淺海環境中流速對高頻聲波傳播之研究

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Institute of Oceanography College of Science

National Taiwan University Master Thesis

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A Study of Ocean Current on High-Frequency Acoustic Propagation in Coastal Water Environments

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Leow Khang Yen

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Advisor: Chen-Fen Huang, Ph.D.

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July, 2013

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Abstract

A ray-tracing program for a moving medium (Current Ray-Tracing program) is developed to study the eﬀects of ocean current on high- frequency acoustic propagation. A higher-order Runge-Kutta integration algorithm with adjustable step size is implemented for high accuracy.

Numerical simulations for the waveguide of constant current velocity or constant current shear agree well with the analytic solutions. For the case of constant current velocity, the average along-ray speed perturbation is the dominant factor in perturbed travel time. The angular variation of perturbed travel time is due to the perturbed ray length.

The magnitude of all perturbed ray properties are very small. For the case of constant current shear, the range variation of vertical component of wavefront normal is inversely proportional to current shear and has less dependence of the wavefront normal angle. The gradual rays are more sensitive to the current shear. For the range-dependent waveguide of deterministic small-scale bathymetric structure, Monte Carlo simulations show that the small bottom slopes have a dramatic eﬀect on ray paths: larger predictive uncertainty is observed for steeper rays or for the case of larger current magnitude. Modeling error of eﬀective sound speed approaches on predicting travel time comes from errors in both ray length and average along-ray speed. When using the ESSP approach (projected current on the sound speed) the dominant error is the ray length prediction. The importance of correct inclusion of ocean current in the ray-tracing program for long-range acoustic propagation is shown using a realistic ocean environment with deterministic irregular bottom topography.

Key words: Runge-Kutta, ray-tracing with current, geometric acoustics, acoustic propagation, underwater sound

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List of Figures

2.1 Concept of a ray path in a moving medium. The associated sound- speed normal is cn, the current velocity is v, and the ray velocity is cn + v. . . 8 2.2 Schematic of 2D ray geometry. . . 11 2.3 Wavefronts and rays in a stationary medium. . . 12 2.4 (a) Sound speed profile (b) Current profile (c) Simplified eﬀective

sound speed profile. . . 13 2.5 Schematic plots of the ray velocity vector modeled by (a) moving-

medium ray-tracing and (b) the eﬀective sound speed approach. . . . 14 3.1 Schematic of ray reflecting from a piecewise linear boundary . . . 19 3.2 Shallow water environment: (a) Shallow water sound channel (b)

Sound speed profile . . . 20 3.3 Modeled environment: Measured bathymetry, sound-speed profiles

(gray heavy line) and current profiles (arrow) from HYCOM model. 21 3.4 (a) Ray paths obtained by Current Ray-Tracing program and BELL-

HOP for θ0 = 15^◦; (b) Diﬀerence in bounce ranges between ray paths obtained by two programs. Colors indicate diﬀerent launching angles;

(c) Diﬀerence in travel time obtained by two programs. . . 22 4.1 Schematic of ray lengths in the presence of current and in the reference

state for a ray of launching angle θ0. . . 27 4.2 Schematic of boundary-bounce ranges in the presence of current and

in the reference state for a launching angle θ0. . . 28 4.3 Horizontal current profiles of various constant current velocity. Black

line indicates reference state, and colored lines indicate current profiles with diﬀerent current magnitudes. . . 30

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4.4 Ray paths for θ0 = −10^◦ in the ocean with diﬀerent current magnitudes. (top) Full view; (middle) enlarged view at the ninth sea surface bounce; (bottom) perturbed boundary-bounce range versus boundary-bounce range in the reference state. . Black line indicates the ray path in the reference state and colored lines are the ray paths for constant current velocity of diﬀerent magnitudes. Solid line indicates the analytical solution and circle indicates the numerical results. 31 4.5 Perturbed ray properties versus current velocity: (a) perturbed ray

length; (b) perturbed average along-ray speed; (c) perturbed travel time. The color of line indicates launching angle of the results. In these plots diﬀerent launching angles are indicated by colors. . . 32 4.6 Perturbed ray properties versus launching angle: (a) perturbed ray

length; (b) perturbed average along-ray speed; (c) perturbed travel time. Lines indicate the results from analytic formula, and circles are the numerical results. . . 33 4.7 Perturbed travel time for v = 1 m/s. Line indicates numerical results.

Circle indicates analytical solution. Asterisk indicates the component due to perturbed average along-ray speed. Plus indicates the component due to perturbed ray length. Note that the y-axis is divided into two sections. . . 34 4.8 Current profiles of various current shear. The color with diﬀerent

shade is used to indicate diﬀerent current shear profile. . . 37 4.9 Analytical solution of derivatives of wavefront normal vector as a func-

tion of θ: (a) Derivative of vertical component of wavefront normal vector, d sin θ/dr. (b) Derivative of horizontal component of wavefront normal vector, d cos θ/dr. . . 37 4.10 Comparison of the ray paths in various current shear for θ0 =−0.5^◦.

Black line indicates the result in the baseline model, and colored lines indicate the results in the presence of current shear: (a) ray path, (b) vertical component and (c) horizontal component of wavefront normal vector. . . 38 4.11 Comparison of results with various launching angle for (left) ∂v/∂z =

−0.01 s ⁻¹ and (right) ∂v/∂z = 0.01 s ⁻¹. colors indicate diﬀerent launching angles: (a) (d) ray path, (b) (e) vertical component of wavefront normal vector and (c) (f) horizontal component of wavefront normal vector. . . 39

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4.12 Ray path in the reference state (black line) and in the presence of constant current velocity (blue/red lines) for θ0 = 5^◦. (a) full view (b) the enlarged view at the fourth bottom bounce. . . 41 4.13 Perturbed ray length versus current magnitude (θ0 = 5^◦). Brown line

indicates the flat-bottom result and the circles indicate the results from the irregular bathymetric seafloor shown in Figure 4.12. Diﬀer- ent colors are for diﬀerent current magnitudes. Note that the y-axis is divided into two section. . . 42 4.14 Perturbed ray length versus current velocity (θ0 = 5^◦). Brown line

indicates the flat-bottom result and colored dots indicate the results from 100 realizations. . . 43 4.15 Mean and the error bar of perturbed ray length versus current velocity

for 100 realizations, θ0 = 5^◦. Brown line indicates the flat-bottom result. . . 43 4.16 Box-and-whisker plot of perturbed ray length versus current velocity

for the launching angle θ₀ = 5^◦. Brown line indicates the flat-bottom result. . . 44 4.17 Box-and-whisker plot of perturbed ray length versus launching angle

for v = 1 m/s. Brown line indicates the flat-bottom result. . . 44 4.18 (a) Ray paths obtained using the Current Ray-Tracing (black line),

