This section investigates the effects of deterministic irregular bottom on the ray length perturbation. Sections 4.1 and 4.2 examines the ray length perturbation in a waveguide with the flat bottom. However, in a real waveguide, the roughness of bathymetry might increase the magnitude of ray displacement and leads to the distortion of ray path in the presence of current. Therefore, the relationship between ray length perturbation and current velocity or launching angle might be much more complicated than what are shown in Figures 4.6(a) and 4.5(a).
The shallow water environment (Section 3.2.1) is used in this section, but the seafloor is modeled as an irregular bottom. The method of generating an irregular bottom is introduced in Section 3.4. In Equation (3.13), the average bathymetry bave(r) is 100 m, and the small-scale bathymetric structure γ(r) is featured with RMS height σ = 1 m, and the correlation length l0 = 1000 m.
Monte Carlo simulations with 100 realizations of random bottom are used to estimate the statistics of perturbed ray length. For each realization, the ray path in the presence of constant current velocity is compared with that in the reference state (the same bottom but no current) to obtain the perturbed ray length.
Figure 4.12(a) shows the ray path in the reference state (black) and in the presence of constant current velocity (colored lines) for θ0 = 5◦ computed from a realization of seafloor. We see that the ray path for negative current velocity (blue) is different from that for positive current (red) after the fourth bottom-bounce (∼ 6.6 km). Figure 4.12(b) shows the enlarged view at the fourth bottom bounce. The ray paths for v = −1, −0.8 and −0.6 m/s hit the bottom whose local slope differs from the other rays (note that the depth of bottom is given in a regular grid, and the depth of bottom within a grid cell is calculated by linear interpolation, therefore the local slope is same within a grid cell, and it is different for cross-grid), resulting in a significantly different path for the latter part of the ray. Figure 4.13 shows the perturbed ray length at 10-km range for this realization (circles). Compared with the result from the flat-bottom (line), we see that the perturbed ray lengths of v =−1, −0.8 and −0.6 m/s are deviated significantly from the flat-bottom case.
The reason is that these rays interact with different local slope of bottom from the reference state [Figure 4.12(b)].
Figure 4.14 shows the dot plot of the perturbed ray length from 100 realizations for θ0 = 5◦ . Each dot is the perturbed ray length from a specific current velocity and a realization of seafloor. There are 1000 dots (points) on the plot. We see that most data points are scattered around the values predicted from the flat bottom,
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Figure 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.
but a few points show significant deviation (outliers). To summarize the statistics of these data, we try either mean & standard-deviation or the box-and-whisker plots [16] to show the distribution of the results from the Monte Carlo simulation.
Figure 4.15 shows the mean and the standard deviation estimated from the data shown in Figure 4.14. Circle indicates the mean, and line indicates one standard deviation. Due to the presence of outliers, e.g. 30-m raylength perturbation for positive current (red dots) in Figure 4.14, the mean and standard deviation may not be a representative summary statistics.
Figure 4.16 shows the box-and-whisker plot. The red line indicates the median, the bottom and top of the blue box are the first Q1 (25th percentiles) and third quartiles Q3 (75th percentiles), respectively. The length of the blue box is the interquartile range (IQR ≡ Q3 − Q1; the difference between the 75th and 25th percentiles). The black dashed line indicates the whisker. The ends of the whisker represent the lowest datum still within 1.5 IQR of the lower quartile (Q1+ 1.5 IQR), and the highest datum still within 1.5 IQR of the upper quartile (Q3 + 1.5 IQR).
The plus signs show the outliers that are defined as the data falling outside of the whisker. The IQR (blue box) and the whisker (black dashed line) indicates the
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Figure 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. Different colors are for different current magnitudes. Note that the y-axis is divided into two section.
spreading of the data. We see that the expected perturbed ray length (median; red line) agrees well with the results in flat bottom (brown line). The IQR (blue box) and the whisker (black dashed line) increase as the magnitude of current velocity increases. The reason is that with increasing current magnitude, one would expect large boundary-bounce range perturbation (see Section 4.1). When the rays interact with the bottom, the ray with larger current magnitude would more likely to hit different patches of bottom compared with the ray in the reference state.
Figure 4.17 shows the box-and-whisker plot of perturbed ray length versus launching angle for v = 1 m/s. We see that the median (red line) agrees well with the results in flat bottom (brown line), and the IQR (length of blue box) and the whisker (black dashed line) increase for steeper rays. The steeper rays interact with the bottom more frequently and thus the uncertainty is larger.
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Figure 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 real-izations.
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Figure 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.
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Figure 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.
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Figure 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.
