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 corre-lation 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 and the Gaussian power spectrum is
P (k) = σ2l02 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 spec-trum 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
hnei2πnk/N.
Because the transform is taken of a Gaussian white noise sequence the Fourier spectrum will be flat (the amplitudes of the |Hk| will be equal).
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 ∆r2π . 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.
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 wave-front 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.
To examine the ray displacement without associating with refraction, we as-sume 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)
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 Lv = �
R2+ zv2
≈ 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≡ Lv − Lref =−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.
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
rv = rref The perturbed boundary-bounce range∆ rbounce is defined as the difference between rv and rref
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 propa-gates in a straight line, the average along-ray speed u is equal to the magnitude of
the ray velocity vray,
u = |vray|.
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≡ uv− uref = 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 propaga-tion speed along the path. Therefore, the travel time perturbapropaga-tion 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 ≡ tv − tref,
= 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.
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∆ rbounce. 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 func-tion of the boundary-bounce range in the reference state rref. It shows that the perturbed boundary-bounce range accumulates over the range. The numerical re-sult is consistent with the analytic solution using Equation (4.8).
0 1 2 3 4 5 6 7 8 9 10 0
20 40 60 80 100 120
/DXQFKLQJDQJOH ïk
Range (km)
Depth (m)
0 1 2 3 4 5 6 7 8 9 10
ï
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.
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
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.
Figure 4.6 plots the perturbed ray properties as a function of launching angle.
Figure 4.6(a) shows that the perturbed ray length increases as the launching angle becomes steeper. This is because the difference between ray tangent and wavefront normal is larger for a steeper launching ray. The black line indicates the result of the reference state. Note that the ray tangent coincides with wavefront normal in the reference state. The numerical result (line) is consistent with the analytic solution (circle) using Equation (4.4).
(a)
(c) (b)
0 5 10 15
ï
0
0 5 10 15
ï
ï
0
1
0 5 10 15
ï
0 5
e0 (deg)
Perturbed travel time (ms)Perturbed ray length (m)Perturbed average speed (m/s)
Figure 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.
For the perturbed average along-ray speed [Figure 4.6(b)], its sign is consistent with the sign of current velocity, whereas its magnitude decreases slightly for steeper rays. The numerical result (line) is consistent with the analytic solution (circle) using Equation (4.10). For the perturbed travel time [Figure 4.6(c)], its sign is opposite to the sign of current, whereas its magnitude increases slightly for steeper rays.
The analytical solution for travel time perturbation, Equation (4.11), states that the perturbed travel time∆ t can be decomposed into the contribution from perturbed average along-ray speed∆ u and that from perturbed ray length∆ L.
Figure 4.7 shows the numerical simulation result for perturbed travel time and the contributions of different factors when v = 1 m/s. We see that first the perturbed travel time from numerical simulation (line) is identical to that from the analytical result (circle). Second,∆ u makes most contribution to the travel time perturbation (asterisk). Third, the angular variation of perturbed travel time is due to∆ L (plus).
ï
ï
v = 1 m/s
5 15
ï
ï
e (deg) ï
6t (ms)
Figure 4.7: Perturbed travel time for v = 1 m/s. Line indicates numerical results.
Circle indicates analytical solution. Asterisk indicates the component due to per-turbed average along-ray speed. Plus indicates the component due to perper-turbed ray length. Note that the y-axis is divided into two sections.