ESSP (gray dashed line) and SESSP (symbols) approaches for θ0 =

−2^◦ (b) Enlarged view at the first sea-surface-bounce of ray path. . . 46 4.19 Diﬀerence between ray properties obtained by eﬀective sound speed

approaches and actual ray properties versus launching angle: (a) ray length (b) average along-ray speed and (c) travel time. Gray dashed line indicates ESSP results and gray solid line indicates SESSP results. . . 48 4.20 Diﬀerence between travel time obtained by eﬀective sound speed ap-

proaches and actual travel time versus launching angle: (a) the component due to ray length diﬀerence and (b) the component due to average speed diﬀerence. Gray dashed line indicates ESSP results and gray solid line indicates SESSP results. . . 49

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4.21 Ray path obtained by ESSP approach (gray dashed line) and actual ray path (black solid line) for θ0 = −15^◦: (a) full view, (b) vertical component of wavefront normal sin θ, and (c) magnitude of sin θ diﬀerence between ESSP approach and actual ray path. In (c), blue line indicates the diﬀerence, and asterisk symbol indicates the actual boundary-bounce range. . . 51 4.22 Enlarged view at the last sea-surface-bounce of ray path obtained by

ESSP approach (gray dashed line) and actual ray path (black solid line) for θ₀ =−15^◦ . . . 52 4.23 Diﬀerence in ray properties between ESSP approach and the Current

Ray-Tracing: (a) ray length difference (b) travel time difference. In (b), black line indicates the total travel time difference, and blue line indicates the difference due to ray length difference. . . 52 4.24 Difference between travel time obtained by SESSP approach and the

Current Ray-Tracing versus launching angle θ₀. Note that the y-axis is divided into two ranges. . . 53 4.25 The ray paths obtained by SESSP approach (dashed line) and actual

ray path (solid line) for θ0 = 11.6^◦ which has the largest travel time diﬀerence. . . 54 4.26 (a) The eigen rays obtained by Current Ray-Tracing (colored line),

Ray 1 (green line) indicates the eigen ray of first arrival, gray dashed line indicates Ray 1 obtained by SESSP approach. (b) Difference between travel time of each eigen ray obtained by SESSP approach and the Current Ray-Tracing (c) The component of time difference due to ray length difference (d) The component of time difference due to average speed difference. . . 55

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Chapter 1 Introduction

1.1 Background and Motivation

Ocean acoustic tomography is a remote sensing method that infers the ocean temperature or current structure from the travel time of reciprocal transmission through the moving ocean.[18]. The modeling of the path and travel time of acoustic propagation in ocean is important for ocean acoustic tomography.

Ray theory is widely used to model the path and travel time of acoustic propagation in ocean. With the high-frequency approximation involved in the derivation of ray equations, ray theory is applicable for high-frequency acoustic propagation. Ray theory assumes that the medium is stationary, however, the ocean as the medium of acoustic propagation is in a variety of motion, such as currents, tides, eddies and internal wave. The acoustic propagation in ocean would be aﬀected by the motion and these eﬀects are not accounted for while applying the ray theory.

The typical value of sound speed c in ocean is between 1480 and 1540 m/s [32], and the maximum velocity of currents v is around 1 m/s [14]. The ratio v/c is of the order of 10⁻³ which is very small and suggested that the eﬀects of current on the characteristics of acoustic propagation might not be significant. Despite that, currents may aﬀect acoustic propagation in the following ways [20]:

1. Significant phase change of acoustic wave. If the range of propagation is large enough, currents might causes significant phase change hence to its travel time.

2. Significant change in amplitude. If there are several rays arriving at the receiver, and the phase change of at least one ray depends on current, the amplitude would also depends on current.

3. Significant refraction of ray path. If the magnitude of current shear is larger

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than the magnitude of sound speed gradient, the refraction of ray path might be induced.

Most studies [7, 8, 9, 29, 30] have been mainly concerned with the eﬀects of current on the phase and amplitude of the total acoustic field on a receiver. However, the total acoustic field is the combination of each of the eigenrays that pass through the receiver. Few had discuss the detailed eﬀects of current on a single ray.

A common approach which is widely used to account for current effects is to add the current directly to the sound speed. This approach interprets current effects in terms of an effective sound speed profile (ESSP) and replaces the real moving medium by a hypothetical stationary medium. However, since the actual ray path and the wavefront normal are not collinear in a moving medium, this approach may induce an error in the modeling of ray path and travel time. Most studies [25, 26]

that apply eﬀective sound speed approach did not quantify the error.

Most ray-tracing models are based on the ray theory for stationary medium.

Therefore it is required to develop a ray-tracing program which is capable of modeling the high-frequency acoustic propagation that account for the current eﬀects.

1.2 A Concise Survey of Literature

The study of acoustics in a moving medium was begin from the study of acoustic propagation in atmosphere. The effects of current velocity on the sound propagation in ocean are analogous to the effects of wind velocity on the sound propagation in atmosphere. The ratio of wind velocity to sound speed in atmosphere is greater than the ratio of current to sound speed in ocean, so the effects of a moving medium would be more significant in the atmosphere compared with in the ocean.

In 1899, Barton [1] first shows that the direction of propagation are diﬀer to the direction of wavefront normal in a moving medium, an equation has been derived to calculate the ray path for constant wind gradient case.

The study of the eﬀects of current and current shear on acoustic propagation in various environment had been carried out. Ray theory is used to examine the ray geometry, travel time, and spreading loss of each arrival and the phase and amplitude of the total acoustic field at a receiver [7, 8, 9, 29, 30]. Besides ray theory, parabolic equation (PE) is also used to examine the received intensity for continuous wave [19, 25, 26] and pulse [19] propagation.

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Idealized environment

In a channel with horizontal boundaries and constant sound speed, Stallworth and Jacobson studied the effects of constant current [29] and constant current shear [30]. The effects of constant current is not so significant on amplitude but is significant on the phase of the total field. The phase shift is affected by the time- varying current and the results have been confirmed by experimental data. The effects of current shear on ray geometry, travel time, and spreading loss of each arrival is significant. The ray paths are approximated by circular helical arcs. Small changes in the current shear cause large changes in amplitude and phase of the total field. Franchi and Jacobson [7] studied the effects of constant current shear in a channel with constant sound speed gradient, the effects of current shear and bottom currents on the phase and amplitude of the total field are important and cannot be ignored.

