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Effects of earth curvature on two-dimensional ray tracing in underwater acoustics
Applied Acoustics 57 (1999) 163±177
Eects of earth curvature on two-dimensional ray tracing in underwater acoustics Jianguo Yan* Applied Sciences and Technology, Inc., PO Box 833152, Miami, FL 33283, USA
Abstract The objective of this paper is to investigate the eects of earth curvature on two-dimensional ray tracing. This is done by comparing numerical solutions of dierent ray equations. From numerical results, it is concluded that (1) at very long ranges, a 2-D ray path constructed without earth curvature correction may have less turning points than an actual ray; (2) computed travel times of eigenrays may be longer than measurements, and this travel-time error will increase with launch angle; (3) the error of launch angles of eigenrays decreases with launch angle; and (4) both the travel-time error and the launch-angle error increase with propagation range. Conclusions (2) and (3) are consistent with a published result in ocean acoustic tomography. Within a 1000 km range, the eect of earth ¯attening is not signi®cant, and spherical earth is a good approximation. A set of 2-D ray equations with spherical earth curvature correction is presented in this paper, which can considerably extend the range of validity of 2-D ray tracing while being as simple as the traditional 2-D equations. # 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction Ray theory is important in underwater acoustics. To construct ray paths, various ray equations can be used. The traditional ray equations are two-dimensional (2-D), which are not valid for long-range transmissions. For long-range transmissions, earth curvature and horizontal sound speed gradient cannot be ignored. These are taken into account by HARPO [1], horizontal ray equations [2,3], and the threedimensional (3-D) ray equations in ellipsoidal coordinates [4]. Qualitatively, we know that earth curvature and horizontal sound speed gradient are important for a long-range propagation problem. But, how long is ``long-range''? What happens if these are not considered in ray tracing? Quantitatively, these are not * Tel.:+1-305-992-3119;fax:+1-954-433-3825;e-mail:[email protected] 0003-682X/99/$Ðsee front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S000 3-682X(98)0005 2-8
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J. Yan/Applied Acoustics 57 (1999) 163±177
quite clear. Mercer et al. [5] have demonstrated that at 1000 km range, horizontal refraction produces signi®cant travel-time variation. The objective of this paper is to quantify the eects of earth curvature. This is done by the comparison of twodimensional numerical solutions of dierent ray equations. In the ®rst Section, we propose a benchmark problem for the numerical comparison. The ray equations used for the numerical ray tracing are given in the second section. In the third section, we present numerical results, and analyze the eects of earth curvature. Conclusions are given in the fourth section. 2. A benchmark problem For the comparison of numerical solutions of dierent ray equations, we propose a benchmark problem as follows. The acoustic source is located at 30 N latitude and at a depth of 1.2 km. The receiver is placed at a depth of 1 km and 1000 km north of the source. At the locations of the source and the receiver, the sound speed pro®les are estimated using Munk's canonical expression [6] c z ca 1 " eÿ ÿ 1;
1
where 2 r ÿ ra =B:
2
Here c is the sound speed; r is the depth; ca and ra are the sound speed and the depth at channel axis, respectively; B is the scale depth; " is the perturbation coecient; and is a dimensionless distance. The canonical parameters at the locations of the source and the receiver are shown in Table 1. Between the source and the receiver, the sound speed is expressed as C r; Cs r
Cr^ r ÿ Cs r
1000
3
where is the range, and subscripts s and r denote the source and the receiver, respectively. This model is similar to that of Georges et al. [7] and Mercer, et al. [5], Table 1 Canonical parameters at the source and the receiver Parameter
Source
Receiver
ca ra B "
1.495 km/s 1.2 km 1.2 km 0.005
1.485 km/s 0.9 km 1 km 0.0057
J. Yan/Applied Acoustics 57 (1999) 163±177
165
the distinction being that it does not take into account the three-dimensional Gaussian perturbation. 3. Ray equations This section presents the ray equations that are used to investigate the eects of earth curvature. 3.1. Traditional 2-D equations Let us start with the 3-D ray equations in ellipsoidal coordinates [4]: d cos cos ds ÿr
4
dl cos sin ds ÿ r cos
5
dr sin ds
6
d cos tan sin 1 1 sin sin cos ÿ ds ÿr ÿr ÿr sin @ cos @ ln N ÿ ÿ r @ ÿ r cos @l cos
7
2 d sin cos2 ÿ cos ds ÿr ÿr sin cos @ sin sin @ @ ÿ cos ln N ÿ ÿ r @ ÿ r cos @l @r
8
and
where ÿ
ÿ a 1 ÿ e2 3=2; 1 ÿ e2 sin2
9
and a ÿ 1=2 2 1 ÿ e sin2
10
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J. Yan/Applied Acoustics 57 (1999) 163±177
are the radius of curvature and the radius of curvature in prime vertical, respectively [8]. In Eqs. (4)±(8), is geographic latitude; l is longitude, east of Greenwich being positive; r is ocean depth, downward positive; is azimuth, measured clockwise from north; is grazing angle; a is the semimajor radius of the reference ellipsoid; e is the eccentricity of the ellipsoid; and N 1=c, where c is sound speed. The travel time can be estimated using dt N; ds and the propagation range is given by the following equation: s 2 2 d cos cos sin ds ÿr ÿr
11
12
The derivation of this equation has been presented in a previous paper [9]. These equations take into account the curvature of an ellipsoidal earth and horizontal refraction. Therefore, they are currently the most accurate ray equations in ocean acoustics. Other ray equations can be considered as the approximations to these equations. If horizontal refraction and the earth curvature are neglected (this implies that depends only on the ocean depth r, !1 and !1), the 3-D Eqs. (4±8) will reduce to the traditional 2-D ray equations: dr sin ds
