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Linear segmentation of velocity profile in sea water
Journal of Sound and
Vibration (1989) 132(l), 161-163
LINEAR SEGMENTATION OF VELOCITY PROFILE IN SEA WATER 1.
INTRODUCTION
It is well known that the velocity of sound in sea water is not constant, and that this causes sound rays to travel in a curved path. The sound velocity depends on temperature, salinity and depth, and because of variations of these quantities from place to place and season to season, the velocity of sound in sea water varies appreciably with location and season. Hence different types of velocity profiles are encountered in practice. Some of them may be approximated by a linear velocity profile, but quite often it may not be possible to fit the experimental data with a single straight line profile without sacrificing accuracy. One way of improving accuracy in such cases is to use segmented linear velocity profiles: i.e., to use several linear profiles instead of a single linear profile [l]. One of the important aspects in the segmentation of velocity profiles is the way in which the segmentation is to be effected. One can think of segmentation with equal divisions on the velocity axis or, alternatively, with equal divisions on the depth axis. Also of interest is knowing how many segments are required to obtain a reasonably good accuracy. In what follows, the above aspects are studied for velocity profiles with positive and negative gradients, and having both concave and convex shapes. 2.
RAY
TRACING
FOR SEVERAL
CASES
OF SEGMENTATION
Ray paths for various profiles have been obtained with an HP2100/5451 B computer by the following procedure. Since the sound speed c is assumed to be linear with respect to the depth z, it may be written as c=c,+gz,
(1)
where co is the initial speed and g is the speed gradient. Then the horizontal distance x for a given z is obtained from the relation [l] x = c(cos 0,-cos
6)/(g sin 13,),
(2)
where the inclination angles of the rays, measured with respect to the vertical, are given by f3= sin-‘[(c sin 6,)/co].
(3)
With an initial angle 0, and an initial speed co, ray paths can be progressively developed on a computer. Ray paths have been obtained by using this procedure with the number of segments n = 2,5,10,20 and 50, and from that range values have been obtained. Obviously, if one segments the profile with a very large number of segments then the approximation will be better. Theoretically, an infinite number of segments would exactly represent the actual profile. Thus the ray path obtained with a larger n (in this case, n = 50) may be taken as reference and then compared with ray paths from smaller values of n. Table 1 shows the range values for different numbers of segments for various cases of velocity profiles with a concave shape. It may be observed from the second and third columns of Table 1 that the difference in the values of ranges for n = 2 and n = 10 is 7360 - 3086 = 4274 m and also the difference for n = 10 and n = 50 is 9114 - 7360 = 1754 m. For constant depth divisioning, however, the difference in the values of ranges for n = 2 161 0022460X/89/130161
+03 $03.00/O
@ 1989 Academic Ress Limited
162
LETTERS
TO
THE EDITOR
TABLE 1 Range values for diflerent number of segments for the case of a concave velocity profile Range Positive gradient
Negative gradient
Number of segments
Constant depth divisioning
Constant velocity divisioning
Constant depth divisioning
Constant velocity divisioning
2 5 10 20 50
9 126 10 773 10 989 11 126 11080
3 086 6 032 7 360 8 204 9 114
8 213 8 655 8 728 8 748 8 755
7 810 8 438 8641 8 730 8 771
and n = 10 is 10989-9126 = 1863 m, and also the difference for n = 10 and n =50 is 11080 - 10989 = 91 m only. Thus one can conclude that constant depth divisioning is better than constant velocity divisioning. The fact that there is not much difference between n = 10 and n = 50 for the constant depth divisioning leads to the very important result that one cannot improve the accuracy simply by increasing the number of segments. After a certain point, not much is gained by increasing the number of segments, and it is observed that even n = 5 gives reasonably good results in most of the cases. This is very important for any practical implementation, since it is desirable to have as small a number of segments as possible without losing much accuracy. Results for the case of negative gradient and concave profile shape are shown in the last two columns of Table 1. These results also lead to the same conclusions. In this case the rays will bend towards the bottom and hence bottom reflections have to be considered in measuring the range. Next, results for the case of positive and negative velocity profiles with convex shape are considered. The ranges obtained for a positive velocity gradient with convex shape for the case of equal depth divisioning and equal velocity divisioning are shown in Table 2. It may be observed for constant depth divisioning that the difference in the values of ranges for n = 2 and n = 10 is 3014- 1211= 1803 m, and also the difference for n = 10 and n = 50 is 1211- 857 = 354 m. However, for constant velocity divisioning the difference 2