Geostrophic flow

The eﬀects of geostrophic flow on the acoustic propagation has been studied by Franchi and Jacobson in the case of along the current [8] and across the current [9].

The sound speed is found to depend upon both the magnitude of the current and the spatial position in a cross section of the current, so that the geostrophic flow not only affects the sound propagation directly, but also affects the sound propagation indirectly via the relationship with sound speed. Experiment had been carried out in the Florida Straits [9] and shows that the phase and amplitude of the received signal are significantly affected by variations in the current.

Reciprocal acoustic transmission

Stallworth [28] studied the relationship between current and travel time and suggested a method for measuring ocean currents via reciprocal acoustic transmission.

Worcester [31] conducted a reciprocal acoustic transmission in a midocean (about 1 km depth) and observed that the travel time and amplitude of sound impulse depends on the direction of sound propagation.

Munk et al. [17, 18] proposed ocean acoustic tomography, a method to estimate the current structure from the travel time diﬀerence of reciprocal transmission using inverse methods.

In a channel with horizontal boundaries and constant sound speed, Robertson

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et al. studied the relative intensity of reciprocal transmission using parabolic equation (PE) with eﬀective sound speed approach in the presence of constant current shear [25] and realistic current shear [26]. The variation of intensity depends on current velocity, source and receiver geometry, and acoustic frequency and no simple characterization of current eﬀects on intensity.

Ngiem-Phu and Tappert [19] developed a PE acoustic model that can be used to account for the eﬀects of current and current shear on transmission loss. Consider the typical Gulf Stream conditions, the eﬀects of current on both travel time and amplitudes of the received signals of the reciprocal transmission are significant.

Internal wave

The effects of internal wave on acoustic propagation have been studied. The magnitude of current shear and sound speed perturbations are estimated using the Garrett–Munk internal wave model and observations. Duda [5] and Colosi [3] suggested that the ray-refracting effects of the vertical shear of the horizontal current are of comparable magnitude to sound speed perturbations. These effects may be significant in distorting the acoustic paths [5] and may be an important source of upper ocean acoustic scattering [3].

Small-scale bathymetric structure

Most studies of the effects of current and current shear on acoustic propagation are considering horizontal boundaries or some simplified boundaries. Without considering current effects, Palmer et al. study the sensitivity of ray path [21, 22] and arrival time pattern [21] to deterministic small-scale bathymetric structure. The resulting chaotic ray path after a few bottom bounces may cause difficulty in the identification of individual rays [22] and proliferation of eigenrays [21]. The arrival time pattern is associated with a group instead of a single eigenray [21].

1.3 Objectives

The objectives of this thesis included:

1. To develop a ray-tracing program which is capable of modeling current eﬀects on the high-frequency acoustic propagation.

2. To study the ray properties, such as ray path, ray length, average along- ray speed and travel time in the presence of current and current shear in an

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idealized waveguide.

3. To study the statistics of ray properties in the presence of current associated with deterministic small-scale bathymertic feature.

4. To examine the applicability of eﬀective sound speed approaches which is widely used to account for current eﬀects.

1.4 Scopes of the Thesis

The major contents of this thesis consist of four chapters. Chapter 2 reviews general ray theory and derives the ray-tracing equations for a moving medium including both current magnitude and current shear eﬀects. A more complete derivation of ray-tracing equations for a moving medium is the subject of Appendix A. A common treatment for currents, called eﬀective sound speed approach, are reviewed for the comparison.

Chapter 3 provides a description of the computer program used to solve the moving-medium ray-tracing equations (the Current Ray-Tracing program). The details of the program and the environments used for the numerical simulations are introduced. The program is validated by comparing the simulated results with those using the BELLHOP ray-tracing model. A method of generating artificial irregular bottom is stated.

Chapter 4 uses the Current Ray-Tracing program to study the eﬀects of constant current velocity on the ray properties (ray path, average speed along ray and travel time) and how the current shear aﬀects the wavefront normal and ray path.

Numerical results using a simplified ocean environment are compared with the analytic solutions. Eﬀect of deterministic small-scale bathymetric feature on ray length perturbation is studied via Monte Carlo simulations. Calculations using eﬀective sound speed approaches are compared with those using the Current Ray-Tracing program in various shallow water environments. Finally, Chapter 5 addresses the conclusions of the thesis and suggestions for future research.

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Chapter 2 Theory

Ray theory for sound propagation in a moving medium is developed in this chapter to study the eﬀects of ocean current on ray path and its travel time. First, the development of ray theory for a steady moving medium yields the general ray tracing equations. Then, the general ray tracing equations is modified for sound propagation in ocean via a few assumptions. The ray equations for a stationary medium is stated. Finally, a common approach which is called eﬀective sound speed approach is introduced to simplify the treatment of ray tracing in a moving medium.

2.1 General Ray Theory in a Moving Medium

This section summarizes the ray-tracing equations in a moving medium. It is based on the material in Chapter 8 of the textbook by A. D. Pierce [23]. The goal here is not to explain in detail but simply to summarize results necessary for the development of ray theory in a moving medium.

Imagine a single frequency oscillating acoustic waves spreading spherically in a moving medium, if we take snapshots at time t, the wavefronts τ (x) = t are surfaces where the phase is the same. As the acoustic waves propagate, the wavefronts move with the velocity vray and a waveform feature (e.g., crests or troughs) is simultaneously received at the wavefront.

If we trace a moving point p that always lies on the wavefront while the acoustic waves propagate (as illustrated in Figure 2.1), the velocity with which the point p travels along the path is the vector sum of the ambient fluid velocity vector v and the sound speed c in the direction normal to the wavefront

dxp

dt = v + cn = vray, (2.1)

where xp(t) is the trajectory of the point p and is the path that acoustic pulse follows.

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n is the unit vector normal to the wavefront. The solution of Equation (2.1) defines the ray paths. Figure 2.1 shows that the direction of ray tangent cn + v diﬀers from the direction of wavefront normal n.

Ray path

Figure 2.1: Concept of a ray path in a moving medium. The associated sound-speed normal is cn, the current velocity is v, and the ray velocity is cn + v.

Note that in the case of a stationary medium we are concerned with the velocity at which the wavefront moves normal to itself cn·n, whereas in a moving medium we are concerned with the magnitude of the velocity of the point p, i.e.,|v^ray| = |v+cn|.

