Travel time inversion is a fundamental method of Ocean Acoustic Tomography, for the estimation of perturbations in sound speed. By discretizing the watercolumn into a system of layers, the method allows to introduce a system of linear equations, relating a known vector of perturbations in travel time, to an unknown vector of perturbations in sound speed, through the so-called \observation matrix". Inverting the system allows to determine a solution, which estimates the perturbation in sound speed in each layer of the watercolumn. However, in most problems of practical interest, the number of unknowns (i.e. the perturbations in sound speed) is larger that the number of equations (which correspond to the number of delays in travel time), which implies that inverting the system of linear equations can be viewed as an ill-posed problem. The discussion presented in this paper illustrates an approach to the problem of inversion, which is based on the usage of theoretical modes. Further, it is shown that for a range-dependent perturbation in sound speed, which corresponds to a superposition of plane waves, the inversion problem can be regularized (i.e. the system of linear equations can be rewritten in order to deal
with more equations than unknowns) by estimating only the amplitudes and phases of the linear waves. Particular examples are given for simulated and real data

Jesus, S. M.

Rodríguez, O. C.

Universidade do Algarve

Proceedings of the Sixth European Conference
on Underwater Acoustics
Gdansk, Poland
June, 2002
RANGE-DEPENDENT REGULARIZATION OF TRAVEL-TIME
TOMOGRAPHY BASED ON THEORETICAL MODES
O.C. Rodr´ıguez1 and S.M. Jesus1
1FCT-Universidade do Algarve (UALG), Campus de Gambelas, 8000, Faro, Portugal
Abstract: Travel time inversion is a fundamental method of Ocean Acoustic Tomo-
graphy, for the estimation of perturbations in sound speed. By discretizing the watercolumn
into a system of layers, the method allows to introduce a system of linear equations, relat-
ing a known vector of perturbations in travel time, to an unknown vector of perturbations
in sound speed, through the so-called “observation matrix”. Inverting the system allows to
determine a solution, which estimates the perturbation in sound speed in each layer of the
watercolumn. However, in most problems of practical interest, the number of unknowns
(i.e. the perturbations in sound speed) is larger that the number of equations (which cor-
respond to the number of delays in travel time), which implies that inverting the system
of linear equations can be viewed as an ill-posed problem. The discussion presented in
this paper illustrates an approach to the problem of inversion, which is based on the us-
age of theoretical modes. Further, it is shown that for a range-dependent perturbation in
sound speed, which corresponds to a superposition of plane waves, the inversion problem
can be regularized (i.e. the system of linear equations can be rewritten in order to deal
with more equations than unknowns) by estimating only the amplitudes and phases of the
linear waves. Particular examples are given for simulated and real data.
1 INTRODUCTION
Travel time inversion is a fundamental method of Ocean Acoustic Tomography, that
allows to introduce a system of linear equations, relating a known vector of perturbations
in travel time (called “travel time delays”), to an unknown vector of perturbations in
sound speed, through the so-called “observation matrix”, which can be calculated from a
system of stable eigenrays. Inverting the system allows to determine the pertubations in
sound speed, by estimating it in each layer of the watercolumn. In most cases inverting
the system of linear equations can be viewed as an ill-posed problem since the number of
unknowns (i.e. the perturbations in sound speed) uses to be larger that the number of
equations, which correspond to the number of travel time delays. It can be shown that
theoretical modes can be used efficiently to regularize the problem of inversion, by allowing
to rewrite the system of linear equations obtaining more equations than unknowns, and
1
that regularization proved to be robust when applied to real acoustic data [1]. The
discussion presented in this paper explores further the regularization based on theoretical
modes by developing a range-dependent inversion of the sound speed field, for the case
of internal plane-wave propagation. In this case one estimates the amplitudes and phases
of the plane waves. The proposed method is tested on both simulated and real acoustic
data.
2 THEORETICAL BACKGROUND
2.1 Travel time inversion
A detailed description of travel time inversion can be found in [2]. Briefly, the
method is based on the fundamental assumption that, for a small change of sound speed,
δc(z) = c(z)− c0(z) c0(z), the perturbation in travel time of an acoustic pulse can be
written as
∆τj = τj − τ 0j =
∫
Γj
ds
c(z)
−
∫
Γj
ds
c0(z)
≈ −
∫
Γj
δc(z)
c20(z)
ds , (1)
where the integral Eq.(1) is taken along the unperturbed eigenray Γj. For a set of T
perturbations in travel time, and discretizing the watercolumn into a system composed
by L layers, one can relate a vector of delays, ∆τ , to a vector of perturbations in sound
speed, δc, through a linear system of equations:
∆τ = Eδc + n , (2)
where ∆τ = [ ∆τ1 , ∆τ2 , . . . , ∆τT ]
t, δc = [ δc1 , δc2 , . . . , δcL ]
t, and each δcl
corresponds to an average of δc(z), in the lth layer; [. . .]t represents the transpose of vector
[. . .]. In Eq.(2) n accounts for rounding errors, and also for statistic contributions from
noise sources. Matrix E, dimension T × L, is known as the Observation Matrix and can
be calculated from unperturbed eigenrays. In most cases of practical interest L  T, so
Eq.(2) corresponds to an undetermined system of equations and therefore has an infinite
number of solutions. Providing that rankE = T one can select the solution that has a
minimum norm, which is given by the expression [3]:
δc# = Et
(
EEt
)−1
∆τ . (3)
2.2 Multiple hydrophones
The treatment of the system of independent equations for a set of N hydrophones
∆τ 1 = E1δc + n1, ∆τ 2 = E2δc + n2, . . ., ∆τN = ENδc + nN, sharing a common
vector of perturbations in sound speed, δc, can be handled by introducing the following
concatenated vectors and matrices:
∆τ =