13
d cos dN ds N dr
14
and
which are identical to Eqs. (5.1.55) and (5.1.58) in the reference [10]. We can rewrite these equations by using the range r as the independent variable to replace the path length s. Considering d =ds cos in this case, the traditional 2-D Eqs. (13) and (14) can be rewritten as dr tan d
15
d 1 dN d N dr
16
and
J. Yan/Applied Acoustics 57 (1999) 163±177
167
The travel time equation becomes dt N d cos
17
3.2. Taking into account the curvature of a spherical earth Now, we take into account the curvature of a spherical earth. Let us assume that N still depends only on the ocean depth r, but the earth curvature is not ignored. For a spherical earth, we have ÿ r ÿ r Rc ÿ r
18
where Rc is the radius of the spherical earth. Consequently, the 3-D Eqs. (4)±(8) will reduce to Eq. (13) and d cos dN cos ÿ ds N dr Rc ÿ r
19
The relation between the range and the path length, in this case, can be derived as follows. Using Eqs. (18) and (12), we can write d Rc cos Rc ÿ r ds
20
Using Eq. (20), we rewrite Eq. (13) and (19) as dr tan 1 ÿ r=Rc ; d
21
d 1 dN 1 1 ÿ r=Rc ÿ : d N dr Rc ÿ r
22
and
The travel time equation in this case is dt N r 1ÿ : d cos Rc
23
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J. Yan/Applied Acoustics 57 (1999) 163±177
Eqs. (21)±(23) are the 2-D ray equations with the correction of spherical earth curvature. We have not seen any published reference to these equations. We shall show that these equations will considerably extend the range of validity of 2-D ray tracing, while they are as simple as the traditional 2-D equations. 3.3. Eect of earth ¯attening For the benchmark problem we considered, we have 0: Thus, using Eqs. (4)± (8), (11) and (12), we get d 1 d ;
24
dr r tan 1 ÿ ; d
25
d 1 1 ÿ tan dN r dN ÿ 1ÿ d N d dr 1 1 dN r dN ÿ tan 1ÿ ÿ ; N d dr
26
where use was made of d d according to Geodesy [8]. Eq. (11) can be rewritten as dt N r 1ÿ 27 d cos Eqs. (24)±(25) are the special case, for the benchmark problem, of the 3-D ray equations in ellipsoidal coordinates. We use these equations to provide accurate numerical solutions for the benchmark problem. 4. Numerical solutions and discussions 4.1. Method In this section, we use three sets of ray equations to produce the numerical solutions for the benchmark problem stated in Section 1. The three sets of ray equations are 1. Eqs. (15)±(17), as the traditional 2-D equations;
J. Yan/Applied Acoustics 57 (1999) 163±177
169
2. Eqs. (21)±(23), as the 2-D equations with spherical curvature correction; and 3. Eqs. (24)±(27), as the accurate equations that take into account the curvature of an ellipsoidal earth. These equations are numerically integrated by using the fourth-order RungerKutta method [11]. The step-size is 0.01 km. To calculate for solving Eqs. (24)± (27), WGS84 reference ellipsoid (a=6378.137 km, and e=0.08181919084262149) is used. For the spherical earth, Rc =6374 km. The comparisons of the numerical solutions are presented as follows. 4.2. Dierence between ray paths Figs. 1 ±3 are the comparison between the ray paths with launch grazing angles 2 , 4 , and 10 , respectively. In these ®gures, the ray paths (T) are constructed by using the traditional 2-D equations. The ray paths (S) are produced by using the ray equations with spherical curvature correction. The ray paths (E) are the accurate results computed using Eqs. (24)±(27), which take into account the curvature of ellipsoidal earth. From Fig. 1, we can see that the ray path (T) is apparently stretched, compared with the path (S) and path (E). This dierence increases with the range, and is increasing as the launch angle increases (see Figs. 2 and 3). This is an eect of earth curvature. From these ®gures, we can infer that at a very long range, a ray constructed using the traditional 2-D equations will have less turning points than an actual ray. There is no apparent dierence between the ray path (S) and the path (E) in Figs. 1± 3, suggesting that within 1000 km range, spherical earth is a good assumption.
4.3. Depth error Let us assume that the numerical solutions produced by Eqs. (24)±(27) are the ``accurate solutions,'' and that the other solutions are the approximations. This assumption should be reasonable, considering that Eqs. (24)±(27) are equivalent to Eqs. (4)±(8), which are valid for very long ranges. Then, we can estimate the errors of ray equations by making comparison between an approximate solution and the ``accurate solution.'' The numerical results are illustrated in Figs. 4±5. From Eq. (4), we can see that the traditional 2-D equations produced large errors (the maximum depth error is about 1.57 km). When the spherical earth curvature is incorporated into the ray equations, the errors are considerably reduced (the maximum depth error is about 0.065 km,) as shown in Fig. 5. In both cases, the errors increased with the propagation range; and the larger the launch grazing angle, the larger the errors. Now, let us discuss: How long is long range? By ``long range,'' we usually mean that beyond such a range the traditional 2-D models are not valid. In this case, there is a poor comparison between the model forecast and experimental data. Obviously, the range of validity of models depends on the accuracy required for forecast. Since sound speed is accurate to 0.0001 km/s, [12] the accuracy of model forecast should
170
J. Yan/Applied Acoustics 57 (1999) 163±177
Fig. 1. Ray-path comparison. Launch angle is 2 . T, S, and E denote the ray path constructed using the traditional 2-D ray equations, the 2-D equations with spherical earth curvature, and the equations with ellipsoidal earth curvature, respectively.
be the same as that of sound speed. If this is the required accuracy, the range of validity of the traditional 2-D equations is less than 20 km, as seen in Fig. 4. The depth error might cause errors in a computed pressure ®eld. How eigenray parameters are aected is discussed as follows.