The ray velocity has two components, one in the direction normal to the wavefront and another along the direction of the current velocity.

Solving for the ray paths via Equation (2.1) is a diﬃcult task because one would need to know n as a function of time. Instead of dealing with n, we define a wave-slowness vector S that is the gradient of the wavefront surface S =∇τ and is parallel to n. From the relationship [See Equation (8-1.2) of Ref. [23]]

∇τ ·dx dt = 1 and Equation (2.1), we can obtain

S· (cn + v) = 1. (2.2)

Since S is parallel to n, substitution of S = an into Equation (2.2) yields

S = n

c + v· n. (2.3)

Thus, the equation for the slowness vector is n = cS

Ω, (2.4)

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where Ω= c/(c + v· n). If Equation (2.4) is squared and rearranged, the eikonal equation is obtained

|∇τ|² = Ω²

c². (2.5)

For time independent flows the equation for the time rate of change of wave- slowness vector S(x) (i.e. the material derivative of S) is

dS dt = dx

dt · ∇S = c(n · ∇)S + (v · ∇)S. (2.6) Substituting Equation (2.4) into Equations (2.1) and (2.6), the ray tracing equations with respect to time are

dx

dt = c²

ΩS + v, (2.7)

dS

dt = −Ω

c∇c − S × (∇ × v) − (S · ∇)v. (2.8) The results heretofore are quite general and are applicable to sound propagation in a moving medium such as atmosphere and ocean. No assumptions have been made about the relative sizes of ambient flow speed and sound speed, about the stratification of the medium, or about the directions of flow and sound propagation.

2.2 Ray Equations for a Moving Medium: Ocean

In this section, we apply the ray tracing equations for a moving medium to account for the current eﬀect on acoustic propagation in the ocean. The magnitude of current velocity|v| is much less than sound speed c in oceans.

Consider a cylindrical coordinate with cylindrical symmetry and uncoupled azimuth approximation x = (r, z), we assume that the sound speed and current velocity fields vary in both horizontal and vertical directions, and neglect the vertical component of the current v = (v, 0). Equations (2.7) and (2.8) can be rewritten as

dr

dt = c²Sr

Ω + v, (2.9)

dz

dt = c²S_z

Ω , (2.10)

dSr

dt = −Ω c

∂c

∂r − S^r∂v

∂r, (2.11)

dSz

dt = −Ω c

∂c

∂z − S^r∂v

∂z. (2.12)

θ is defined as the angle of unit vector normal to wavefront measured with respect

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to the horizontal axis, therefore

n = (cos θ, sin θ), (2.13)

S = (Sr, Sz) =

� cos θ

c + v cos θ, sin θ c + v cos θ,

�

(2.14)

Ω = c

c + v· n = c

c + v cos θ. (2.15)

Equations (2.9)–(2.12) become dr

dt = c cos θ + v, (2.16)

dz

dt = c sin θ, (2.17)

d(_{c+v cos θ}^{cos θ} )

dt = − 1

c + v cos θ

∂c

∂r − cos θ c + v cos θ

∂v

∂r, (2.18)

d(_{c+v cos θ}^{sin θ} )

dt = − 1

c + v cos θ

∂c

∂z − cos θ c + v cos θ

∂v

∂z. (2.19)

In order to solve Equations (2.16)–(2.19) using existing numerical integrators such as the Runge-Kutta method, the right-hand side of these diﬀerential equations are required to formulate in terms of Sr and Sz. However it might not be an easy task to perform. Furthermore to avoid the calculation of the transcendental functions, the ray equations are expressed in cos θ and sin θ instead of θ. Since the distance of source-receiver is given but the travel time is an unknown variable during the numerical integration, the above ray equations are transformed into a set of diﬀerential equations with the ray range r as the independent variable. [6]

Via a few manipulations (See Appendix A), we can obtain Equations (2.16)- (2.19), to the first order in v/c, as:

dz

dr= c sin θ

c cos θ + v, (2.20)

d cos θ

dr =− tan θ c cos θ + v

�

sin θ∂c

∂r−cos θ∂c

∂z + sin θ cos θ∂v

∂r − cos²θ∂v

∂z

�

, (2.21) d sin θ

dr = 1

c cos θ + v

�

sin θ∂c

∂r − cos θ∂c

∂z + sin θ cos θ∂v

∂r − cos²θ∂v

∂z

�

, (2.22) where (r, z) the ray trajectory, and (cos θ, sin θ) the wavefront normal vector, with θ the angle of wavefront normal measured with respect to the horizontal axis as depicted in Figure 2.2. The positive angles are for down-going rays so that θ is a declination angle. The initial conditions for z, cos θ, sin θ are

z(0) = z0, cos θ(0) = cos θ₀,

sin θ(0) = sin θ0,

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for a ray starting at the source position (0, z0) with a specified launching angle θ0. Then, the travel time t can be integrated along the horizontal range

t =

� R 0

1

c cos θ + vdr, (2.23)

where R the ray range.

Equation (2.20) is identical to the ray tangent (ray slope). Note that the ray angle is diﬀerent from the angle of the wavefront normal, i.e., tan^{−1 dz}_dr �= θ.

Equation (2.21) and (2.22) describe the horizontal and vertical components of the wavefront normal, respectively.

Ray Range

Depth

Source

Wavefront normal

Figure 2.2: Schematic of 2D ray geometry.

2.3 Ray Equations for a Stationary Medium

The ray tracing in a stationary medium is relatively simple compared to the case of the moving medium. For v = 0, as depicted in Figure 2.3, the direction of ray tangent coincides with the direction of wavefront normal. The magnitude of ray velocity vector|v^ray| equals to the local sound-speed c and its direction depends on the wavefront normal vector n. Set v = 0 in Equation (2.5), the eikonal equation for a stationary medium is

|∇τ(x)|² = 1

c². (2.24)

This nonlinear ordinary diﬀerential equation can be reduced to linear ordinary diﬀerential equations by transforming to the ray coordinate with arclength s as an

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rays

wavefronts source

Figure 2.3: Wavefronts and rays in a stationary medium.

independent parameter. Then x(s) and∇τ(s) satisfy dx

ds = c∇τ, (2.25)

d

ds∇τ = −1

c²∇c. (2.26)

The ray-tracing equations become [13]:

dr

ds = cξ(s), (2.27)

dz

ds = cζ(s), (2.28)

dξ

ds = −1 c²

∂c

∂r, (2.29)

dζ

ds = −1 c²

∂c

∂z, (2.30)

where (r, z) is the ray trajectory in the range-depth plane, and (cξ, cζ) = ∇τ the ray tangent vector. For a given launching angle θ₀, the initial conditions are

r(0) = 0, z(0) = z0, ξ(0) = cos θ0

c₀ , ζ(0) = sin θ0

c0

,

where (0, z₀) is the source position, and c₀ is the sound speed at the source position.

