DETERMINATION OF AVERAGE OCEAN DEPTHS
FROM BATHYMETRIC DATA
b y M. D i s h o n
D ep artm en t o f A p p lied M athem atics, T he W eizm a n n In stitu te
R ehovoth, Israel
1. Introduction
T he problem of d eterm in in g average ocean depths for a region o f the
w orld -ocean , or for all oceans togeth er, is en cou n tered in th eoretical
ocean ograp h y (su ch as d eterm in in g th e ocean tid es th eo r etica lly on the
b asis o f L a p l a c e ’ s tid al eq u a tio n s), w h ere n u m erical co m p u ta tio n s h ave to
be carried out. In th ese ca se s it is u su a lly requ ired to rep resen t th e actual
variable d ep th s by average v a lu es tak en over som e regu lar grid. It is the
p u rpose o f th is p aper to describe a m eth od o f co m p u tin g su ch average
d ep th s from b ath y m etric sou n d in gs m ade at irregu larly spaced station s
in the ocean. T he m eth od is applied in order to d eterm in e average d ep th s
for th e w orld oceans in a grid o f one degree.
2. Statem ent o f th e P roblem
A rough in sp ectio n o f available b ath ym etric ch arts w ill illu str a te th e
problem and its d ifficu lties. T he ch arts con tain d ep th s derived from
so u n d in g s m ade at iso la ted poin ts, as w e ll as con tou r lin e s d ed uced from
th e in d ivid u al sou n d in gs. T he coverage is by no m ean s u n ifo rm . A lon g th e
p opu lated coasts, and in the N orth A tlan tic O cean at o n e extrem e, the
coverage is rea so n a b ly d en se. In th e oth er extrem e, there are v a st region s
o f the A rctic and of th e Southern H em isp h ere O ceans w h ere, excep t for
p oin ts on sin g le track s, th ere are m an y on e-d egree sq u ares o f ocean area
w h ich are co m p letely em p ty o f sou n d in gs. It seem s th a t th is prob lem has
not yet been given proper atten tion in ocean ograp h ic literatu re.
T he approach to th e so lu tio n o f th is p rob lem su ggested here, d raw s
from an a n alogy in m eteorology, w h ere a sim ilar situ a tio n ex ists. T he
m eteorological a n a logy referred to is k n o w n as th e p rocess o f “ ob jective
an a ly sis ”. By th is, m eteo ro lo g ists u nd erstan d th e p rep aration and tr a n sfo r­
m ation of w id ely scattered elem en ts o f ob servation s, su ch a s p ressu re,
tem perature, w in d s, etc. in to in p u t data at regular grid -p oin ts for u se in
th e com p u tation o f w ea th er forecasts. F or a typ ical a n a ly sis o f th e N orthern
H em isphere, there are m an y m ore reporting sta tio n s over la n d areas th an
over the oceans, and the in com in g inform ation has to be transferred to a
regular grid on w h ich the eq u ations, describing the behaviour of the
atm osphere, are defined. Since u su a lly not m uch tim e is available from the
tim e o f receipt o f the data to the tim e of release of the forecast, the
m a th em a tical structure of the an a lysis has to be as sim p le as possible,
aim in g to achievc the greatest am ount of accuracy w ith the least am ount
o f com putation.
3. The Numerical Procedure
In line w ith th is analogy, the follow in g num erical procedure for the
com p u tation of average ocean depths has been developed.
j-2
i —I
j-l
„x
>
j+2
J
X
X
X
X
X
'
y
1
*
X
X
x x
X
X X XX
X
L+1
X
X
X
L+2
1
--------1>
X
—
<
'
* )
<1
F i g . 1. — The Numerical Grid,
x Observation points (mostly along tracks)
o Grid points (where averages are required)
Initial Step : Assignment of Starting Values h t at every Grid Point.
A rectangular grid is su perim p osed on the ocean area under con sidera­
tion where d ep th -sou n d in gs are given at irregularly spaced observation
p oin ts. An in itia l trial valu e h t, let us say zero, is assigned to every grid
p oint (fig. 1). T hen th e com p u tation w ill go on in a 3-step repetitive cycle
o f scans through the area. At the first step, an interpolated depth h int is
calcu lated at every observation p oin t from values of the 4 surrounding
grid points (fig. 2 ) :
j
!