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171
Fig. 2. Ray-path comparison. Launch angle is 4 . T, S, and E denote the ray path constructed using the traditional 2-D ray equations, the 2-D equations with spherical earth curvature, and the equations with ellipsoidal earth curvature, respectively.
4.4. Errors of eigenray parameters We now analyze the errors of eigenray parameters. We use the Successive Shooting method [13] to construct eigenrays. A ray is determined as an eigenray, if it passes within 0.0005 km of the receiver's depth. The parameters of some sample eigenrays at various ranges are listed in Tables 2±6. In these tables, T, S, and E
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J. Yan/Applied Acoustics 57 (1999) 163±177
Fig. 3. Ray-path comparison. Launch angle is 10 . T, S, and E denote the ray path constructed using the traditional 2-D ray equations, the 2-D equations with spherical earth curvature, and the equations with ellipsoidal earth curvature, respectively.
denote the numerical solutions produced by the traditional 2-D equations, the 2-D equations with spherical curvature, and the 2-D equations with ellipsoidal curvature, respectively. Let us assume that the numerical results (denoted by E) are the ``accurate'' solutions. Then we estimate the errors of other numerical results (T and S) by subtracting each from the ``accurate'' one (E). The eigenray identi®er in Tables 2±6 signi®es the total number of ray-path turning points between the source
J. Yan/Applied Acoustics 57 (1999) 163±177
173
Fig. 4. Depth-error of ray paths computed using the traditional 2-D ray equations. The numbers in the legend denote launch angles.
Fig. 5. Depth-error of ray paths computed using the 2-D ray equations with spherical earth curvature. The numbers in the legend denote launch angles.
and the receiver. Its sign indicates whether the ray left the source at an angle above (ÿ) or below (+) the horizontal. Table 2 is a comparison of the parameters of the eigenrays at 1000 km range. As seen from this table, the travel times (T) calculated using the traditional 2-D equations are 100±200 ms longer than the accurate solutions (E). When the spherical earth curvature is incorporated into the 2-D equations, the travel-time errors are
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J. Yan/Applied Acoustics 57 (1999) 163±177
Fig. 6. Travel-time error of the eigenrays computed using the traditional 2-D ray equations.
Fig. 7. Launch-angle error of the eigenrays computed using the traditional 2-D ray equations.
reduced to 1 ms and less (see E±S in Table 2). This suggests that these travel time errors are caused mainly by ignoring the earth curvature, and that the 2-D equations with spherical earth curvature correction are very accurate in this case. This conclusion is true also for the eigenrays at the ranges 100, 200, 300, and 600 km, respectively, as can be seen from Tables 3±6.
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175
Table 2 Comparison of the parameters of eignrays at 1000 Km range Eigenray identi®er
43 42 41 40 39 38 37 36 35 34
Grazing angle at source, deg
Travel time to receiver, s
T
S
E
E-T
E-S
T
S
E
E-T
E-S
0.7276 2.1328 3.6481 4.8483 5.5806 6.6111 7.1431 8.1262 8.5601 9.5493
1.1836 2.8008 4.0469 5.1968 5.8896 6.9053 7.4219 8.4019 8.8282 9.8203
1.1875 2.8066 4.05452 5.2051 5.8994 6.9150 7.4324 8.4122 8.8394 9.8311
0.4599 0.6738 0.4061 0.3596 0.13188 0.3039 0.2893 0.2860 0.2793 0.2818
0.0039 0.0058 0.0073 0.0083 0.0098 0.0097 0.0105 0.0103 0.0112 0.0108
671.0543 671.0525 671.0219 670.9737 670.8971 670.7821 670.6608 670.4613 670.2945 669.9891
670.9444 670.9377 670.8983 670.8409 670.7557 670.6291 670.4990 670.2855 670.1092 669.7866
670.9441 670.9374 670.8979 670.8405 670.7552 670.6287 670.4986 670.2850 670.1086 669.7860
0.110 0.115 0.124 0.133 0.142 0.153 0.162 0.176 0.186 0.203
0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.001 0.001 0.001
Table 3 Comparison of the parameters of eigenrays at 600 Km range Eigenray identi®er 24 23 22 21 20 19
Grazing angle at source, deg
Travel time to receiver, s
T
S
E
E-T
E-S
T
S
E
E-T
E-S
2.2148 4.1748 6.2432 7.1294 8.9150 9.5647
2.8691 4.5688 6.5669 7.4268 9.2048 9.8469
2.8730 4.5742 6.5728 7.4331 9.2109 9.8535
0.6582 0.3994 0.3296 0.3037 0.2959 0.2888
0.0039 0.0054 0.0059 0.0063 0.0061 0.0066
402.0846 402.0573 401.9626 401.8645 401.1617 401.4437
402.0119 401.9764 401.8697 401.7627 401.4986 401.3152
402.0116 401.9762 401.8694 401.7624 401.4986 401.3148
0.073 0.081 0.093 0.102 0.337 0.129
0.000 0.000 0.000 0.000 0.000 0.000
Table 4 Comparison of the parameters of eigenrays at 300 Km range Eigenray identi®er
12 11 10 >9
Grazing angle at source, deg
Travel time to receiver, s
T
S
E
E-T
E-S
T
S
E
E-T
E-S
0.7813 3.8477 8.0303 9.3457
0.9844 4.2891 8.3384 9.6372
0.9844 4.2930 8.3418 9.6406
0.2031 0.4453 0.3115 0.2949
0.0000 0.0039 0.0034 0.0034
200.8504 200.8412 200.6934 200.5552
200.8142 200.7995 200.6372 200.4891
200.8140 200.7993 200.6370 200.4889
0.036 0.042 0.056 0.066