∆τ 1
∆τ 2
...
∆τN
 , E =

E1
E2
...
EN
 and n =

n1
n2
...
nN
 , (4)
and further applying the solution Eq.(3).
2
2.3 Theoretical modes
For the hydrostatic linear rotationless case the set of theoretical modes, Ψm, can be
calculated by solving a Sturm-Liouville problem of the form [4]:
d2Ψm
dz2
+
N2
C2m
Ψm = 0 + Boundary Conditions , (5)
where N2(z) represents the buoyancy frequency. N2(z) can be calculated from tempera-
ture T and salinity S [4], although a satisfactory approximation can be obtained using
only temperature [5]. In Eq.(5) Cm represents the propagation velocity of the linear
wave associated to each Ψm; for a fixed frequency of internal wave propagation, ω, the
wavenumber will be given by km = ω/Cm. Under homogeneous top and bottom bound-
ary conditions the Ψm(z) form an orthogonal basis of functions, i.e. 〈Ψm |N2|Ψn〉 = 0 for
m 6= n, where the “inner product” is defined as 〈f1 |f2| f3〉 =
D∫
0
f1f2f3 dz.
2.4 Plave wave propagation
For internal plane wave propagation the range-dependent field of perturbation on
sound speed can be represented as
δc (z, r) =
dc0
dz
M∑
m=1
Ψm(z) [αm sin (kmr cos θ) + βm cos (kmr cos θ)] (6)
where θ represents the direction of propagation related to the acoustic path (see Fig.1(a)).
In Eq.(6) M represents the number of relevant theoretical modes.
2.5 Regularization using theoretical modes
Substituting Eq.(6) into Eq.(1) one can obtain a system of linear equations of the
form
∆τ = Px + n , (7)
where
P = [S R] , S =