Then, the travel time t can be calculated along a ray:

t =

�

L

ds

c(s), (2.31)

where L is the total ray length.

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2.4 Eﬀective Sound Speed Approaches

A common approach to account for current effects is to use an effective sound speed c_eff, which is to add the current directly to the sound speed profile, which interprets current effects in terms of an effective sound speed profile (ESSP). This approach is often used in the literatures [25, 26].

The non-rigorous nature of introducing ceff is because the refractive index of a moving medium is not as simple as stationary medium. When introducing effective sound speed, the real moving medium is often replaced by a hypothetical stationary medium with sound speed equal to effective sound speed. By this approach, the general ray equations in a stationary medium is applicable, which simplify the treatment of ray-tracing in a moving medium. And because the typical current velocity is much less than sound speed in ocean, the sound propagation is dominate by sound speed, this approach seems to be persuasive.

Generally, the eﬀective sound speed ceﬀ is defined as [20]

ceﬀ = c + vR, (2.32)

where c is the real sound speed, vR= v· n is the component of the current velocity along the direction of propagation.

For some cases, e.g. the direction of sound propagation coincides with the direction of current. A further simplification can be made that

cs-eﬀ = c + v, (2.33)

where the current magnitude is added directly to the sound speed, without projecting to n.

1500 1501

0

50

(m/s)

Depth (m)

c_s-eff

0 1

v (m/s) 0

50

Depth (m)

1500 c (m/s) 0

50

Depth (m)

(a) (b) (c)

Figure 2.4: (a) Sound speed profile (b) Current profile (c) Simplified eﬀective sound speed profile.

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However, it is not physically correct to account for current effect using the effective sound speed approach. Figure 2.5(a) shows the ray velocity vector in a moving medium and Figure 2.5(b) shows how the effective sound speed approach accounts for the ray velocity. As depicted in Figure 2.5(a), the ray velocity in a moving medium is cn + v, the sum of two vectors. Whereas in the effective sound speed approach, the ray velocity is approximated by c + v · n. There are two differences between the effective sound speed approach and a correct ray-tracing for a moving medium. First, the wavefront normal vector n and the actual ray path cn + v are not collinear, which means that the effective sound speed cannot account for the shifting of wavefront by current. Second, the magnitude of ray velocity in a moving medium|cn + v| is different from the effective sound speed c + v · n.

It can be shown, via the Cauchy-Schwarz and triangular inequalities, that the relationship between average sound speeds calculated by correctly including the current eﬀects in the ray tracings and those by two simplified approaches is

c + v· n

� ��

c_eﬀ

≤ |cn + v| ≤ c + v� ��

c_s-eﬀ

. (2.34)

Equation (2.34) shows that the eﬀective sound speed approach will under-estimate the actual average speed along ray path, whereas the simplified eﬀective sound speed approach will over-estimate it.

xp v

cn

cn + v

cn

Horizontal xp

v v · n

c

(a) (b)

Figure 2.5: Schematic plots of the ray velocity vector modeled by (a) moving- medium ray-tracing and (b) the eﬀective sound speed approach.

Introducing effective sound speed has greatly simplified the treatment of ray- tracing in a moving medium. However, the effective sound speed approach is not applicable for all conditions. Ostashev [20] had discussed for the conditions that the rigorous introduction of ceff is possible. It is valid only for short-range propagation.

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Section 4.4 shows the numerical results using eﬀective sound speed approach and discusses the modeling errors in ray proeperties such as ray length, average speed along ray and travel time for a moving medium.

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Chapter 3 Numerical Simulation

This chapter describes the details of numerical simulation, including the Cur- rent Ray-Tracing program, the environments used in the simulation, the validation of the Current Ray-Tracing program and perturbations of bathymetry.

3.1 Current Ray-Tracing Program

This section states the details of Current Ray-Tracing program. The computer program is written in double precision using matlab. The program is based on the ray equations for a moving medium, Equations (2.20)–(2.23) [Section 2.2]. Given the source position and launching angle θ0 as the initial conditions, the ray trajectory and travel time can be obtained by integrating Equations (2.20)–(2.23). A higher- order Runge-Kutta method with variable step size (ode45 solver) is used to perform the range integration. The derivatives of depth, wavefront normal and travel time with respect to range are calculated at each step.

3.1.1 Integration of the Diﬀerential Equations

The ode45 solver [10] in matlab is used to perform the integration of Equa- tions (2.20)–(2.23). The ode45 solver is based on an explicit Runge-Kutta(4,5) formula, the Dormand-Prince pair [4], which combines a fourth order method and a fifth order method. Given a diﬀerential equation written as:

y^� = f (x, y), (3.1)

y(xn) = yn. (3.2)

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To predict a new value yn+1 by a step size h, we first compute four auxiliary quan- tities k1, k2, k3, k4 and use them to compute yn+1

k1 = hf (xn, yn), (3.3)

k2 = hf (xn+ h

2, yn+k1

2), (3.4)

k3 = hf (xn+ h

2, yn+k2

2), (3.5)

k4 = hf (xn+ h, yn+ k3), (3.6) yn+1 = yn+ 1

6(k1+ 2k2+ 2k3+ k4). (3.7) The modified Runge-Kutta varies the step size h at each step to ensure the error is small enough. The error tolerance |e^k| is controlled by two parameters: relative tolerance RelT ol and absolute tolerance AbsT ol, as follows:

|e^k| ≤ max(RelT ol × |y^k|, AbsT ol) (3.8) The relative and absolute tolerances are set as 1× 10⁻⁹ in the program for the consideration of accuracy and computational eﬃciency.

3.1.2 Sound Speed and Horizontal Current Velocity

The sound speed and horizontal current velocity of the waveguide vary in depth and range. These environmental parameters are given in gridded data. The values of sound speed and horizontal current velocity within a grid cell are obtained using cubic spline interpolation. The diﬀerentiation terms �_∂c

∂r, _∂z^∂c, ^∂v_∂r, & ^∂v_∂z�

in Equa- tions (2.21) and (2.22) are evaluated using central diﬀerence.

