +V1
V1
 +2
1tan 

−1−ℎ1V1+1𝜑
1+

×
 𝑙

𝑚=1 𝑚
𝑚−𝑚+1
𝑘
,
(A.)
samedenitionsareappliedin(A.a)∼(A.c)and(A.a)∼
(A.c). en the Lagrangian function (,,V;𝑎,𝑎,V𝑎;p)
can be written as
,,V;𝑎,𝑎,V𝑎;p
=,,V;p
−𝑎1V𝑖+1/2,𝑗,11+1/2
𝑖+1/2,𝑗,1+1/2 1−2
+𝑎1V𝑖+1/2,𝑗,11+1/2
𝑖+1/2,𝑗,1+1/2 V1−V2−1
+⋅⋅⋅
−𝑎𝑘V𝑖+1/2,𝑗,𝑘𝑘+1/2
𝑖+1/2,𝑗,𝑘+1/2 𝑘−𝑘+1
−𝑎𝑘V𝑖+1/2,𝑗,𝑘−1𝑘−1/2
𝑖+1/2,𝑗,𝑘−1/2 𝑘−1 −𝑘
−V𝑎𝑘V𝑖+1/2,𝑗,𝑘𝑘+1/2
𝑖+1/2,𝑗,𝑘+1/2 V𝑘−V𝑘+1
−V𝑎𝑘V𝑖+1/2,𝑗,𝑘−1𝑘−1/2
𝑖+1/2,𝑗,𝑘−1/2 V𝑘−1 −V𝑘−𝑘
+⋅⋅⋅
+𝑎𝑙V𝑖+1/2,𝑗,𝑙−1𝑙−1/2
𝑖+1/2,𝑗,𝑙−1/2 𝑙−1 −𝑙
+𝑎𝑙V𝑖+1/2,𝑗,𝑙−1𝑙−1/2
𝑖+1/2,𝑗,𝑙−1/2 V𝑙−1 −V𝑙−𝑙,
(A.)
where V𝑖+1/2,𝑗,𝑘 =(V𝑖,𝑗,𝑘V𝑖+1,𝑗,𝑘)/2,𝑖+1/2,𝑗,𝑘+1/2 =(
𝑖,𝑗,𝑘 +
𝑖+1,𝑗,𝑘 +𝑖,𝑗,𝑘+1 +𝑖+1,𝑗,𝑘+1)/4, and functions 1,𝑘,and𝑙are,
respectively, dened as
1=𝑎1 ⋅(A.1a)+𝑎11⋅(A.1b)+1𝜆
1
+V𝑎11⋅(A.1c)+1𝜑
1, (A.)