A y(htj j
i
hi
j)
-)- A i(hi + i. j — hi j)
+ A i A /(/i; + i, j + i — hi + i. j — K j + i + K P
d)
T hen the difference is form ed betw een the interpolated and the observed
depth at th is point :
hc — h 0j)8
h^nf
(2)
H ere h„ht! and h lHt denote the observed and interpolated depth at the obser­
v ation point. At the second step, all grid-points are scanned su ccessively
in a sy stem atic m anner, and a correction W - h, is com puted. \V is a
H
h i j
A
h i,j+i
t+ l
Fig. 2. — First Step of Iteration : Computation of Interpolated
Depths at Observation Points.
hint — h A -}- A * (hB — h A)
—
/
+
A j (/w, j + i
A* A j
h* , j )
( ^ t + 1,/ + 1
+
^ « + 1, j
A* (f it + : , / —
h
Difference : h c = h„it — h inl
F
ig .
3. — A Typical Weighting Function.
Ra — dW = ----------------
Rs + da
*, j)
/ i *i / )
w e ig h tin g factor w h ich has the value 1 at the considered grid-point, and
goes dow n m on oton ically to zero at a radius of R from the grid-point. All
observation points insid e the circle o f radius R are considered, and for
every observation point a w eighted m ean is determ ined, w h ich is used to
im prove the initial trial value, h t, — or the result of p reviou s scan — , by
I (W
h,.)
ad din g the correction of ----- ^ ----- . Such a w eigh tin g fu n ction is sh ow n
in fig. 3. Then, at the third step, all grid-points are scanned again, and a
sm ooth in g operator is applied w ith the purpose of filterin g out sm all w ave
len g th s of the order o f about 2 grid-lengths, w h ile leaving larger w ave
len g th s (associated w ith the overall b ottom -form ation) su b stan tially
unchan ged . (The d evelop m en t of such a sm ooth in g procedure is given by
S h u m a n [1] ). T he operator actually used has the form :
h<
1
, =
~8
+
16
[A*
[ h j - l. ;
h t + l. j 4"
j - l 4 - ht
i] +
(3)
i,
j
+
hi
l. ; + i
‘i + l. i — l +
ht +
i • j+
J
-o
N
4 -I
F ig . 4
a"
=
[ - T
[1 — cos (kd)
]
kd = ----- d
The increase of the selectivity of a smoothing operator q (expressing the ratio of
smoothed to unsmoothed amplitudes, k is the wave number, d the mesh length
of the grid. L the wave length, q the number of successive applications of the
operator).
The selectiv ity o f the operator w ith regard to w ave length can be increased
by u sin g it rep etitiou sly, as sh ow n in fig. 4, w here the ratio o f th e a m p li­
tu d es before and after sm o o th in g is sh ow n. T h is 31-step iteration cycle is
repeated u n til th e resu lts o f con secu tive iteration s do not differ at any
grid-point by m ore th a n a predeterm ined arbitrary sm all con stan t.
4. Computations in a Trial Area
To test th is m ethod, a trial area o f ocean has been ch o sen in the
Southern P acific, in a region w here sou nd ings are scarce and concentrated
m o stly along sh ip p in g track s. T he area is bounded in latitu de by the equator
and the parallel of 60° South, and in longitud e by the m erid ian s 90° W est
and 150° W est (fig. 5). From all available data on ly th ose given on U.S.
N avy O ceanographic O ffice ch arts N os. 823 and 824 have been used as inp u t
data, m ak ing a total of 1131 poin ts. T he output grid, in M ercator coordi­
nates, has about 4700 p oin ts.
F ig . 5 .
— Trial Area for Computation of Average Depths.
A s a quick and con ven ien t visu al m ean s of d isp lay it is u sefu l to draw
contour lines from th e averages obtained at the grid -p oints. T h is is sh ow n
for th e region under con sid eration in fig. 6 . A k n ow n m apped large-scale
feature of the ocean bottom m ay then be used for com p arison . In th e trial
area under con sideration , w e have the P acific-A n tarctic R idge, d efin ed on
U.S. N avy O ceanographic O ffice Chart of th e W orld No. 1262A by th e 2 000
fathom s con tou rs. T he agreem en t b etw een th e m apped and p lotted con tou rs
is quite good (see fig. 7), con sid erin g the in com p leten ess o f th e in p u t data
used. T he overall sm ooth in g effect is evid en t, and is even m ore im p ressive
on a larger scale m ap o f con tou r lines o f 100 fath om s sp acin gs.