St1
St2
...
StT
 , R =

Rt1
Rt2
...
RtT
 ,
x =
[
α
β
]
, α =

α1
α2
...
αM
 , β =

β1
β2
...
βM
 ,
(8)
Stj = [ Sj1 , Sj2 , . . . , SjM ] ,
Rtj = [ Rj1 , Rj2 , . . . , RjM ] ,
(9)
3
and [ Sjm
Rjm
]
= −
∫
Γj
dc0
dz
Ψm(z)
c20
ds
[
sin (kmr cos θ)
cos (kmr cos θ)
]
. (10)
Providing that 2M < T one can regularize the original system of equations, Eq.(2), and
write the solution of the system Eq.(7) as [3]
x# = (PtP)
−1
Pt∆τ . (11)
The solution given by Eq.(11) allows to determine the modal amplitudes x, made of two
components α and β, which determine uniquely the range-dependent field of sound speed
c(z, r) = c0(z) + δc(z, r) through Eq.(6). A multiple hydrophone system can be efficiently
handled concatenating once more the corresponding systems of equations.
3 SIMULATIONS
The range-dependent regularization of travel time inversion was tested on simulated
data, following the experimental scenarion of the INTIMATE’96 experiment, described
in detail in [1]. Theoretical modes were calculated from temperature records acquired
during the experiment (see Fig.1(b)). Further, as discussed in that reference analysis of
hydrographic data acquired near the Vertical Line Array (hereafter VLA) allow to consider
that in average M = 3. The propagation geometry for simulations correspondend to a
source at depth of 90 m, a VLA with three hydrophones (at depths of 35, 105 and
115 m), a transmission distance R = 5,6 km and a bottom depth D = 135 m. Stable
eigenrays were calculated using the background sound speed profile c0(z), derived from
CTD records. The range-dependent field c(z, r) = c0(z) + δc(z, r) was calculated for
an arbitrary set of realistic modal amplitudes αm and βm, and with θ = 75
◦, which is
an estimate of the direction of propapagation of the internal tide in the INTIMATE’96
scenario [5]. Perturbed travel times were calculated using a range-dependent ray-tracing
procedure, which used a discretized distribution of c(z, r) along range, with a reduced
number of intervals. The first test of inversion using Eq.(11) failed providing unrealistic
values of x, which in part can be related to a mismatch between “truth” perturbed arrivals,
and the arrivals obtained with the mentioned procedure. Additional tests optimized the
solution Eq.(11), using different range discretizations, and calculating the matrix P over
a commom range interval for each discretization. This optimization provided different
solutions x#, most of which were unrealistic, except the one that corresponded to the
reliable estimation of x.
4 APPLICATION TO REAL DATA
Tests with real data were based on a particular set of simultaneous data on both
acoustic records and hydrographic measurements taken near the acoustic source and the
VLA. Due to synchronization problems during the experiment absolute arrivals were not
available directly. The lack of synchronization could be compensated through an accurate
estimation of waveguide geometry, and by developing an accurate match of the inverse
solution with direct observations taken near the VLA, i.e. by minimizing
∣∣∣∣∣∣δc# − δc∣∣∣∣∣∣,
where δc# represented a range-independent estimate of the truth solution, δc (see [1]).
4
This matching was repeated once more for the set of simultaneous data, but based on
range dependent regularization and taking θ = 75◦. As indicated by simulations different
range discretizations allowed to optimize the matching. However, it was also noticed that