 Mathematical Problems in Engineering
𝑘=𝑎𝑘 ⋅(A.2a)+𝑎𝑘𝑘⋅(A.2b)−(𝑘−1)𝜆 −𝑘𝜆
1
+V𝑎𝑘𝑘⋅(A.2c)−(𝑘−1)𝜑 −𝑘𝜑
𝑘, (A.)
𝑙=𝑎𝑙 ⋅(A.3a)+𝑎𝑙𝑙⋅(A.3b)−(𝑙−1)𝜆
𝑙
+V𝑎𝑙𝑙⋅(A.3c)−(𝑙−1)𝜑
𝑙. (A.)
Note that (A.)∼(A.)donotcontainthevariableV,which
means 1
V𝑖,𝑗,𝑘 =𝑘
V𝑖,𝑗,𝑘 =𝑙
V𝑖,𝑗,𝑘 =0. (A.)
e Lagrangian function can be written as
,,V;𝑎,𝑎,V𝑎;p
=,,V;p
−𝑎1,𝑗,1V𝑖+1/2,𝑗,11+1/2
𝑖+1/2,𝑗,1+1/2 𝑖,𝑗,1 −𝑖,𝑗,2
+𝑎1,𝑗,1V𝑖+1/2,𝑗,11+1/2
𝑖+1/2,𝑗,1+1/2 V𝑖,𝑗,1 −V𝑖,𝑗,2−1
+⋅⋅⋅
−𝑎𝑖,𝑗,𝑘V𝑖+1/2,𝑗,𝑘𝑘+1/2
𝑖+1/2,𝑗,𝑘+1/2 𝑖,𝑗,𝑘 −𝑖,𝑗,𝑘+1
−𝑎𝑖,𝑗,𝑘V𝑖+1/2,𝑗,𝑘−1𝑘−1/2
𝑖+1/2,𝑗,𝑘−1/2 𝑖,𝑗,𝑘−1 −𝑖,𝑗,𝑘
−V𝑎𝑖,𝑗,𝑘V𝑖+1/2,𝑗,𝑘𝑘+1/2
𝑖+1/2,𝑗,𝑘+1/2 V𝑖,𝑗,𝑘 −V𝑖,𝑗,𝑘+1
−V𝑎𝑘V𝑖+1/2,𝑗,𝑘−1𝑘−1/2
𝑖+1/2,𝑗,𝑘−1/2 V𝑖,𝑗,𝑘−1 −V𝑖,𝑗,𝑘−𝑘
+⋅⋅⋅
+𝑎𝑖,𝑗,𝑙V𝑖+1/2,𝑗,𝑙−1𝑙−1/2
𝑖+1/2,𝑗,𝑙−1/2 𝑖,𝑗,𝑙−1 −𝑖,𝑗,𝑙
+𝑎𝑖,𝑗,𝑙V𝑖+1/2,𝑗,𝑙−1𝑙−1/2
𝑖+1/2,𝑗,𝑙−1/2 V𝑖,𝑗,𝑙−1 −V𝑖,𝑗,𝑙−𝑙.
(A.)
Finally, according to the typical theory of Lagrangian
multiplier method, we have the following rst-order derivate
of Lagrangian function with respect to the control parameter
V:

V𝑖,𝑗,𝑘 =0, (A.)
then the gradient of the cost function with respect to the
variable Vcanbededucedfrom(A.):

V𝑖,𝑗,𝑘
=
𝑛𝑛
𝑎𝑖,𝑗,𝑘 𝑘+1/2
𝑖+1/2,𝑗,𝑘+1/2 𝑛
𝑖,𝑗,𝑘 −𝑛
𝑖,𝑗,𝑘+1
+𝑛
𝑎𝑖−1,𝑗,𝑘 𝑘+1/2
𝑖−1/2,𝑗,𝑘+1/2 𝑛
𝑖−1,𝑗,𝑘 −𝑛
𝑖−1,𝑗,𝑘+1
−
𝑛𝑛
𝑎𝑖,𝑗,𝑘+1 𝑘+1/2
𝑖+1/2,𝑗,𝑘+1/2 𝑛
𝑖,𝑗,𝑘 −𝑛
𝑖,𝑗,𝑘+1
+𝑛
𝑎𝑖−1,𝑗,𝑘+1 𝑘+1/2
𝑖−1/2,𝑗,𝑘+1/2 𝑛
𝑖−1,𝑗,𝑘 −𝑛
𝑖−1,𝑗,𝑘+1
+
𝑛V𝑛
𝑎𝑖,𝑗,𝑘 𝑘+1/2
𝑖+1/2,𝑗,𝑘+1/2 V𝑛
𝑖,𝑗,𝑘 −V𝑛
𝑖,𝑗,𝑘+1
+V𝑛
𝑎𝑖−1,𝑗,𝑘 𝑘+1/2
𝑖−1/2,𝑗,𝑘+1/2 V𝑛
𝑖−1,𝑗,𝑘 −V𝑛
𝑖−1,𝑗,𝑘+1
−
𝑛V𝑛
𝑎𝑖,𝑗,𝑘+1 𝑘+1/2
𝑖+1/2,𝑗,𝑘+1/2 V𝑛
𝑖,𝑗,𝑘 −V𝑛
𝑖,𝑗,𝑘+1
+V𝑛
𝑎𝑖−1,𝑗,𝑘+1 𝑘+1/2
𝑖−1/2,𝑗,𝑘+1/2 V𝑛
𝑖−1,𝑗,𝑘 −V𝑛
𝑖−1,𝑗,𝑘+1
=𝑘+𝑘+1
×
𝑛
𝑛
𝑖,𝑗,𝑘 −𝑛
𝑖,𝑗,𝑘+1𝑛
𝑎𝑖,𝑗,𝑘 −𝑛
𝑎𝑖,𝑗,𝑘+1
𝑖,𝑗,𝑘 +𝑖+1,𝑗,𝑘 +𝑖,𝑗,𝑘+1 +𝑖+1,𝑗,𝑘+1
+𝑛
𝑖−1,𝑗,𝑘 −𝑛
𝑖−1,𝑗,𝑘+1𝑛
𝑎𝑖−1,𝑗,𝑘 −𝑛
𝑎𝑖−1,𝑗,𝑘+1
𝑖−1,𝑗,𝑘 +𝑖,𝑗,𝑘 +𝑖−1,𝑗,𝑘+1 +𝑖,𝑗,𝑘+1 