F ig . 6. — C o n to u r lin e s o f d e p th f r o m n u m e r ic a l c o m p u ta tio n b y th e p ro c e d u r e
d e s c rib e d in t h i s p a p e r , u s in g 1131 s o u n d in g s fro m H.O. C h a r ts 823 a n d 824.
D e p th s in f a th o m s .
5. Average Depths of the World Oceans
B y the m ethod described above averages of depth have been determ ined
on a grid, in M ercator coordin ates t, superim posed on the w orld oceans,
w h ere
■z = In tan
O +
.
(4)
H ere ¢) den otes geographical latitu de in radians, and t denotes Mercator
la titu d e in radians.
F ig . 7. — C o n to u r lin e s o f D e p th f r o m H.O. C h a r t 1262A , 1 0 th e d it i o n , J a n . 1961.
D e p th s i n f a t h o m s .
To obtain t in degrees, as given in tab le 2 and in fig. 8, w e m u st of
course use the form ula
180
,
- .
x = ------- In tan ( — O + — ) .
(5)
(
v
t
)
IT
As inp u t data, so u n d in gs at th e in tersection s of on e-degree geograph ical
grid -lines have been ch osen from th e set o f th e GEBCO, of th e IHB, M onaco,
or from U.S. N avy O ceanographic Office ch arts, w h erever a so u n d in g at an
in tersection or near an in tersection w as available. F or a con sid erab le part
o f in tersection s no inp u t data at all are at p resent available. A com p ilation
o f the input data, w h ich w ere read m an u ally from the ch arts, is given in
table 1, arranged b y m erid ian s. T he resu ltin g averages o f depths at gridp o in ts o f on e M ercator degree are given in table 2.
(F o u r p a g e s o f T a b l e 2 a r e reproduced, h e r e u n d e r )
T
able
2
A vera g e d e p th s o f w orld-oceans
L a titu d e s a n d lo n gitu des in degrees, d e p th s in m e tr e s
E = E ast, W = W est, N = North, S = South
C = P oin t on con tin en t, P aren th eses ( ) = L ocation o f islan d or p en in su la
P hi = G eographical latitude, Tau = M ercator latitude
LONGITUDE
PHI
TAU
0
1 E
2 E
3 E
4 E
5 E
6 E
7 E
8 E
9 E
3103
3047
2964
2861
2740
2607
2465
2318
2170
2023
3073
2993
2883
2751
2600
2436
2265
2092
1921
1756
3011
2 910
27 78
26 20
2443
2253
2056
1859
1668
1486
2919
2800
2 649
2471
2273
2061
1844
1628
1419
1224
2796
2666
2501
2309
2095
1867
1634
1403
1182
976
2647
2509
2336
2135
1911
1673
1430
1189
959
748
70.
69.
6 9.
69.
68.
68.
68.
67.
67.
66.
19
85
50
15
79
43
06
68
30
91
100
99
98
97
96
95
94
93
92
91
N
N
N
N
N
N
N
N
N
N
2 943
2986
3018
3039
3047
3042
3022
2986
2930
2854
3018
3039
3045
3037
3016
2981
2934
2872
2795
2704
3073
3069
3046
3006
2952
2885
2807
2717
2618
2508
3102
3072
30 19
2947
2859
2758
2648
2530
2406
2278
66.
66.
65.
65.
64.
64.
64.
63.
63.
62 .
51
11
70
29
87
44
00
56
11
66
90
89
88
87
86
85
84
83
82
81
N
N
N
N
N
N
N
N
N
N
2756
2638
2499
2342
2168
1981
1782
1576
1366
1158
2596
2473
2335
2183
2018
1843
1659
1468
1273
1079
2389
2261
2124
1977
1824
1663
1496
1325
1150
976
2146
2012
1874
1735
1592
1448
1302
1153
1003
853
1878
1737
1600
1467
1337
1210
1085
962
839
716
1598
1451
1314
1187
1070
961
859
761
666
572
1318
1166
1029
909
805
714
635
563
496
431
1047
891
756
644
553
480
423
377
337
300
793
635
503
400
323
268
233
211
197
186
561
403
277
183
120
84
70
71
C
C
62.
61.
61.
60.
60.
59.
59.
58.
58.
57.