0.000 0.000 0.000 0.000
Fig. 6 is a comparison of travel-time errors of the eigenrays at various ranges, and Fig. 7 shows the corresponding launch-angle errors. We can see that both the traveltime error and the launch-angle error of the eigenrays constructed using the traditional 2-D equations increased with the range. At a given range, the travel-time error increased with the launch angle, but the launch-angle error decreased as the launch angle increased. The former suggests that early arriving rays, predicted using traditional
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J. Yan/Applied Acoustics 57 (1999) 163±177
Table 5 Comparison of the Parameters of eigenrays at 200 Km range Eigenray identi®er 8 7 ÿ7
Grazing angle at source, deg T
S
E
E-T
Travel time to receiver, s E-S
T
S
E
E-T
E-S
1.0391 1.1875 1.1875 0.1484 0.0000 133.8588 133.8341 133.8341 0.025 0.000 4.5664 4.9609 4.9629 0.3965 0.0020 133.8467 133.8168 133.8168 0.030 0.000 ÿ8.0771 ÿ8.4131 ÿ8.4170 0.3399 0.0039 133.7760 133.7405 133.7402 0.036 0.000
Table 6 Comparison of the Parameters of eigenrays at 100 Km range Eigenray identi®er 4 3
Grazing angle at source, deg
Travel time to receiver,s
T
S
E
E-T
E-S
T
S
E
1.6875 6.7520
1.8750 7.0605
1.8750 7.0625
0.1875 0.3105
0.0000 0.0020
66.9081 66.8763
66.8954 66.8574
66.8953 66.8573
E-T
E-S
0.013 0.000 0.019 0.000
2-D models, may have larger travel-time error than late arrival. The latter implies that the late arrival may have larger intensity error considering the intensity depends on cos [14]. This inference appears in good agreement with a 300-km path west of Bermuda [15]. 5. Conclusions From the numerical results, we can conclude that if a traditional 2-D ray model is used without earth curvature correction; 1. a computed ray path will be stretched, 2. computed travel times of eigenrays may be longer than measurements, and this travel-time error will increase with the launch angle, 3. the launch-angle error of eigenrays will decrease with the launch angle, and 4. Both the travel-time error and the launch-angle error increase with propagation range. The ®rst conclusion implies that at very long ranges, a 2-D ray path constructed without earth curvature correction may have less turning points than an actual path. The second and the third conclusions may explain the dierences, in travel time and intensity, between an acoustic measurement and a prediction using the traditional 2D equations. Within 1000 km range, the eect of earth ¯attening is not signi®cant, and spherical earth is a good approximation for the tested problem. The 2-D ray equations with spherical earth curvature correction, which we present in this paper, can considerably extend the range of validity of 2-D ray tracing, while as simple as the traditional 2-D equations.
J. Yan/Applied Acoustics 57 (1999) 163±177
177
Acknowledgements This work was supported by the Oce of Naval Research (Grant No. N00014-9510443) while the author was visiting Florida International University. References [1] Jones RM, Riley JP, Georges TM. HARPOÐA versatile three-dimensional Hamiltonian ray-tracing program for acoutic waves in an ocean with irregular bottom. Boulder, CO: NOAA Report, Environmetal Research Laboratories, 1986. [2] Munk WH, O'Reilly WC, Reid JL. Australia-Bermuda sound transmission experiment (1960) revisited. J Phys Oceanogr 1991;18:1876±98. [3] Heaby KD, Kuperman WA, McDonald BE. Perth-Bermuda sound propagation (1960): Adiabatic mode interpretation. J Acoust Soc Am 1991;90:2586±94. [4] Yan J, Yen K. A derivation of three-dimensional ray equations in ellipsoidal coordinates. J acount Soc Am 1538;97:1995. [5] Mercer JA, Felton WJ, Booker JR. Three-dimensional eigenrays through ocean mesoscale structure J Acoust Soc Am. 78 1985;157±63. [6] Munk WH. Sound channel in an exponentially straiti®ed ocean with applications to SOFAR. J Acoust Soc Am 220;55(2):1974. [7] Georges TM, Jones RM, Riley JP. Simulating ocean acoustic tomography measurements with Hamiltonian ray tracing. IEEE J Oceanic Eng 1986;OE-1 1:58±71. [8] Bomford G, Geodesy. London: Oxford U. P., 1971, pp. 107±112, 562±6. [9] Yan J, Yen K. Constructing three-dimensional ray paths for underwater sound from Heard Island to Ascension Island. In Brebbia CA et al. Computational acoustics and its environmental applications II Southampton: Computational Mechanics Publications, 1997: 41±8. [10] Boyles CA. Acoustic waveguides: applications to oceanic science. New York: Wiley, 1984, 191 pp. [11] Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical recipes, the art of scienti®c Computing. New York: Cambridge U. P., 1988: 554. [12] Clay CS, Medwin H. Acoustical oceanography, principles and applications. New York: John Wiley and Sons, 1977: 4. [13] Carnahan, B. Applied numerical methods. New York: Wiley, 1969: 405±8. [14] Jensen FB, Kuperman WA, Porter MB, Schmodt, H. Computational ocean acoustics. New York: AIP Press, 1994: 158±9. [15] Munk W, Worcester P, Wunsch C. Ocean acoustic tomography. New York: Cambridge University Press, 1995: 10.