+𝑘+𝑘+1
×
𝑛
V𝑛
𝑖,𝑗,𝑘 −V𝑛
𝑖,𝑗,𝑘+1V𝑛
𝑎𝑖,𝑗,𝑘 −V𝑛
𝑎𝑖,𝑗,𝑘+1
𝑖,𝑗,𝑘 +𝑖+1,𝑗,𝑘 +𝑖,𝑗,𝑘+1 +𝑖+1,𝑗,𝑘+1
+V𝑛
𝑖−1,𝑗,𝑘 −V𝑛
𝑖−1,𝑗,𝑘+1V𝑛
𝑎𝑖−1,𝑗,𝑘 −V𝑛
𝑎𝑖−1,𝑗,𝑘+1
𝑖−1,𝑗,𝑘 +𝑖,𝑗,𝑘 +𝑖−1,𝑗,𝑘+1 +𝑖,𝑗,𝑘+1 
.
(A.)
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.

Mathematical Problems in Engineering 
Acknowledgments
Partial support for this research was provided by the
National Natural Science Foundation of China through Grant
, the State Ministry of Science and Technology of
China through Grant AA, and the Fundamental
Research Funds for the Central Universities  and
.
References
[] M. Rattray, “On the coastal generation of internal tides,” Tellus,
vol.,no.,pp.–,.
[] P. G. Baines, “e generation of internal tides by at-bump
topography,” Deep-Sea Research and Oceanographic Abstracts,
vol.,no.,pp.–,.
[] T. H. Bell, “Topographically generated internal waves in the
open ocean,” Journal of Geophysical Research,vol.,pp.–
, .
[] P. G. Baines, “On internal tide generation models,” Deep Sea
Research A: Oceanographic Research Papers,vol.,no.,pp.
–, .
[] P. D. Craig, “Solutions for internal tidal generation over coastal
topography,” Journal of Marine Research, vol. , pp. –,
.
[] T. Gerkema, “A unied model for the generation and ssion of
internal tides in a rotating ocean,” Journal of Marine Research,
vol.,no.,pp.–,.
[] S. G. Llewellyn Smith and W. R. Young, “Conversion of the
barotropic tide,” JournalofPhysicalOceanography,vol.,no.
, pp. –, .
[] S. K. Kang, M. G. G. Foreman, W. R. Crawford, and J. Y.
Cherniawsky, “Numerical modeling of internal tide generation
along the Hawaiian Ridge,” JournalofPhysicalOceanography,
vol. , no. , pp. –, .
[] Y. Niwa and T. Hibiya, “Numerical study of the spatial distri-
bution of the M2internal tide in the Pacic Ocean,” Journal
of Geophysical Research C: Oceans,vol.,no.,pp.–
, .
[] P. F. Cummins, J. Y. Cherniawsky, and M. G. G. Foreman,
“North Pacic internal tides from the Aleutian ridge: altimeter
observations and modeling,” Journal of Marine Research,vol.,
no. , pp. –, .
[] Y. Niwa and T. Hibiya, “ree-dimensional numerical sim-
ulation of M2internal tides in the East China Sea,” Journal
of Geophysical Research C: Oceans,vol.,no.,ArticleID
C, .
[] S.Jan,C.-S.Chern,J.Wang,andS.-Y.Chao,“Generationof
diurnal K internal tide in the Luzon Strait and its inuence
on surface tide in the South China Sea,” Journal of Geophysical
Research C: Oceans,vol.,no.,ArticleIDC,.
[] R. C. Pacanowski and S. G. H. Philander, “Parameterization of
vertical mixing in numerical models of tropical oceans,” Journal
of Physical Oceanography, vol. , no. , pp. –, .
[] B. Henderson-Sellers, “A simple formula for vertical eddy
diusion coecients under conditions of nonneutral stability,”
Journal of Geophysical Research,vol.,pp.–,.
[] N. S. Heaps, “ree-dimensional model for tides and surges
with vertical eddy viscosity prescribed in two layers—I. Math-
ematical formulation,” Geophysical Journal of the Royal Astro-
nomical Society,vol.,pp.–,.
[] N. S. Heapsand and J. E. Jones, “ree-dimensional model for