20
73
25
76
27
77
26
75
23
70
80
79
78
77
76
75
74
73
72
71
N
N
N
N
N
N
N
N
N
N
955
765
591
439
311
210
134
82
51
34
890
711
547
403
282
186
114
66
37
23
806
644
496
365
255
168
104
61
37
26
706
566
438
325
230
155
101
66
48
41
596
481
376
283
206
146
104
78
66
63
480
393
312
242
184
141
112
96
89
90
368
C
C
C
C
140
124
118
117
120
C
C
C
C
C
C
C
C
C
C
C
149
153
C
C
186
188
c
c
c
c
c
c
c
221
227
224
57.
56.
56.
55.
54.
54.
53.
53.
52.
51.
16
61
06
49
92
34
76
16
56
95
70
69
68
67
66
65
64
63
62
61
N
N
N
N
N
N
N
N
N
N
18
17
16
12
4
0
0
0
0
0
24
26
29
29
27
18
10
6
7
15
41
45
50
52
51
39
30
24
25
33
65
70
73
74
72
57
45
36
34
42
93
96
97
94
87
77
54
40
33
32
123
123
118
109
96
79
55
36
24
C
153
147
136
119
97
73
50
23
C
C
183
170
150
123
93
60
29
2
( 2 1 2)
( 1 9 1)
( 1 6 0)
C
C
42
3
C
26
21
14
2
0
0
0
0
0
(0)
C
C
C
c
C
c
c
T
able
2
A vera g e d e p th s o f w o rld-ocea n s
L a titu d e s a n d lon gitu d es in d e g re e s , d e p th s in m e tr e s
E = E ast, W = W est, N = N orth, S = S outh
C = P o in t on con tin en t, P aren th eses ( ) = L ocation of islan d or p en in su la
P h i = G eographical latitu de, T au = M ercator latitu d e
LONGI TU D E
PHI
TAU
0
1E
2 E
3 E
4 E
5 E
6 E
7 E
8 E
9 E
23
43
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
51.
50.
50.
49.
48.
48.
47.
46.
46.
45.
33
70
06
41
76
09
42
74
05
35
60
59
58
57
56
55
54
53
52
51
N (0)
N 0
N 0
N 0
N c
N c
N c
N c
N c
N c
0
0
0
0
0
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
44.
43.
43.
42.
41.
40.
40.
39.
38.
37.
65
93
21
47
73
98
22
45
68
89
50
49
48
47
46
45
44
43
42
41
N
N
N
N
N
N
N
N
N
N
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
37.
36.
35.
34.
33.
33.
32.
31.
30.
29.
10
30
49
67
84
01
16
31
46
59
40
39
38
37
36
35
34
33
32
31
N
N
N
N
N
N
N
N
N
N
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
28.
27.
26.
26.
25.
24.
23.
22.
72
84
95
05
15
24
33
41
30
29
28
27
26
25
24
23
N
N
N
N
N
N
N
N
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
21. 48
20. 55
22 N
21 N
T
able
2
A v e r a g e d e p t hs o f w o r l d - o c e a n s
L a t i t u d e s a n d l ong i t udes in degrees, de p t hs in m e t r e s
E = E ast, W = W e s t, N = N o rth , S = South
C = P o in t on continent, Paren th eses ( ) = Lo catio n o f islan d or peninsula
P h i = G eograph ical latitude, T a u = M ercator latitude
LONGITUDE
PHI
TA U
19. 61
20 N
18. 66
19 N
17. 71
18 N
16. 76
17 N
15. 80
16 N
14. 83
15 N
13. 86
14 N
12. 89
13 N
11. 91
12 N
10. 93
11 N
9. 95
10 N
8. 96
9 N
7. 97
8 N
6. 98
7 N
0
1 E
2
E
3E
C
C
C
C
C
C
C
C
C
C
4E
C
C
5
E
6
E
C
C
C
C
C
C
C
C
C
C
C
C
c
c
c
c
c
c
c
c
c
c
C
C
C
C
c
c
c
c
c
c
c
c
3130
3173
3269
3407
3579
3775
2973
3051
3177
3340
3532
2760
2752
2808
2917
3067
3251
2513
2 543
2632
2767
2939
3743
3962
4179
4386
4576
4746
4894
5020
5129
5225
3456
3673
3890
4100
4295
4474
4633
4774
4899
5011
3136
3347
3560
3768
3965
4148
4316
4468
4607
4735
4269
4409
5. 99
6 N
3428
3289
4. 99
5 N
34 97
3348
4. 00
4 N
3612
3456
3. 00
3 N
3762
3603
2. 00
2 N
3937
3778
C
C
c
c