the accuracy could be significantly improved by taking M = 4. The optimized match can
be seen on Fig.2(a). In fact it corresponds to an accurate estimation of β, but not of α.
Additional tests of optimization for slight variations of θ around 75◦ provided unrealistic
estimates of both α and β. Additionally, a realistic estimate of α was found by optimizing
once more the estimate
α# = (StS)−1St
(
∆τ −Rβ#
)
, (12)
for different range discretizations, and choosing the estimate with a minimum norm. The
expected profile δc(z, R), calculated using α# and β#, can be seen on Fig.2(b), and reveals
a good agreement with the direct estimation of the sound speed perturbation, calculated
from a temperature record acquired near the position of the acoustic source.
5 CONCLUSIONS
The feasibility of range dependent regularization based on theoretical modes was
tested on simulated and real data for the case of internal plane-wave propagation in a
shallow water environment. Inversion results proved to be accurate, and the inversion
procedure was found to be robust to the presence of noise in real data, and able to resolve
high order theoretical modes present in the sound speed field. However, the accurate
numerical evaluation of the range-dependent matrix P still remains an open question,
and the inversion procedure can be only applied to the propagation of internal plane-
waves, at a fixed direction, across the scenario of acoustic propagation.
References
[1] Rodr´ıguez O. C. Application of Ocean Acoustic Tomography to the estimation of
internal tides on the continental platform. PhD. Thesis, Faculdade de Cieˆncias
e Tecnologia–Universidade do Algarve (FCT-UALG), Faro, Portugal (available in
ftp://ftp.ualg.pt/users/siplab/doc/thesis.pdf), December, 2000.
[2] Munk W., Worcester P., and Wunsch C. Ocean Acoustic Tomography. Cambridge
Monographs on Mechanics, Cambridge, University Press, 1995.
[3] Menke W. Geophysical Data Analysis: Discrete Inverse Theory. Academic Press, Inc,
San Diego, California, 1989.
[4] Apel J.R. Principles of Ocean Physics, volume 38. Academic Press, International
Geophysics Series, London, 1987.
[5] Rodr´ıguez O.C., Jesus S., Stephan Y., De´moulin X., Porter M., and Coelho E. Internal
tide acoustic tomography: reliability of the normal modes expansion as a possible
basis for solving the inverse problem. In Proc. of the 4th. European Conference on
Underwater Acoustics, pages 587–592, Rome, Italy, 21-25 September 1998.
5
(a) (b)
PSfrag replacements
Source
Receiver
θ
P
ro
pa
ga
tin
g
w
av
e
1
0 0.5 1
0
20
40
60
80
100
120
D
e
p
th
 (m
)
−1 0 1
0
20
40
60
80
100
120
−1 0 1
0
20
40
60
80
100
120
−1 0 1
0
20
40
60
80
100
120
PSfrag replacements
Ψ1(z) Ψ2(z) Ψ3(z) Ψ4(z)
1
Figure 1: (a) Propagating direction of the linear internal wave related to the acoustic
path; (b) theoretical modes.
(a) (b)
−1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4
0
20
40
60
80
100
120
140
D
e
p
th
 (m
)
Perturbation on sound speed (m/s)
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
20
40
60
80
100
120
140
D
e
p
th
 (m
)
Perturbation on sound speed (m/s)
Figure 2: (a) Best match of sound speed perturbations near the Vertical Line Arrray
(VLA); (b) extrapolated profile of sound speed perturbations near the position of the
acoustic source. In both cases the continues line corresponds to the estimated value,
while the dashed line represents direct measurements.
6