3.1.3 Boundaries and Reflection

The top of the medium is assumed to be pressure-release boundary, and the bottom is assumed to be hard boundary. The sea surface is assumed to be flat. For an irregular bottom, the bathymetry is given in a regular grid. The depth of an irregular bottom within a grid cell is calculated by linear interpolation.

At each step the program detects if a boundary reflection occurs in the next range step. If an impact has occurred, the initial ray position, travel time and θ for the next step are modified accordingly. The initial ray position and travel time are the values when the previous ray section hits the boundary. As for the initial θ, it is calculated using the reflection law to change the sign of θ after reflection. For a

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flat bottom, the sin θ and cos θ after reflection are modified as follows

sin θr = − sin θi (3.9)

cos θr = cos θi (3.10)

where θr is the new initial θ for the next step, and θi is θ before reflection. For the case of irregular bottom, Figure 3.1, according to the law of reflection, θi−α = θr+α, the sin θ and cos θ after reflection are modified as follows

sin θr = − sin(θi− 2α) (3.11)

cos θr = cos(θi− 2α) (3.12)

where α is the angle of the bottom with respect to the horizontal. The slope of the interface is interpolated from the neighboring grid points of the impact point.

r i

Boundary normal

Horizontal Incident ray

5HÁHFWHGUD\

Bottom

surface

Figure 3.1: Schematic of ray reflecting from a piecewise linear boundary

3.2 Environments

The environments used in the numerical study are featured with diﬀerent water depths, bottom topographies and sound-speed profiles. The first environment is an idealized environment which is called shallow water environment. The second environment is a realistic environment which is called modeled environment.

3.2.1 Shallow Water Environment

A shallow water sound channel is depicted in Figure 3.2(a). The channel depth is 100 m. The bottom is flat. Figure 3.2(b) shows the range-independent sound- speed profile used in the environment. The sound speed is taken to be constant

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(c = 1500 m/s) throughout the water. The horizontal current velocity will be modified for diﬀerent cases. The source is located at 80 m depth, and the distance of source-receiver is 10 km.

(a) (b)

1500 1501

0 20 40 60 80 100

c (m/s)

Depth (m)

0 2 4 6 8 10

0 20 40 60 80 100

Range (km)

Depth (m)

Figure 3.2: Shallow water environment: (a) Shallow water sound channel (b) Sound speed profile

3.2.2 Modeled Environment

A range dependent modeled sound channel is depicted in Fig. 3.3. The channel depth is around 1200 m. The source is located at a depth of 1040 m, and the distance of source-receiver is 48.08 km. The bathymetry obtained from a separate multi-beam survey [15] is used as the bottom topography data with a horizontal grid size length of 100 m. The bottom is featured by irregular surface with a seamount of 300 m height and 30 km width. The HYCOM ocean model data provided by the Naval Research Laboratory [2] is adopted for range-dependent sound-speed profiles and current profiles in this case. The sound-speed profiles are constructed by temperature and salinity data from the HYCOM model. The range between profiles is around 9 km. The maximum current velocity is around 0.7 m/s. The waveguide can be devided into two horizontal layer. The upper layer (0 to 200 m depth) is featured with strong negative current where mean velocity is−0.35 m/s, the deep layer (200 to 1200 m depth) is featured with weak positive current where mean velocity is 0.01 m/s.

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0 10 20 30 40 0

200 400 600 800 1000 1200 1400

Depth (m)

v=0.5 m/s 1470 1510 1550

Sound speed (m/s)

Range (km)

Figure 3.3: Modeled environment: Measured bathymetry, sound-speed profiles (gray heavy line) and current profiles (arrow) from HYCOM model.

3.3 Validation with BELLHOP Ray-Tracing Pro- gram

The validation of the Current Ray-Tracing program is carried out by comparing the simulated results with these from the BELLHOP ray tracing program.

BELLHOP is a widely used and highly eﬃcient ray-tracing program, written by Michael Porter as part of the Acoustic Toolbox [24, 27]. The ray equations implemented in BELLHOP are Equations (2.27)–(2.30), and the travel time is obtained by Equation (2.31).

The ray equations in BELLHOP cannot account for the current effects, therefore the shallow water environment (Section 3.2.1) without current velocity is used in the validation. Figure 3.4(a) shows the ray paths obtained by the Current Ray- Tracing (red) and BELLHOP(blue), both ray paths are in good agreement. Fig- ure 3.4(b) shows the difference between boundary-bounce ranges obtained by Cur- rent Ray-Tracing and BELLHOP for launching angles from−15^◦ to +15^◦ (indicated by colors). We observe that the maximum range difference is less than 6 mm, which is sufficiently small at a range of 10 km. Figure 3.4(c) shows the difference in travel times obtained by Current Ray-Tracing program and BELLHOP. We observe that the maximum travel time difference is less than 3 ms. In general, the results obtained by the Current Ray-Tracing are consistent with those by BELLHOP for the environment without current effects.

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(b) (a)

ï

0

6x 10^ï

Bounce range (km)

Difference of bounce range (m)

ï

0

10

0 6 8 10

0

60 80 100

Depth (m)

Range (km)

ï ï ï 0 10

ï

0

V₀ (deg)

)t (ms)

(c)

Figure 3.4: (a) Ray paths obtained by Current Ray-Tracing program and BELLHOP for θ0 = 15^◦; (b) Difference in bounce ranges between ray paths obtained by two programs. Colors indicate different launching angles; (c) Difference in travel time obtained by two programs.

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3.4 Perturbations in the Bathymetry

The bathymetry is represented as sum of two terms,

b(r) = bave(r) + γ(r), (3.13)

where b(r) is the ocean depth as a function of range r. The term bave(r) represents the large-scale, average bathymetry and γ(r) represents the small-scale seafloor feature, i.e., seafloor roughness.

Assume the seafloor randomness γ(r) is a stationary Gaussian process, the randomness can be represented using height probability density function and correlation function. The height probability density function describes the spreading of heights about the mean seafloor height and the correlation function describes the variation of these heights along the sea floor. Assume γ(r) is mean zero and has the correlation function C(∆r) = E[γ(r)γ(r + ∆r)], realizations of the seafloor height is generated by making use of the spectral representation of the random process γ(r):

The height probability density function is Gaussian:

p(γ) = 1 σ√

2π exp

�

−(γ)² 2σ²,

�

(3.14) and the Gaussian power spectrum is

P (k) = σ²l₀² 4π exp

�

−k²l²₀ 4

�

(3.15) where l0 is the surface correlation length in the horizontal direction and σ the root mean square (RMS) height.