Eects of earth curvature on two-dimensional ray tracing in underwater acoustics Jianguo Yan* Applied Sciences and Technology, Inc., PO Box 833152, Miami, FL 33283, USA
Abstract The objective of this paper is to investigate the eects of earth curvature on two-dimensional ray tracing. This is done by comparing numerical solutions of dierent ray equations. From numerical results, it is concluded that (1) at very long ranges, a 2-D ray path constructed without earth curvature correction may have less turning points than an actual ray; (2) computed travel times of eigenrays may be longer than measurements, and this travel-time error will increase with launch angle; (3) the error of launch angles of eigenrays decreases with launch angle; and (4) both the travel-time error and the launch-angle error increase with propagation range. Conclusions (2) and (3) are consistent with a published result in ocean acoustic tomography. Within a 1000 km range, the eect of earth ¯attening is not signi®cant, and spherical earth is a good approximation. A set of 2-D ray equations with spherical earth curvature correction is presented in this paper, which can considerably extend the range of validity of 2-D ray tracing while being as simple as the traditional 2-D equations. # 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction Ray theory is important in underwater acoustics. To construct ray paths, various ray equations can be used. The traditional ray equations are two-dimensional (2-D), which are not valid for long-range transmissions. For long-range transmissions, earth curvature and horizontal sound speed gradient cannot be ignored. These are taken into account by HARPO [1], horizontal ray equations [2,3], and the threedimensional (3-D) ray equations in ellipsoidal coordinates [4]. Qualitatively, we know that earth curvature and horizontal sound speed gradient are important for a long-range propagation problem. But, how long is ``long-range''? What happens if these are not considered in ray tracing? Quantitatively, these are not * Tel.:+1-305-992-3119;fax:+1-954-433-3825;e-mail:[email protected] 0003-682X/99/$Ðsee front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S000 3-682X(98)0005 2-8
164
J. Yan/Applied Acoustics 57 (1999) 163±177
quite clear. Mercer et al. [5] have demonstrated that at 1000 km range, horizontal refraction produces signi®cant travel-time variation. The objective of this paper is to quantify the eects of earth curvature. This is done by the comparison of twodimensional numerical solutions of dierent ray equations. In the ®rst Section, we propose a benchmark problem for the numerical comparison. The ray equations used for the numerical ray tracing are given in the second section. In the third section, we present numerical results, and analyze the eects of earth curvature. Conclusions are given in the fourth section. 2. A benchmark problem For the comparison of numerical solutions of dierent ray equations, we propose a benchmark problem as follows. The acoustic source is located at 30 N latitude and at a depth of 1.2 km. The receiver is placed at a depth of 1 km and 1000 km north of the source. At the locations of the source and the receiver, the sound speed pro®les are estimated using Munk's canonical expression [6] c z ca 1 " eÿ ÿ 1;
1
where 2 r ÿ ra =B:
2
Here c is the sound speed; r is the depth; ca and ra are the sound speed and the depth at channel axis, respectively; B is the scale depth; " is the perturbation coecient; and is a dimensionless distance. The canonical parameters at the locations of the source and the receiver are shown in Table 1. Between the source and the receiver, the sound speed is expressed as C r; Cs r
Cr^ r ÿ Cs r
1000
3
where is the range, and subscripts s and r denote the source and the receiver, respectively. This model is similar to that of Georges et al. [7] and Mercer, et al. [5], Table 1 Canonical parameters at the source and the receiver Parameter
Source
Receiver
ca ra B "
1.495 km/s 1.2 km 1.2 km 0.005
1.485 km/s 0.9 km 1 km 0.0057
J. Yan/Applied Acoustics 57 (1999) 163±177
165
the distinction being that it does not take into account the three-dimensional Gaussian perturbation. 3. Ray equations This section presents the ray equations that are used to investigate the eects of earth curvature. 3.1. Traditional 2-D equations Let us start with the 3-D ray equations in ellipsoidal coordinates [4]: d cos cos ds ÿr
4
dl cos sin ds ÿ r cos
5
dr sin ds
6
d cos tan sin 1 1 sin sin cos ÿ ds ÿr ÿr ÿr sin @ cos @ ln N ÿ ÿ r @ ÿ r cos @l cos
7
2 d sin cos2 ÿ cos ds ÿr ÿr sin cos @ sin sin @ @ ÿ cos ln N ÿ ÿ r @ ÿ r cos @l @r
8
and
where ÿ
ÿ a 1 ÿ e2 3=2; 1 ÿ e2 sin2
9
and a ÿ 1=2 2 1 ÿ e sin2
10
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J. Yan/Applied Acoustics 57 (1999) 163±177
are the radius of curvature and the radius of curvature in prime vertical, respectively [8]. In Eqs. (4)±(8), is geographic latitude; l is longitude, east of Greenwich being positive; r is ocean depth, downward positive; is azimuth, measured clockwise from north; is grazing angle; a is the semimajor radius of the reference ellipsoid; e is the eccentricity of the ellipsoid; and N 1=c, where c is sound speed. The travel time can be estimated using dt N; ds and the propagation range is given by the following equation: s 2 2 d cos cos sin ds ÿr ÿr
11
12
The derivation of this equation has been presented in a previous paper [9]. These equations take into account the curvature of an ellipsoidal earth and horizontal refraction. Therefore, they are currently the most accurate ray equations in ocean acoustics. Other ray equations can be considered as the approximations to these equations. If horizontal refraction and the earth curvature are neglected (this implies that depends only on the ocean depth r, !1 and !1), the 3-D Eqs. (4±8) will reduce to the traditional 2-D ray equations: dr sin ds
13