tides and surges with vertical eddy viscosity prescribed in two
layers—II.IrishSeawithbedfrictionlayer,”Geophysical Journal
of the Royal Astronomical Society,vol.,pp.–,.
[] V. P. Kochergin, “ree-dimensional prognostic models. In:
three-dimensional coastal ocean models,” Coastal and Estuarine
Science, vol. , pp. –, .
[] T. Pohlmann, “Calculating the annual cycle of the vertical eddy
viscosity in the North Sea with a three-dimensional baroclinic
shelf sea circulation model,” Continental Shelf Research,vol.,
no. , pp. –, .
[] A. F. Bennett and P. C. McIntosh, “Open ocean modeling as an
inverse problem: tidal theory,” JournalofPhysicalOceanography,
vol. , no. , pp. –, .
[] L. Yu and J. J. O’Brien, “Variational estimation of the wind stress
drag coecient and the oceanic eddy viscosity prole,” Journal
of Physical Oceanography,vol.,pp.–,.
[] R. W. Lardner, “Optimal control of open boundary conditions
for a numerical tidal model,” Computer Methods in Applied
Mechanics and Engineering,vol.,no.,pp.–,.
[] U. Seiler, “Estimation of open boundary conditions with the
adjoint method,” Journal of Geophysical Research, vol. , no. ,
pp. –, .
[] I. M. Navon, “Practical and theoretical aspects of adjoint
parameter estimation and identiability in meteorology and
oceanography,” Dynamics of Atmospheres and Oceans,vol.,
no. –, pp. –, .
[] N. Ayoub, “Estimation of boundary values in a North Atlantic
circulation model using an adjoint method,” Ocean Modelling,
vol.,no.-,pp.–,.
[] J. Zhang and X. Lu, “Inversion of three-dimensional tidal
currents in marginal seas by assimilating satellite altimetry,”
Computer Methods in Applied Mechanics and Engineering,vol.
, no. –, pp. –, .
[] J. C. Zhang and X. Q. Lu, “Parameter estimation for a three-
dimensional numerical barotropic tidal model with adjoint
method,” International Journal for Numerical Methods in Fluids,
vol.,no.,pp.–,.
[] H. Chen, C. Miao, and X. Lv, “A three-dimensional numerical
internal tidal model involving adjoint method,” International
Journal for Numerical Methods in Flui ds,vol.,no.,pp.–
, .
[] J. Zhang and H. Chen, “Semi-idealized study on estimation of
partly and fully space varying open boundary conditions for
tidal models,” Abstrac t and Applied Analysi s,vol.,Article
ID,pages,.
[] A. Cao, H. Chen, J. Zhang, and X. Lv, “Optimization of open
boundary conditions in a D internal tidal model with the
adjoint method around Hawaii,” Abstrac t and Applied Analysis,
vol. , Article ID ,  pages, .
[] H. Chen, C. Miao, and X. Lv, “Estimation of open boundary
conditions for an internal tidal model with adjoint method:
a comparative study on optimization methods,” Mathematical
Problems in Engineering,vol.,ArticleID,pages,
.
[] X. Lu and J. Zhang, “Numerical study on spatially varying
bottom friction coecient of a D tidal model with adjoint
method,” Continental Shelf Research, vol. , no. , pp. –
, .
[] J. Zhang, X. Lu, P. Wang, and Y. P. Wang, “Study on linear and
nonlinear bottom friction parameterizations for regional tidal