The technique of generating the artificial random surfaces based upon the wavenumber power spectrum is as follows. Given a one-dimensional power spectrum P (k), [11, 12]

1. Consider N incremental steps of length∆ r. Each step is given a random value hn based on Gaussian probability distribution. Note that this is a Gaussian white noise sequence and adjacent values are totally uncorrelated.

2. A discrete Fourier transform is taken of the random values. The Fourier coef- ficients are given by

Hk= ∆r

N�−1 n=0

hne^i2πnk/N.

Because the transform is taken of a Gaussian white noise sequence the Fourier spectrum will be flat (the amplitudes of the |H^k| will be equal).

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3. The resulting Fourier coeﬃcients Hk are filtered using the relation Yk =�

PkHk

where Pk represents the discrete form of the power spectrum P (k) with the sampling∆ k = _{N ∆r}^2π . The square-root is taken because the power spectral density is proportional to the amplitude squared.

4. An inverse discrete Fourier transform is taken to result in a sequence of number representing the rough surface:

yn= 1 N ∆r

N−1

�

k=0

Yke^−i2πkn/N.

The sequence generated by the above procedure has the prescribed power spectrum as the spectral coeﬃcients of the random process. In this study the RMS height σ is 1 m, and the correlation length l0 is 1000 m.

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Chapter 4 Results and Discussions

This chapter summarizes the results of the numerical simulation. In order to understand the current eﬀects via analytic formulae, we mainly discuss various ray properties for a fan of rays, instead of searching for eigen rays connecting at a receiver. According to Equations (2.20)–(2.22), the currents aﬀect the ray path in the following ways:

1. The ray displacement¹ due to constant current velocity v.

2. The ray refraction due to the current shear ^∂v_∂z and ^∂v_∂r.

The first effect comes from the relative motion of moving medium (for example, a boat in current), resulting in the change of the ray trajectory directly. The second effect comes from the spatial variation of the current, resulting in the change of the direction of wavefront normal. The first effect always exists in a moving medium, but the second effect exists only in the presence of the current shear.

In a stationary medium, wavefront propagates along the direction of wavefront normal, and thus the direction of ray tangent coincides with the direction of wavefront normal. In a moving medium, as depicted in Figure 2.1, while wavefront propagates along the direction of wavefront normal, a point on the wavefront will be also in a motion following the current velocity v, therefore the direction of ray tangent does not coincide with the direction of wavefront normal. If we compare the actual ray path in a moving medium with that in a stationary medium, the actual ray path would look like being oﬀset by the current along the current direction.

However, because v � c in the ocean, the ray displacement from the shifting eﬀect is not big enough to alter the ray path significantly.

1This is diﬀerent from the “ray displacement” used to describe the ray traveling along the water-seafloor interface.

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To examine the ray displacement without associating with refraction, we assume that the sound speed and current are both constant, the direction of the wavefront normal remains constant. Without refraction, the ray will propagates as a straight line.

4.1 Current Velocity

To obtain the analytical formulas, a few assumptions are made: 1. the current is in horizontal direction only, i.e., v = (v, 0), 2. the current velocity is constant throughout the water, i.e., ^∂v_∂r = ^∂v_∂z = 0, 3. the boundaries are flat. The eﬀects of current velocity v on ray properties, such as the boundary-bounce range, ray length, average along-ray speed and travel time, are investigated in this section. Here, we define a reference state as

cref ≡ c, vref ≡ 0,

which neglects the current velocity and models the ocean as a stationary medium.

The ray properties in the presence of current are compared with those in the reference state. The purpose of this section is to develop analytical formulas of ray properties perturbation in a simplified ocean waveguide and to understand how these ray properties in a moving medium diﬀer from those in the reference state.

Ray length perturbation

The current-induced ray displacement and ray refraction would result in the ray length perturbation. The ray length perturbation would directly aﬀect the travel time. Here, we will focus on the ray length perturbation due to ray displacement (in a moving media of constant current velocity).

As depicted in Figure 4.1, for a sound ray launching at the angle θ0, the ray path in the reference state is indicated by the black line, whereas the ray path in the presence of constant current velocity is indicated by the gray line. Note that θ₀ is the angle of wavefront normal at the initial position, which does not necessarily coincide with the angle of ray tangent.

In the reference state, the ray path (black line in Figure 4.1) reaches to the depth zref after traveling distance R. The total ray length can be expressed as

Lref = R

cos θ₀. (4.1)

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Reference state In the presence of current

Source

0

Wavefront normal

Figure 4.1: Schematic of ray lengths in the presence of current and in the reference state for a ray of launching angle θ₀.

In the presence of constant current velocity, at the same traveling distance R, due to ray displacement the ray (gray line in Figure 4.1) would reach to the depth zv. According to Equation (2.20), the relationship between zv and R is

zv

R = c sin θ₀

c cos θ0+ v, (4.2)

and the ray length Lv can be calculated as L_v = �

R²+ z_v²

≈ R

cos θ0

� 1− v

c sin²θ0

cos θ0

�

. (4.3)

to first order in v/c. The diﬀerence between Lv and Lref gives the perturbed ray length

∆L≡ L^v − L^ref =−vR

c tan²θ0. (4.4)

Equation (4.4) shows that the perturbed ray length is inversely proportional to current velocity and range in the presence of ray displacement.

Boundary-bounce range perturbation

For horizontal current, the ray displacement can be quantified by the boundary- bounce range perturbation. As depicted in Figure 4.2, for a source located at depth z with launching angle θ0, the ray path in the reference state is indicated by the black line, whereas the ray path in the presence of current is indicated by the gray line. Note that θ0 is the angle of wavefront normal at the initial position.

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Boundary

Reference state In the presence of current

Source

0

Wavefront normal

Figure 4.2: Schematic of boundary-bounce ranges in the presence of current and in the reference state for a launching angle θ₀.

In the reference state, the ray path (black line) first hits the boundary on range rref. In the presence of constant current velocity, the ray path (gray line) would first hits the boundary at range rv due to ray displacement. According to Equation (2.20), the relationship between z and rv is

z rv

= c sin θ0

c cos θ0+ v, (4.5)

The relationship between z and rref is z rref

= tan θ0, (4.6)

Solving Equations (4.5) and (4.6) gives the relationship between rv and rref

r_v = r_ref

� 1 + v

c 1 cos θ0

�

(4.7) The perturbed boundary-bounce range∆ r_bounce is defined as the diﬀerence between rv and rref

∆rbounce ≡ rv − rref = rref

�v c

1 cos θ0

�

. (4.8)

Equation (4.8) shows that the perturbed boundary-bounce range is proportional to current velocity and boundary-bounce range in the reference state in the presence of ray displacement.