d cos dN ds N dr
14
and
which are identical to Eqs. (5.1.55) and (5.1.58) in the reference [10]. We can rewrite these equations by using the range r as the independent variable to replace the path length s. Considering d =ds cos in this case, the traditional 2-D Eqs. (13) and (14) can be rewritten as dr tan d
15
d 1 dN d N dr
16
and
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167
The travel time equation becomes dt N d cos
17
3.2. Taking into account the curvature of a spherical earth Now, we take into account the curvature of a spherical earth. Let us assume that N still depends only on the ocean depth r, but the earth curvature is not ignored. For a spherical earth, we have ÿ r ÿ r Rc ÿ r
18
where Rc is the radius of the spherical earth. Consequently, the 3-D Eqs. (4)±(8) will reduce to Eq. (13) and d cos dN cos ÿ ds N dr Rc ÿ r
19
The relation between the range and the path length, in this case, can be derived as follows. Using Eqs. (18) and (12), we can write d Rc cos Rc ÿ r ds
20
Using Eq. (20), we rewrite Eq. (13) and (19) as dr tan 1 ÿ r=Rc ; d
21
d 1 dN 1 1 ÿ r=Rc ÿ : d N dr Rc ÿ r
22
and
The travel time equation in this case is dt N r 1ÿ : d cos Rc
23
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Eqs. (21)±(23) are the 2-D ray equations with the correction of spherical earth curvature. We have not seen any published reference to these equations. We shall show that these equations will considerably extend the range of validity of 2-D ray tracing, while they are as simple as the traditional 2-D equations. 3.3. Eect of earth ¯attening For the benchmark problem we considered, we have 0: Thus, using Eqs. (4)± (8), (11) and (12), we get d 1 d ;
24
dr r tan 1 ÿ ; d
25
d 1 1 ÿ tan dN r dN ÿ 1ÿ d N d dr 1 1 dN r dN ÿ tan 1ÿ ÿ ; N d dr
26
where use was made of d d according to Geodesy [8]. Eq. (11) can be rewritten as dt N r 1ÿ 27 d cos Eqs. (24)±(25) are the special case, for the benchmark problem, of the 3-D ray equations in ellipsoidal coordinates. We use these equations to provide accurate numerical solutions for the benchmark problem. 4. Numerical solutions and discussions 4.1. Method In this section, we use three sets of ray equations to produce the numerical solutions for the benchmark problem stated in Section 1. The three sets of ray equations are 1. Eqs. (15)±(17), as the traditional 2-D equations;
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2. Eqs. (21)±(23), as the 2-D equations with spherical curvature correction; and 3. Eqs. (24)±(27), as the accurate equations that take into account the curvature of an ellipsoidal earth. These equations are numerically integrated by using the fourth-order RungerKutta method [11]. The step-size is 0.01 km. To calculate for solving Eqs. (24)± (27), WGS84 reference ellipsoid (a=6378.137 km, and e=0.08181919084262149) is used. For the spherical earth, Rc =6374 km. The comparisons of the numerical solutions are presented as follows. 4.2. Dierence between ray paths Figs. 1 ±3 are the comparison between the ray paths with launch grazing angles 2 , 4 , and 10 , respectively. In these ®gures, the ray paths (T) are constructed by using the traditional 2-D equations. The ray paths (S) are produced by using the ray equations with spherical curvature correction. The ray paths (E) are the accurate results computed using Eqs. (24)±(27), which take into account the curvature of ellipsoidal earth. From Fig. 1, we can see that the ray path (T) is apparently stretched, compared with the path (S) and path (E). This dierence increases with the range, and is increasing as the launch angle increases (see Figs. 2 and 3). This is an eect of earth curvature. From these ®gures, we can infer that at a very long range, a ray constructed using the traditional 2-D equations will have less turning points than an actual ray. There is no apparent dierence between the ray path (S) and the path (E) in Figs. 1± 3, suggesting that within 1000 km range, spherical earth is a good assumption.
4.3. Depth error Let us assume that the numerical solutions produced by Eqs. (24)±(27) are the ``accurate solutions,'' and that the other solutions are the approximations. This assumption should be reasonable, considering that Eqs. (24)±(27) are equivalent to Eqs. (4)±(8), which are valid for very long ranges. Then, we can estimate the errors of ray equations by making comparison between an approximate solution and the ``accurate solution.'' The numerical results are illustrated in Figs. 4±5. From Eq. (4), we can see that the traditional 2-D equations produced large errors (the maximum depth error is about 1.57 km). When the spherical earth curvature is incorporated into the ray equations, the errors are considerably reduced (the maximum depth error is about 0.065 km,) as shown in Fig. 5. In both cases, the errors increased with the propagation range; and the larger the launch grazing angle, the larger the errors. Now, let us discuss: How long is long range? By ``long range,'' we usually mean that beyond such a range the traditional 2-D models are not valid. In this case, there is a poor comparison between the model forecast and experimental data. Obviously, the range of validity of models depends on the accuracy required for forecast. Since sound speed is accurate to 0.0001 km/s, [12] the accuracy of model forecast should
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Fig. 1. Ray-path comparison. Launch angle is 2 . T, S, and E denote the ray path constructed using the traditional 2-D ray equations, the 2-D equations with spherical earth curvature, and the equations with ellipsoidal earth curvature, respectively.
be the same as that of sound speed. If this is the required accuracy, the range of validity of the traditional 2-D equations is less than 20 km, as seen in Fig. 4. The depth error might cause errors in a computed pressure ®eld. How eigenray parameters are aected is discussed as follows.