Range-dependent regularization of travel-time tomography based on theoretical modes

Sapientia

Proceedings of the Sixth European Conferenceon Underwater AcousticsGdansk, PolandJune, 2002RANGE-DEPENDENT REGULARIZATION OF TRAVEL-TIMETOMOGRAPHY BASED ON THEORETICAL MODESO.C. Rodr´ıguez1 and S.M. Jesus11FCT-Universidade do Algarve (UALG), Campus de Gambelas, 8000, Faro, PortugalAbstract: Travel time inversion is a fundamental method of Ocean Acoustic Tomo-graphy, for the estimation of perturbations in sound speed. By discretizing the watercolumninto a system of layers, the method allows to introduce a system of linear equations, relat-ing a known vector of perturbations in travel time, to an unknown vector of perturbationsin sound speed, through the so-called “observation matrix”. Inverting the system allows todetermine a solution, which estimates the perturbation in sound speed in each layer of thewatercolumn. However, in most problems of practical interest, the number of unknowns(i.e. the perturbations in sound speed) is larger that the number of equations (which cor-respond to the number of delays in travel time), which implies that inverting the systemof linear equations can be viewed as an ill-posed problem. The discussion presented inthis paper illustrates an approach to the problem of inversion, which is based on the us-age of theoretical modes. Further, it is shown that for a range-dependent perturbation insound speed, which corresponds to a superposition of plane waves, the inversion problemcan be regularized (i.e. the system of linear equations can be rewritten in order to dealwith more equations than unknowns) by estimating only the amplitudes and phases of thelinear waves. Particular examples are given for simulated and real data.1 INTRODUCTIONTravel time inversion is a fundamental method of Ocean Acoustic Tomography, thatallows to introduce a system of linear equations, relating a known vector of perturbationsin travel time (called “travel time delays”), to an unknown vector of perturbations insound speed, through the so-called “observation matrix”, which can be calculated from asystem of stable eigenrays. Inverting the system allows to determine the pertubations insound speed, by estimating it in each layer of the watercolumn. In most cases invertingthe system of linear equations can be viewed as an ill-posed problem since the number ofunknowns (i.e. the perturbations in sound speed) uses to be larger that the number ofequations, which correspond to the number of travel time delays. It can be shown thattheoretical modes can be used efficiently to regularize the problem of inversion, by allowingto rewrite the system of linear equations obtaining more equations than unknowns, and1that regularization proved to be robust when applied to real acoustic data [1]. Thediscussion presented in this paper explores further the regularization based on theoreticalmodes by developing a range-dependent inversion of the sound speed field, for the caseof internal plane-wave propagation. In this case one estimates the amplitudes and phasesof the plane waves. The proposed method is tested on both simulated and real acousticdata.2 THEORETICAL BACKGROUND2.1 Travel time inversionA detailed description of travel time inversion can be found in [2]. Briefly, themethod is based on the fundamental assumption that, for a small change of sound speed,δc(z) = c(z)− c0(z) c0(z), the perturbation in travel time of an acoustic pulse can bewritten as∆τj = τj − τ 0j =∫Γjdsc(z)−∫Γjdsc0(z)≈ −∫Γjδc(z)c20(z)ds , (1)where the integral Eq.(1) is taken along the unperturbed eigenray Γj. For a set of Tperturbations in travel time, and discretizing the watercolumn into a system composedby L layers, one can relate a vector of delays, ∆τ , to a vector of perturbations in soundspeed, δc, through a linear system of equations:∆τ = Eδc + n , (2)where ∆τ = [ ∆τ1 , ∆τ2 , . . . , ∆τT ]t, δc = [ δc1 , δc2 , . . . , δcL ]t, and each δclcorresponds to an average of δc(z), in the lth layer; [. . .]t represents the transpose of vector[. . .]. In Eq.(2) n accounts for rounding errors, and also for statistic contributions fromnoise sources. Matrix E, dimension T × L, is known as the Observation Matrix and canbe calculated from unperturbed eigenrays. In most cases of practical interest L  T, soEq.