Average along-ray speed perturbation

For a waveguide of constant sound speed and constant current in which ray propagates in a straight line, the average along-ray speed u is equal to the magnitude of

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the ray velocity vray,

u = |v^ray|.

In the reference state, vref = 0, therefore the average along-ray speed uref is equal to sound speed c. In the presence of current, the average along-ray speed uv

equals to the magnitude of the ray velocity vray as described in Equation (2.1), uv = �

(c cos θ + v)² + (c sin θ)²

= √

c²+ 2cv cos θ + v²

≈ c + v cos θ, (4.9)

to the first order in v/c. The diﬀerence between uv and uref gives the perturbed average along-ray speed

∆u≡ u^v− u^ref = v cos θ. (4.10)

Equation (4.10) shows that the perturbed average along-ray speed is equal to the current velocity projection on the direction of wavefront normal.

Travel time perturbation

The ray travel time depends on how long the ray path is, and how fast the propagation speed along the path. Therefore, the travel time perturbation is the combining eﬀect of perturbed ray length and perturbed along-ray speed.

The diﬀerence between the travel time in the presence of current, tv, and the travel time in the reference state, tref, defines the perturbed travel time

∆t ≡ t^v − t^ref,

= Lv

uv −Lref

uref

,

= Lref+ ∆L

c + v cos θ − Lref

c ,

≈ −Lrefv cos θ

c² +∆L

c −∆Lv cos θ

c² . (4.11)

to the first order in v/c. The first term of Equation (4.11) indicates the component due to perturbed average along-ray speed∆ u for ray length Lref, the second term is the component due to perturbed ray length∆ L, and the third term is negligible if

∆L is much less than Lref.

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Numerical results

The shallow water environment (Section 3.2.1) is used in this section. The current profiles is set to be constant throughout the water to exclude the eﬀect of current shear. 10 current profiles are used in this section. The current magnitude of these profiles varied from −1 m/s to 1 m/s with 0.2 m/s increment, as illustrated in Figure 4.3(a).

ï ï5 5

v (m/s)

Depth (m)

Figure 4.3: Horizontal current profiles of various constant current velocity. Black line indicates reference state, and colored lines indicate current profiles with diﬀerent current magnitudes.

Figure 4.4(a) shows the ray paths for θ0 = −10^◦ in the presence of current (colored lines) and in the reference state (black line). Note that there are 11 ray paths in this figure. However, these rays seem to be similar to each other because the diﬀerence between these ray paths is too small to notice in this scale. Thus, a close look at the ninth top bounce is shown in Figure 4.4(b). We see that the perturbed boundary-bounce range is proportional to current magnitude and the sign of∆ rbounce follows the current direction, i.e., a positive current velocity (red line) shows a positive∆ r_bounce. It is due to the ray path shifted by the positive current toward +r direction.

Figure 4.4(c) shows the perturbed boundary-bounce range∆ rbounce as a function of the boundary-bounce range in the reference state rref. It shows that the perturbed boundary-bounce range accumulates over the range. The numerical result is consistent with the analytic solution using Equation (4.8).

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0 1 2 3 4 5 6 7 8 9 10 0

20 40 60 80 100 120

/DXQFKLQJDQJOH ï^k

Range (km)

Depth (m)

0 1 2 3 4 5 6 7 8 9 10

ï

0 5 10

Perturbed boundary-bounce range (m)

Boundary-bounce range in the reference state (km)

9515 9520 9525 9530 9535 9540

0

0.2

0.4

0.6

0.8

1

Range (m)

Depth (m)

Figure 4.4: Ray paths for θ0 =−10^◦ in the ocean with diﬀerent current magnitudes.

(top) Full view; (middle) enlarged view at the ninth sea surface bounce; (bottom) perturbed boundary-bounce range versus boundary-bounce range in the reference state. . Black line indicates the ray path in the reference state and colored lines are the ray paths for constant current velocity of diﬀerent magnitudes. Solid line indicates the analytical solution and circle indicates the numerical results.

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Figure 4.5 plots the perturbed ray properties as a function of current velocity.

Figure 4.5(a) shows that the perturbed ray length∆ L is inversely proportional to current velocity. Figure 4.5(b) shows that the perturbed average along-ray speed is proportional to current velocity as expected. Figure 4.5(c) shows that the perturbed travel time is inversely proportional to current velocity.

(a)

(c) (b)

Perturbed travel time (ms)Perturbed ray length (m)Perturbed average speed (m/s)

ï ï

ï

ï ï

ï

ï ï

ï

v (m/s)

ï

Figure 4.5: Perturbed ray properties versus current velocity: (a) perturbed ray length; (b) perturbed average along-ray speed; (c) perturbed travel time. The color of line indicates launching angle of the results. In these plots diﬀerent launching angles are indicated by colors.

淺海環境中流速對高頻聲波傳播之研究

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Institute of Oceanography College of Science

National Taiwan University Master Thesis

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A Study of Ocean Current on High-Frequency Acoustic Propagation in Coastal Water Environments

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Leow Khang Yen

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Advisor: Chen-Fen Huang, Ph.D.

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July, 2013

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Abstract

Contents

List of Figures

Chapter 1 Introduction

1.1 Background and Motivation

1.2 A Concise Survey of Literature

1.3 Objectives

1.4 Scopes of the Thesis

Chapter 2 Theory

2.1 General Ray Theory in a Moving Medium

2.2 Ray Equations for a Moving Medium: Ocean

2.3 Ray Equations for a Stationary Medium

rays

wavefronts source

2.4 Eﬀective Sound Speed Approaches

Chapter 3

Numerical Simulation

3.1 Current Ray-Tracing Program

3.1.1 Integration of the Diﬀerential Equations

3.1.2 Sound Speed and Horizontal Current Velocity

3.1.3 Boundaries and Reflection

3.2 Environments

3.2.1 Shallow Water Environment

3.2.2 Modeled Environment

3.3 Validation with BELLHOP Ray-Tracing Pro- gram

3.4 Perturbations in the Bathymetry

Chapter 4

Results and Discussions

4.1 Current Velocity

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