c
c
c
c
C
C
C
C
2953
c
c
c
c
c
c
c
c
c
c
c
c
C
C
c
c
7 E
8
E
9
E
c
C
C
C
C
C
C
C
C
C
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
C
C
2260
c
1735
1477
2 263
1975
1687
141 1
c
c
c
c
c
2553
C
C
C
C
C
C
C
C
C
C
C
c
c
c
c
c
c
c
c
c
C
c
C
c
C
c
c
c
2 331
2023
1717
1427
2450
2126
1805
1504
2608
2270
1936
1622
1 N
4128
3974
0
4327
41 82
1s
45 2 4
43 92
2. 00
2 S
4 712
4595
3. 00
3 S
4 881
47 82
4. 00
4 S
5 028
4948
4. 99
5 S
5149
5089
5. 99
6 S
5 244
52 05
6. 98
7 S
5315
52 96
7. 97
8 S
5367
5367
8. 96
9 S
5409
54 27
3987
4204
4416
4616
4796
4854
5087
5197
5287
5365
9. 95
10 S
5446
5479
5435
5311
5115
4854
4540
10. 93
11 s
5480
55 27
5 497
5390
5209
4962
11. 91
12. 89
12 S
5407
5448
5423
5328
5167
4946
13 S
5 339
5373
5348
5261
51 14
4913
4667
4388
4091
3793
13. 86
14 S
53 05
5334
5308
5226
5088
4900
4670
4407
4128
3 8 47
3862
1. 00
. 00
1. 00
2794
2443
20 95
1768
2995
2632
2271
1930
3201
28 27
2454
2101
34 0 4
3021
2639
2 276
3598
3212
28 23
2452
2 628
3783
33 96
30 04
3 957
3572
3180
2803
4118
37 39
3350
2973
3898
35 14
3138
4048
3670
3 297
4189
3818
3448
4661
4320
3956
35 90
4675
4367
4039
37 10
14. 83
15 S
5280
5302
52 73
5191
50 58
4877
4655
44 03
4134
15. 80
16 S
52 47
5256
5218
5131
4997
4818
4602
4357
4095
3829
16. 76
17 S
5197
51 8 8
5135
5037
4897
4717
4 504
4263
4007
3744
17. 71
18 S
5135
5104
5033
4922
4772
4588
4373
4135
3880
3618
18. 66
19 S
5070
5019
4931
4806
4647
4457
4 238
3 998
3740
3475
T able 2
A v e r a g e de p t h s o f w o r l d - o c e a n s
L a t i t u d e s a n d l ongi t udes in degr ees, dept hs in m e t r e s
E = East, W = W e s t, N = N o rth , S = South
C = P o in t on continent, Paren th eses ( ) = Location of islan d o r p en in su la
P h i = G eograph ical latitude, T a u = M ercator latitude
LO NGITUDE
PHI
TAU
0
1 E
2 E
3 E
4 E
5 E
6 E
7 E
8 E
9 E
19. 61
20 S
5007
4942
4842
4708
45 43
43 47
4124
3879
3615
3341
2 0 . 55
21 S
4946
48 75
4771
4635
4467
4270
4044
37 94
3523
32 3 7
21. 48
22 S
4888
48 17
4717
4585
4422
4227
4001
37 47
3470
3173
22. 41
23 S
4835
4771
4680
4558
4403
4215
3993
3739
34 56
3148
2 3 . 33
24 S
4791
47 40
4663
45 57
4415
4237
4 021
37 67
3477
3156
2 4 . 24
25 S
4707
46 74
4619
4535
4416
4259
4061
3822
3 544
32 32
2 5 . 15
26 S
4630
4616
4584
4524
4430
4 295
4117
3895
36 31
3330
26. 05
27 S
4562
45 68
4560
4525
4455
43 4 4
41 86
3983
3735
3447
26. 95
28 S
4505
4531
4545
4535
4490
4402
4267
4082
38 51
3577
2 7 . 84
29 S
4457
4502
4538
4552
45 31
4466
43 51
4186
3 972
3714
3849
2 8 . 72
30 S
4419
4480
4535
4570
4570
4527
44 32
4286
4090
2 9 . 59
31 S
4 387
4461
4530
4582
46 01
4576
4500
4 372
4194
3970
3 0 . 46
32 S
4 356
443 5
4511
4571
4 604
4598
4546
4447
4302
4116
3 1 . 31
33 S
4333
4415
4494
4562
4605
4614
45 82
4506
4387
4229
3 2 . 16
34 S
4 320
4 405
4490
4 564
4617
4640
4623
4565
4465
4326
3 3 . 01
35 S
4322
4 414
4506
4591
4657
4694
4693
4649
45 63
4437
33. 84
36 S
4339
4444
45 51
4653
4740
4798
4815
4788
4715
4597
34. 67
37 S
4 365
4491
46 22
4750
4863
4950
4997
4995
4939
482 8
4896
35. 49