(2) corresponds to an undetermined system of equations and therefore has an infinitenumber of solutions. Providing that rankE = T one can select the solution that has aminimum norm, which is given by the expression [3]:δc# = Et(EEt)−1∆τ . (3)2.2 Multiple hydrophonesThe treatment of the system of independent equations for a set of N hydrophones∆τ 1 = E1δc + n1, ∆τ 2 = E2δc + n2, . . ., ∆τN = ENδc + nN, sharing a commonvector of perturbations in sound speed, δc, can be handled by introducing the followingconcatenated vectors and matrices:∆τ =∆τ 1∆τ 2...∆τN , E =E1E2...EN and n =n1n2...nN , (4)and further applying the solution Eq.(3).22.3 Theoretical modesFor the hydrostatic linear rotationless case the set of theoretical modes, Ψm, can becalculated by solving a Sturm-Liouville problem of the form [4]:d2Ψmdz2+N2C2mΨm = 0 + Boundary Conditions , (5)where N2(z) represents the buoyancy frequency. N2(z) can be calculated from tempera-ture T and salinity S [4], although a satisfactory approximation can be obtained usingonly temperature [5]. In Eq.(5) Cm represents the propagation velocity of the linearwave associated to each Ψm; for a fixed frequency of internal wave propagation, ω, thewavenumber will be given by km = ω/Cm. Under homogeneous top and bottom bound-ary conditions the Ψm(z) form an orthogonal basis of functions, i.e. 〈Ψm |N2|Ψn〉 = 0 form 6= n, where the “inner product” is defined as 〈f1 |f2| f3〉 =D∫0f1f2f3 dz.2.4 Plave wave propagationFor internal plane wave propagation the range-dependent field of perturbation onsound speed can be represented asδc (z, r) =dc0dzM∑m=1Ψm(z) [αm sin (kmr cos θ) + βm cos (kmr cos θ)] (6)where θ represents the direction of propagation related to the acoustic path (see Fig.1(a)).In Eq.(6) M represents the number of relevant theoretical modes.2.5 Regularization using theoretical modesSubstituting Eq.(6) into Eq.(1) one can obtain a system of linear equations of theform∆τ = Px + n , (7)whereP = [S R] , S =St1St2...StT , R =Rt1Rt2...RtT ,x =[αβ], α =α1α2...αM , β =β1β2...βM ,(8)Stj = [ Sj1 , Sj2 , . . . , SjM ] ,Rtj = [ Rj1 , Rj2 , . . . , RjM ] ,(9)3and [ SjmRjm]= −∫Γjdc0dzΨm(z)c20ds[sin (kmr cos θ)cos (kmr cos θ)]. (10)Providing that 2M < T one can regularize the original system of equations, Eq.(2), andwrite the solution of the system Eq.(7) as [3]x# = (PtP)−1Pt∆τ . (11)The solution given by Eq.(11) allows to determine the modal amplitudes x, made of twocomponents α and β, which determine uniquely the range-dependent field of sound speedc(z, r) = c0(z) + δc(z, r) through Eq.(6). A multiple hydrophone system can be efficientlyhandled concatenating once more the corresponding systems of equations.3 SIMULATIONSThe range-dependent regularization of travel time inversion was tested on simulateddata, following the experimental scenarion of the INTIMATE’96 experiment, describedin detail in [1]. Theoretical modes were calculated from temperature records acquiredduring the experiment (see Fig.1(b)). Further, as discussed in that reference analysis ofhydrographic data acquired near the Vertical Line Array (hereafter VLA) allow to considerthat in average M = 3. The propagation geometry for simulations correspondend to asource at depth of 90 m, a VLA with three hydrophones (at depths of 35, 105 and115 m), a transmission distance R = 5,6 km and a bottom depth D = 135 m. Stableeigenrays were calculated using the background sound speed profile c0(z), derived fromCTD records. The range-dependent field c(z, r) = c0(z) + δc(z, r) was calculated foran arbitrary set of realistic modal amplitudes αm and βm, and with θ = 75◦, which isan estimate of the direction of propapagation of the internal tide in the INTIMATE’96scenario [5]. Perturbed travel times were calculated using a range-dependent ray-tracingprocedure, which used a discretized distribution of c(z, r) along range, with a reducednumber of intervals. The first test of inversion using Eq.(11) failed providing unrealisticvalues of x, which in part can be related to a mismatch between “truth” perturbed arrivals,and the arrivals obtained with the mentioned procedure. Additional tests optimized thesolution Eq.