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Fig. 2. Ray-path comparison. Launch angle is 4 . T, S, and E denote the ray path constructed using the traditional 2-D ray equations, the 2-D equations with spherical earth curvature, and the equations with ellipsoidal earth curvature, respectively.
4.4. Errors of eigenray parameters We now analyze the errors of eigenray parameters. We use the Successive Shooting method [13] to construct eigenrays. A ray is determined as an eigenray, if it passes within 0.0005 km of the receiver's depth. The parameters of some sample eigenrays at various ranges are listed in Tables 2±6. In these tables, T, S, and E
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Fig. 3. Ray-path comparison. Launch angle is 10 . T, S, and E denote the ray path constructed using the traditional 2-D ray equations, the 2-D equations with spherical earth curvature, and the equations with ellipsoidal earth curvature, respectively.
denote the numerical solutions produced by the traditional 2-D equations, the 2-D equations with spherical curvature, and the 2-D equations with ellipsoidal curvature, respectively. Let us assume that the numerical results (denoted by E) are the ``accurate'' solutions. Then we estimate the errors of other numerical results (T and S) by subtracting each from the ``accurate'' one (E). The eigenray identi®er in Tables 2±6 signi®es the total number of ray-path turning points between the source
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Fig. 4. Depth-error of ray paths computed using the traditional 2-D ray equations. The numbers in the legend denote launch angles.
Fig. 5. Depth-error of ray paths computed using the 2-D ray equations with spherical earth curvature. The numbers in the legend denote launch angles.
and the receiver. Its sign indicates whether the ray left the source at an angle above (ÿ) or below (+) the horizontal. Table 2 is a comparison of the parameters of the eigenrays at 1000 km range. As seen from this table, the travel times (T) calculated using the traditional 2-D equations are 100±200 ms longer than the accurate solutions (E). When the spherical earth curvature is incorporated into the 2-D equations, the travel-time errors are
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Fig. 6. Travel-time error of the eigenrays computed using the traditional 2-D ray equations.
Fig. 7. Launch-angle error of the eigenrays computed using the traditional 2-D ray equations.
reduced to 1 ms and less (see E±S in Table 2). This suggests that these travel time errors are caused mainly by ignoring the earth curvature, and that the 2-D equations with spherical earth curvature correction are very accurate in this case. This conclusion is true also for the eigenrays at the ranges 100, 200, 300, and 600 km, respectively, as can be seen from Tables 3±6.
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Table 2 Comparison of the parameters of eignrays at 1000 Km range Eigenray identi®er
43 42 41 40 39 38 37 36 35 34
Grazing angle at source, deg
Travel time to receiver, s
T
S
E
E-T
E-S
T
S
E
E-T
E-S
0.7276 2.1328 3.6481 4.8483 5.5806 6.6111 7.1431 8.1262 8.5601 9.5493
1.1836 2.8008 4.0469 5.1968 5.8896 6.9053 7.4219 8.4019 8.8282 9.8203
1.1875 2.8066 4.05452 5.2051 5.8994 6.9150 7.4324 8.4122 8.8394 9.8311
0.4599 0.6738 0.4061 0.3596 0.13188 0.3039 0.2893 0.2860 0.2793 0.2818
0.0039 0.0058 0.0073 0.0083 0.0098 0.0097 0.0105 0.0103 0.0112 0.0108
671.0543 671.0525 671.0219 670.9737 670.8971 670.7821 670.6608 670.4613 670.2945 669.9891
670.9444 670.9377 670.8983 670.8409 670.7557 670.6291 670.4990 670.2855 670.1092 669.7866
670.9441 670.9374 670.8979 670.8405 670.7552 670.6287 670.4986 670.2850 670.1086 669.7860
0.110 0.115 0.124 0.133 0.142 0.153 0.162 0.176 0.186 0.203
0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.001 0.001 0.001
Table 3 Comparison of the parameters of eigenrays at 600 Km range Eigenray identi®er 24 23 22 21 20 19
Grazing angle at source, deg
Travel time to receiver, s
T
S
E
E-T
E-S
T
S
E
E-T
E-S
2.2148 4.1748 6.2432 7.1294 8.9150 9.5647
2.8691 4.5688 6.5669 7.4268 9.2048 9.8469
2.8730 4.5742 6.5728 7.4331 9.2109 9.8535
0.6582 0.3994 0.3296 0.3037 0.2959 0.2888
0.0039 0.0054 0.0059 0.0063 0.0061 0.0066
402.0846 402.0573 401.9626 401.8645 401.1617 401.4437
402.0119 401.9764 401.8697 401.7627 401.4986 401.3152
402.0116 401.9762 401.8694 401.7624 401.4986 401.3148
0.073 0.081 0.093 0.102 0.337 0.129
0.000 0.000 0.000 0.000 0.000 0.000
Table 4 Comparison of the parameters of eigenrays at 300 Km range Eigenray identi®er
12 11 10 >9
Grazing angle at source, deg
Travel time to receiver, s
T
S
E
E-T
E-S
T
S
E
E-T
E-S
0.7813 3.8477 8.0303 9.3457
0.9844 4.2891 8.3384 9.6372
0.9844 4.2930 8.3418 9.6406
0.2031 0.4453 0.3115 0.2949
0.0000 0.0039 0.0034 0.0034
200.8504 200.8412 200.6934 200.5552
200.8142 200.7995 200.6372 200.4891
200.8140 200.7993 200.6370 200.4889
0.036 0.042 0.056 0.066
0.000 0.000 0.000 0.000