4.4 Effective Sound Speed Approaches
This section considers different numerical approaches to account for the current effect and compare the ray properties including ray path, average along-ray speed and travel time. As introduced in Section 2.4, the effective sound speed approaches include a general form and a simplified form. The general form is referred to as effective sound speed approach (ESSP), as described in Equation (2.32),
ceff(z) = c(z) + vR(z),
which adds vR = v· n (the component of the current velocity in the direction of the wavefront normal n) to the sound speed c. The simplified form is referred to as simplified effective sound speed approach (SESSP), Equation (2.33),
cs-eff = c + v,
which adds the magnitude of current velocity to the sound speed, without projecting onto n.
The shallow water environment (Section 3.2.1) is used for the simulation in order to simplify the problem and to clarify the fundamental difference among the results obtained by ESSP, SESSP and the Current Ray-Tracing. The horizontal current velocity is 1 m/s and is uniform throughout the water.
Ray path
Figure 4.18(a) shows the actual ray path computed from the Current Ray-Tracing (black line) and those obtained by ESSP (gray line) and SESSP (asterisk sym-bols) approaches for θ0 = −2◦. We see that the ray paths obtained by ESSP and SESSP approaches are in good agreement with the actual ray path in this scale.
Figure 4.18(b) shows the enlarged view at the first sea-surface-bounce of ray path.
We see that the ray paths obtained by ESSP and SESSP approaches are different from the actual ray path. This is because the ray displacement (see Section 4.1) is not accounted by both ESSP and SESSP approaches. In either ESSP or SESSP ap-proach, the direction of ray tangent coincides with the direction of wavefront normal n. The ray path obtained by ESSP and SESSP approaches are consistent because in this case there is no ray refraction. In general, the difference between the actual ray path and those obtained by either ESSP or SESSP approach is very small as v � c.
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Figure 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.
Ray length, average along-ray speed and travel time
The ray properties difference between either ESSP or SESSP approach and the Current Ray-Tracing are calculated:
δL = LESSP− Lv (4.14)
δu = uESSP− uv (4.15)
δt = tESSP− tv (4.16)
where L is ray length, u is average along-ray speed, t is travel time, subscript ESSP denote ray properties obtained by either ESSP or SESSP approach, and subscript v denote ray properties obtained by the Current Ray-Tracing. Figure 4.19 shows (a) ray length, (b) average along-ray speed and (c) travel time difference between either ESSP or SESSP approach and the Current Ray-Tracing. Solid line indicates the difference of results between SESSP approach and the Current Ray-Tracing and dashed line is for the ESSP. The ray length difference from both approaches
increases for steeper launching rays [Figure 4.19(a)], since the discrepancy between the directions of ray tangent and wavefront normal is greater for steeper launching rays. The ray length differences of ESSP (dashed line) and SESSP (solid line) are the same as these two approaches predict the same ray path [Figure 4.18].
Figure 4.19(b) shows that the average along-ray speed difference: for the ESSP approach (dashed line), the difference doesn’t vary for different launching angles and is equal to zero; for the SESSP approach (solid line) the difference increases as launching angle becomes steeper. This is consistent with our previous discussion, the actual average along-ray speed is c + v cos θ [Equation (4.9)], the speed obtained by ESSP is c + v cos θ, and the average speed obtained by SESSP is c + v. For this simplified environment, we found that the relationship of the ray velocities modeled by two effective approaches and the Current Ray-Tracing is
c + v· n ≈ |cn + v| ≤ c + v. (4.17) which is consistent with Equation (2.34).
Figure 4.19(c) shows the travel time difference: for both approaches the differ-ences increase for steeper launching rays but with different magnitude. The travel time difference (error) comes from errors in both the ray length and the average along-ray speed. To understand the contribution of ray length difference and speed difference to travel time difference, the following equations are used:
δtL = δL
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δtu = −Lvδu
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where δtL is the component of travel time difference due to ray length difference and δtu is the component of travel time difference due to average along-ray speed difference. Figure 4.20(a)&(b) shows the travel time difference due to the ray length difference and average speed difference [Figure 4.19(a)&(b)], Figure 4.20(c) is same as Figure 4.19(c). For ESSP (dashed line), the travel time difference depends only on the ray length difference. For SESSP (solid line), the positive ray length difference results in a positive difference in travel time, whereas the positive speed difference gives a negative difference in travel time. Thus a smaller travel time difference is observed compared with the ESSP.
In general, the results obtained by ESSP and SESSP approaches for the angles of interest (small grazing angle) are in good agreement with those by the Current Ray-Tracing. In this simplified environment, either ESSP or SESSP approach can be applied to account for current effect.
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Figure 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.
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Figure 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.