(11), using different range discretizations, and calculating the matrix P overa commom range interval for each discretization. This optimization provided differentsolutions x#, most of which were unrealistic, except the one that corresponded to thereliable estimation of x.4 APPLICATION TO REAL DATATests with real data were based on a particular set of simultaneous data on bothacoustic records and hydrographic measurements taken near the acoustic source and theVLA. Due to synchronization problems during the experiment absolute arrivals were notavailable directly. The lack of synchronization could be compensated through an accurateestimation of waveguide geometry, and by developing an accurate match of the inversesolution with direct observations taken near the VLA, i.e. by minimizing∣∣∣∣∣∣δc# − δc∣∣∣∣∣∣,where δc# represented a range-independent estimate of the truth solution, δc (see [1]).4This matching was repeated once more for the set of simultaneous data, but based onrange dependent regularization and taking θ = 75◦. As indicated by simulations differentrange discretizations allowed to optimize the matching. However, it was also noticed thatthe accuracy could be significantly improved by taking M = 4. The optimized match canbe seen on Fig.2(a). In fact it corresponds to an accurate estimation of β, but not of α.Additional tests of optimization for slight variations of θ around 75◦ provided unrealisticestimates of both α and β. Additionally, a realistic estimate of α was found by optimizingonce more the estimateα# = (StS)−1St(∆τ −Rβ#), (12)for different range discretizations, and choosing the estimate with a minimum norm. Theexpected profile δc(z, R), calculated using α# and β#, can be seen on Fig.2(b), and revealsa good agreement with the direct estimation of the sound speed perturbation, calculatedfrom a temperature record acquired near the position of the acoustic source.5 CONCLUSIONSThe feasibility of range dependent regularization based on theoretical modes wastested on simulated and real data for the case of internal plane-wave propagation in ashallow water environment. Inversion results proved to be accurate, and the inversionprocedure was found to be robust to the presence of noise in real data, and able to resolvehigh order theoretical modes present in the sound speed field. However, the accuratenumerical evaluation of the range-dependent matrix P still remains an open question,and the inversion procedure can be only applied to the propagation of internal plane-waves, at a fixed direction, across the scenario of acoustic propagation.References[1] Rodr´ıguez O. C. Application of Ocean Acoustic Tomography to the estimation ofinternal tides on the continental platform. PhD. Thesis, Faculdade de Cieˆnciase Tecnologia–Universidade do Algarve (FCT-UALG), Faro, Portugal (available inftp://ftp.ualg.pt/users/siplab/doc/thesis.pdf), December, 2000.[2] Munk W., Worcester P., and Wunsch C. Ocean Acoustic Tomography. CambridgeMonographs on Mechanics, Cambridge, University Press, 1995.[3] Menke W. Geophysical Data Analysis: Discrete Inverse Theory. Academic Press, Inc,San Diego, California, 1989.[4] Apel J.R. Principles of Ocean Physics, volume 38. Academic Press, InternationalGeophysics Series, London, 1987.[5] Rodr´ıguez O.C., Jesus S., Stephan Y., De´moulin X., Porter M., and Coelho E. Internaltide acoustic tomography: reliability of the normal modes expansion as a possiblebasis for solving the inverse problem. In Proc. of the 4th. European Conference onUnderwater Acoustics, pages 587–592, Rome, Italy, 21-25 September 1998.5(a) (b)PSfrag replacementsSourceReceiverθPropagatingwave10 0.5 1020406080100120Depth (m)−1 0 1020406080100120−1 0 1020406080100120−1 0 1020406080100120PSfrag replacementsΨ1(z) Ψ2(z) Ψ3(z) Ψ4(z)1Figure 1: (a) Propagating direction of the linear internal wave related to the acousticpath; (b) theoretical modes.(a) (b)−1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4020406080100120140Depth (m)Perturbation on sound speed (m/s)−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4020406080100120140Depth (m)Perturbation on sound speed (m/s)Figure 2: (a) Best match of sound speed perturbations near the Vertical Line Arrray(VLA); (b) extrapolated profile of sound speed perturbations near the position of theacoustic source. In both cases the continues line corresponds to the estimated value,while the dashed line represents direct measurements.6

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Range-dependent regularization of travel-time tomography based on theoretical modes

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