TABLE
Range values for di$erent number of segments for the case of a convex velocity projile Range Negative gradient
Positive gradient Number of segments 2 5 10 20 50
Contant depth divisioning ’
3 014 1751 1211 860 857
Constant velocity divisioning
Constant depth divisioning
Constant velocity divisioning
1444 993 914 870 868
5 888 5531 5 430 5 386 5 363
5 700 5 425 5 376 5 361 5 356
LETTERS
TO THE
EDITOR
163
in the values of ranges for n = 2 and n = 10 is 1444-914= 530 m, and the difference for n = 10 and n = 50 is 914 - 868 = 46 m only. Thus it is evident that in the case of convex profiles constant velocity divisioning is better than constant depth divisioning. Results for the case of negative gradient shown in the last two columns of Table 2 also lead to the same conclusion. Many realistic ocean profiles will have a velocity profile which has both concave and convex shapes. In view of the results obtained above it is obvious that only constant depth divisioning or only constant velocity divisioning will not lead to the most accurate result, The best way is to use constant depth divisioning for the convex shape and constant velocity divisioning for the concave shape. 3.
CONCLUSIONS
Two important questions, the method of segmenting a realistic ocean velocity profile and the number of segments to be used for a reasonably accurate result, have been considered. It is found that equal depth divisioning gives good results for profiles with a concave shape, and equal velocity divisioning gives good results for profiles with a convex shape. Thus, if there is a profile with both convex and concave shapes only one type of divisioning will not lead to the most accurate result. It is also observed that only a small number of segments (provided that a proper method of divisioning is chosen) will give a reasonably good accuracy. Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560012, India
D.
NARAYANA
Dm-r
(Receioed 23 Nooember 1988) REFERENCE 1. C. B. OFFICER
1958 Introduction to the Theory of Sound Transmission. New York:
McGraw-Hill.
Vibration (1989) 132(l), 161-163
LINEAR SEGMENTATION OF VELOCITY PROFILE IN SEA WATER 1.
INTRODUCTION
It is well known that the velocity of sound in sea water is not constant, and that this causes sound rays to travel in a curved path. The sound velocity depends on temperature, salinity and depth, and because of variations of these quantities from place to place and season to season, the velocity of sound in sea water varies appreciably with location and season. Hence different types of velocity profiles are encountered in practice. Some of them may be approximated by a linear velocity profile, but quite often it may not be possible to fit the experimental data with a single straight line profile without sacrificing accuracy. One way of improving accuracy in such cases is to use segmented linear velocity profiles: i.e., to use several linear profiles instead of a single linear profile [l]. One of the important aspects in the segmentation of velocity profiles is the way in which the segmentation is to be effected. One can think of segmentation with equal divisions on the velocity axis or, alternatively, with equal divisions on the depth axis. Also of interest is knowing how many segments are required to obtain a reasonably good accuracy. In what follows, the above aspects are studied for velocity profiles with positive and negative gradients, and having both concave and convex shapes. 2.
RAY
TRACING
FOR SEVERAL
CASES
OF SEGMENTATION
Ray paths for various profiles have been obtained with an HP2100/5451 B computer by the following procedure. Since the sound speed c is assumed to be linear with respect to the depth z, it may be written as c=c,+gz,
(1)
where co is the initial speed and g is the speed gradient. Then the horizontal distance x for a given z is obtained from the relation [l] x = c(cos 0,-cos
6)/(g sin 13,),
(2)
where the inclination angles of the rays, measured with respect to the vertical, are given by f3= sin-‘[(c sin 6,)/co].