 Mathematical Problems in Engineering
models using data assimilation,” Continental Shelf Research,vol.
,no.,pp.–,.
[] R. W. Lardner and S. K. Das, “Optimal estimation of eddy vis-
cosity for a quasi-three-dimensional numerical tidal and storm
surge model,” International Journal for Numerical Methods in
Fluids, vol. , no. , pp. –, .
[] J. E. Richardson and V. G. Panchang, “A modied adjoint
method for inverse eddy viscosity estimation for use in coastal
circulation models,” in Proceedings of the 2nd International
Conference on Estuarine and Coastal Modeling, pp. –,
November .
[] G. P. Cressman, “An operational objective analysis system,”
Monthly Weather Review,vol.,no.,pp.–,.
[] S. J. Wright and J. Nocedal, Numerical optimization,Springer,
New York, NY, USA, .
[] W. F. Mascarenhas, “e BFGS method with exact line searches
fails for non-convex objective functions,” Mathematical Pro-
gramming,vol.,no.,pp.–,.
[] A.K.Alekseev,I.M.Navon,andJ.L.Steward,“Comparisonof
advanced large-scale minimization algorithms for the solution
of inverse ill-posed problems,” Optimization Methods and So-
ware,vol.,no.,pp.–,.
[] X. Zou, I. M. Navon, M. Berger, K. H. Phua, T. Schlick, and F.
Le Dimet, “Numerical experience with limited-memory quasi-
Newton and truncated Newton methods,” SIAM Journal on
Optimization,vol.,no.,pp.–,.
[] J. Nocedal, “Updating quasi-Newton matrices with limited
storage,” Mathematics of Computation,vol.,no.,pp.–
, .
[] R. Malouf, “A comparison of algorithms for maximum entropy
parameter estimation,” in Proceedings of the 6th Conference on
Natural Language Learning (CoNLL ’02),pp.–,.
[] G. Andrew and J. Gao, “Scalable training of L-regularized
log-linear models,” in Proceedings of the 24th International
Conference on Machine Learning (ICML ’07),pp.–,June
.
[] A.K.Alekseev,I.M.Navon,andJ.L.Steward,“Comparisonof
advanced large-scale minimization algorithms for the solution
of inverse ill-posed problems,” Optimization Methods & So-
ware,vol.,no.,pp.–,.
[] D. C. Liu andJ. Nocedal, “On the limited memory BFGS method
for large scale optimization,” Mathematical Programming,vol.
,no.–,pp.–,.
[] L. J. Nocedal, “BFGS subroutine, soware for large-scale
unconstrained optimization,” http://www.ece.northwestern
.edu/∼nocedal/lbfgs.html.