38 S
4373
4503
4637
4769
4887
4980
5036
5043
4996
36. 30
39 S
4376
4506
4640
4773
4 894
49 92
5055
5072
5037
4951
4883
4984
5053
5080
5059
4989
5006
37. 10
40 S
4374
4500
4 632
4762
37. 89
41 S
4367
4488
4614
4739
4856
4 956
5029
5065
50 58
38. 68
42 S
43 58
44 72
4589
4 704
4813
4909
49 82
50 26
5032
4 998
39. 45
43 S
4351
4454
4558
4661
4757
4844
4913
4959
4976
4961
4 0 . 22
44 S
4 345
44 3 7
4525
4611
4691
4763
4823
4868
48 92
4893
40. 98
45 S
4343
4420
4 491
4 557
4617
4671
4719
4757
47 8 5
4798
41. 73
46 S
4339
4 402
4455
4501
4541
45 76
4608
4637
4662
4684
4 2 . 47
47 S
4328
4377
4415
4443
4465
4482
4499
4517
4538
4563
4 3 . 21
48 S
4304
4 343
4368
4383
4391
4394
43 98
4405
4420
4444
43. 93
49 S
42 64
4295
4312
4318
4317
4311
4306
4305
4 313
4 334
4 4 . 65
50 S
4205
4230
4243
4245
4239
4229
4219
4213
4217
4234
45. 35
51 S
4126
4148
4159
4160
4153
4143
4131
4 124
4125
4139
4 6 . 05
52 S
40 27
4048
4058
4060
4055
4 046
4037
4031
40 32
4044
4 6 . 74
53 S
3910
3 930
3941
3945
3 944
3939
3 9 33
3930
3934
3 945
47. 42
54 S
37 76
3796
3 809
3 817
3820
3821
3 8 21
3823
3830
3842
48. 09
55 S
3 628
3 649
3667
3 680
3690
3697
3704
37 12
3 722
3735
4 8 . 76
56 S
3470
3495
3518
3 538
3557
3573
3588
3603
3617
3633
49. 41
57 S
3 309
3 3 39
3369
3399
3427
3454
3479
3502
3 523
3543
50. 06
58 S
3 149
3186
3226
32 67
3307
3346
3383
34 16
3445
3472
50. 70
59 S
2999
3 0 45
3095
3149
3203
3256
3306
33 52
3393
3 428
U s in g the present com putations, a bathym etric m ap of the oceans in
M ercator coordinates h as been constructed (fig. 8). T h e irre gu la r coastlines
have been replaced b y straight sections w h ic h are m ultiples of the unit
g rid length o f one degree. W it h a view to applications to n um erical analysis
the coastline w a s chosen so that every grid-poin t in the ocean has at least
one n eigh b or in each direction.
In d ra w in g the coastline, the fo llo w in g com putational policy has been
adopted.
T h e large continental m asses, in clud in g A u stra lia and Greenland, h ave
been regard ed as rea l b arriers. T h e rem aining islan ds and some peninsulas
have been included, w h enever their extent is com parable w ith the basic
one degree squ are o f the M ercator grid, but have been disregarded in
c a rry in g out the com putations leadin g to the average depths at their posi­
tions. T h is w a s done fo r the reason that w h e n a la rg er grid-step than 1°
w ill be used, som e or all o f the islands w ill not be recognizable in the
coarser grid. A ccordin gly, these islan ds and peninsulas are sh ow n as lan dareas on the m ap (fig. 8 ) , but the com puted depths at corresponding g ridpoints have also been retained, and are given in table 2 (indicated by
paren th eses).
A t som e points o f the larger islan ds or near the continental coasts, the
com putation yielded negative depth-values resulting fro m the rising trend
o f the ocean bottom tow ard s the coast. W h e r e v e r this happened a value o f
zero has been substituted in table 2 .