(3)
With an initial angle 0, and an initial speed co, ray paths can be progressively developed on a computer. Ray paths have been obtained by using this procedure with the number of segments n = 2,5,10,20 and 50, and from that range values have been obtained. Obviously, if one segments the profile with a very large number of segments then the approximation will be better. Theoretically, an infinite number of segments would exactly represent the actual profile. Thus the ray path obtained with a larger n (in this case, n = 50) may be taken as reference and then compared with ray paths from smaller values of n. Table 1 shows the range values for different numbers of segments for various cases of velocity profiles with a concave shape. It may be observed from the second and third columns of Table 1 that the difference in the values of ranges for n = 2 and n = 10 is 7360 - 3086 = 4274 m and also the difference for n = 10 and n = 50 is 9114 - 7360 = 1754 m. For constant depth divisioning, however, the difference in the values of ranges for n = 2 161 0022460X/89/130161
+03 $03.00/O
@ 1989 Academic Ress Limited
162
LETTERS
TO
THE EDITOR
TABLE 1 Range values for diflerent number of segments for the case of a concave velocity profile Range Positive gradient
Negative gradient
Number of segments
Constant depth divisioning
Constant velocity divisioning
Constant depth divisioning
Constant velocity divisioning
2 5 10 20 50
9 126 10 773 10 989 11 126 11080
3 086 6 032 7 360 8 204 9 114
8 213 8 655 8 728 8 748 8 755
7 810 8 438 8641 8 730 8 771
and n = 10 is 10989-9126 = 1863 m, and also the difference for n = 10 and n =50 is 11080 - 10989 = 91 m only. Thus one can conclude that constant depth divisioning is better than constant velocity divisioning. The fact that there is not much difference between n = 10 and n = 50 for the constant depth divisioning leads to the very important result that one cannot improve the accuracy simply by increasing the number of segments. After a certain point, not much is gained by increasing the number of segments, and it is observed that even n = 5 gives reasonably good results in most of the cases. This is very important for any practical implementation, since it is desirable to have as small a number of segments as possible without losing much accuracy. Results for the case of negative gradient and concave profile shape are shown in the last two columns of Table 1. These results also lead to the same conclusions. In this case the rays will bend towards the bottom and hence bottom reflections have to be considered in measuring the range. Next, results for the case of positive and negative velocity profiles with convex shape are considered. The ranges obtained for a positive velocity gradient with convex shape for the case of equal depth divisioning and equal velocity divisioning are shown in Table 2. It may be observed for constant depth divisioning that the difference in the values of ranges for n = 2 and n = 10 is 3014- 1211= 1803 m, and also the difference for n = 10 and n = 50 is 1211- 857 = 354 m. However, for constant velocity divisioning the difference 2
TABLE
Range values for di$erent number of segments for the case of a convex velocity projile Range Negative gradient
Positive gradient Number of segments 2 5 10 20 50
Contant depth divisioning ’
3 014 1751 1211 860 857
Constant velocity divisioning
Constant depth divisioning
Constant velocity divisioning
1444 993 914 870 868
5 888 5531 5 430 5 386 5 363
5 700 5 425 5 376 5 361 5 356
LETTERS
TO THE
EDITOR
163
in the values of ranges for n = 2 and n = 10 is 1444-914= 530 m, and the difference for n = 10 and n = 50 is 914 - 868 = 46 m only. Thus it is evident that in the case of convex profiles constant velocity divisioning is better than constant depth divisioning. Results for the case of negative gradient shown in the last two columns of Table 2 also lead to the same conclusion. Many realistic ocean profiles will have a velocity profile which has both concave and convex shapes. In view of the results obtained above it is obvious that only constant depth divisioning or only constant velocity divisioning will not lead to the most accurate result, The best way is to use constant depth divisioning for the convex shape and constant velocity divisioning for the concave shape. 3.
CONCLUSIONS
Two important questions, the method of segmenting a realistic ocean velocity profile and the number of segments to be used for a reasonably accurate result, have been considered. It is found that equal depth divisioning gives good results for profiles with a concave shape, and equal velocity divisioning gives good results for profiles with a convex shape. Thus, if there is a profile with both convex and concave shapes only one type of divisioning will not lead to the most accurate result. It is also observed that only a small number of segments (provided that a proper method of divisioning is chosen) will give a reasonably good accuracy. Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560012, India
D.
NARAYANA
Dm-r
(Receioed 23 Nooember 1988) REFERENCE 1. C. B. OFFICER
1958 Introduction to the Theory of Sound Transmission. New York:
McGraw-Hill.