Fig. 6 is a comparison of travel-time errors of the eigenrays at various ranges, and Fig. 7 shows the corresponding launch-angle errors. We can see that both the traveltime error and the launch-angle error of the eigenrays constructed using the traditional 2-D equations increased with the range. At a given range, the travel-time error increased with the launch angle, but the launch-angle error decreased as the launch angle increased. The former suggests that early arriving rays, predicted using traditional
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Table 5 Comparison of the Parameters of eigenrays at 200 Km range Eigenray identi®er 8 7 ÿ7
Grazing angle at source, deg T
S
E
E-T
Travel time to receiver, s E-S
T
S
E
E-T
E-S
1.0391 1.1875 1.1875 0.1484 0.0000 133.8588 133.8341 133.8341 0.025 0.000 4.5664 4.9609 4.9629 0.3965 0.0020 133.8467 133.8168 133.8168 0.030 0.000 ÿ8.0771 ÿ8.4131 ÿ8.4170 0.3399 0.0039 133.7760 133.7405 133.7402 0.036 0.000
Table 6 Comparison of the Parameters of eigenrays at 100 Km range Eigenray identi®er 4 3
Grazing angle at source, deg
Travel time to receiver,s
T
S
E
E-T
E-S
T
S
E
1.6875 6.7520
1.8750 7.0605
1.8750 7.0625
0.1875 0.3105
0.0000 0.0020
66.9081 66.8763
66.8954 66.8574
66.8953 66.8573
E-T
E-S
0.013 0.000 0.019 0.000
2-D models, may have larger travel-time error than late arrival. The latter implies that the late arrival may have larger intensity error considering the intensity depends on cos [14]. This inference appears in good agreement with a 300-km path west of Bermuda [15]. 5. Conclusions From the numerical results, we can conclude that if a traditional 2-D ray model is used without earth curvature correction; 1. a computed ray path will be stretched, 2. computed travel times of eigenrays may be longer than measurements, and this travel-time error will increase with the launch angle, 3. the launch-angle error of eigenrays will decrease with the launch angle, and 4. Both the travel-time error and the launch-angle error increase with propagation range. The ®rst conclusion implies that at very long ranges, a 2-D ray path constructed without earth curvature correction may have less turning points than an actual path. The second and the third conclusions may explain the dierences, in travel time and intensity, between an acoustic measurement and a prediction using the traditional 2D equations. Within 1000 km range, the eect of earth ¯attening is not signi®cant, and spherical earth is a good approximation for the tested problem. The 2-D ray equations with spherical earth curvature correction, which we present in this paper, can considerably extend the range of validity of 2-D ray tracing, while as simple as the traditional 2-D equations.
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Acknowledgements This work was supported by the Oce of Naval Research (Grant No. N00014-9510443) while the author was visiting Florida International University. References [1] Jones RM, Riley JP, Georges TM. HARPOÐA versatile three-dimensional Hamiltonian ray-tracing program for acoutic waves in an ocean with irregular bottom. Boulder, CO: NOAA Report, Environmetal Research Laboratories, 1986. [2] Munk WH, O'Reilly WC, Reid JL. Australia-Bermuda sound transmission experiment (1960) revisited. J Phys Oceanogr 1991;18:1876±98. [3] Heaby KD, Kuperman WA, McDonald BE. Perth-Bermuda sound propagation (1960): Adiabatic mode interpretation. J Acoust Soc Am 1991;90:2586±94. [4] Yan J, Yen K. A derivation of three-dimensional ray equations in ellipsoidal coordinates. J acount Soc Am 1538;97:1995. [5] Mercer JA, Felton WJ, Booker JR. Three-dimensional eigenrays through ocean mesoscale structure J Acoust Soc Am. 78 1985;157±63. [6] Munk WH. Sound channel in an exponentially straiti®ed ocean with applications to SOFAR. J Acoust Soc Am 220;55(2):1974. [7] Georges TM, Jones RM, Riley JP. Simulating ocean acoustic tomography measurements with Hamiltonian ray tracing. IEEE J Oceanic Eng 1986;OE-1 1:58±71. [8] Bomford G, Geodesy. London: Oxford U. P., 1971, pp. 107±112, 562±6. [9] Yan J, Yen K. Constructing three-dimensional ray paths for underwater sound from Heard Island to Ascension Island. In Brebbia CA et al. Computational acoustics and its environmental applications II Southampton: Computational Mechanics Publications, 1997: 41±8. [10] Boyles CA. Acoustic waveguides: applications to oceanic science. New York: Wiley, 1984, 191 pp. [11] Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical recipes, the art of scienti®c Computing. New York: Cambridge U. P., 1988: 554. [12] Clay CS, Medwin H. Acoustical oceanography, principles and applications. New York: John Wiley and Sons, 1977: 4. [13] Carnahan, B. Applied numerical methods. New York: Wiley, 1969: 405±8. [14] Jensen FB, Kuperman WA, Porter MB, Schmodt, H. Computational ocean acoustics. New York: AIP Press, 1994: 158±9. [15] Munk W, Worcester P, Wunsch C. Ocean acoustic tomography. New York: Cambridge University Press, 1995: 10.