T h e rad iu s R o f the w eigh tin g function W , described in section 3, has
been chosen as 15 M ercator grid-units everywhere, except in the Atlantic
Ocean, w h ere, due to the relative abundance o f input data, a value o f 12
units w a s deem ed sufficient.
F o r the actual choice o f a p a rtic u la r w eigh tin g function W there exist
m an y possibilities. O ne w o u ld like to account for k n o w n special features
o f the ocean bottom in the vicinity of the point considered. T h is means that
besides w eigh tin g w ith respect to distance only, a n gu la r w eigh tin g needs
to be considered, and the function becom es tw o-dim ensional. The w eigh tin g
fun ction itself, and p a rtic u la rly its m ax im u m radius R o f applicability, m ay
v a ry fro m point to point. T h ese refinem ents w ill undoubtedly be valuable
and w ill be used in m ore advanced stages o f the com putational program .
A t present w e do not yet possess enough detailed structural inform ation
o f the ocean bottom topograph y on a w o rld -w id e basis to w a rra n t the
construction o f a m ore sophisticated w eigh tin g function than the one given
in fig. 3.
T o c a rry out the actual calculations o f the averages, the ocean area of
fig. 8 h as been divided in 35 com putational squ are regions, i.e. each region
is bou n ded b y tw o m eridians an d tw o p a ra lle ls o f latitudes. A bout 50
iterations w e re req u ired to obtain an average at every grid-point, accurate
to tw o fraction al decim al units, as defined in section 3. T h e sm oothing
operator (3 ) has been applied q ( — 5) times in each iteration. A s expected,
slight discontinuities developed in areas w h ere the com putational regions
jo in , alth o u g h some overlapp in g has been provided in choosing the regions.
T o com pensate fo r this effect, the sets o f depth-values at the overlapping
jo in ts w e re first averaged, and then a fu rth e r sm oothing operation w a s
180®
(20*
90*
60“
30*
0°
30°
60»
90*
F ig . 8. — Bathymetric map of the world — oceans with contour lines of average
depths from numerical computation by the procedure described in this paper.
Depths in metres.
150°
120°
150*
IftO
carried out. T h e fin a l results o f these com putations are given in table 2 u p
to 100“ M ercator latitude i N o rth an d 99° M ercator latitude z South,
corresp on din g to abou t 70 geograph ical degrees N o rth and South. Most o f
the A rctic an d A n tarctic regions are not covered. F ig. 8 contains contour
lines o f depth b ased on the results given in table 2 .
A ll com putations have been carried out on the CD C 1604-A com puter
o f the W e iz m a n n Institute o f Science, Rehovoth, Israel.
6. Remarks on Smoothing
A fin a l point to be m entioned concerns sm oothing. It m ay be a rgu ed
that sm oothing in this case is m ore o f a convenience than a necessity. T h e
actual bottom c on figu ration o f the oceans, even on a large scale, is quite
rugged, w ith plenty o f m ou n tain ranges, ridges, valleys, basins and trenches.
T h o u g h in a v e ra g in g itself the sm oothing effect is inherent, the pattern o f
the averages m a y be quite irre gu la r. N o w , m ost o f the m athem atical
app licatio n s fo r w h ose sake the averages have been com puted, and espe­
cially those w h ic h instigated this research, require a smooth field, p articu ­
la rly if derivatives o f the depth-field are needed. These have little m eaning
w h e n sh arp local v ariation s are preserved. It has also been fou n d that the
com putation of the average depth is m ore stable, and convergence is reached
m ore quickly, i f sm oothing is present. F o r both these reasons a sm oothing
cycle h as been includ ed in the com putation scheme.
T h e p a rticu lar sm oothing operator used preserves the overall average
fo r the w o rld oceans as a w h o le and fo r bounded regions as w ell.
7. Conclusion
In conclusion, it sh ould be pointed out that the bathym etric averages
presen ted in this p a p er are based on sparse, irre g u la rly spaced, and
po ssibly not too reliable input data. Nevertheless, the results obtained m ay
serve as a tentative m odel-ocean fo r num erical oceanographic com putations.
T h e y fo rm the first presentation o f its kind.
Acknowledgements
T h e auth or expresses his gratitude to P ro fe sso r C. L . P e k e r i s fo r advice
an d help.
M r. A m ir P n u e l i aided considerably in the later stages o f the w o rk ,
by speedin g up the p rogram m es an d by c a rryin g out the fin al com pilations
o f the entire program m e.
T h is research w a s supported b y the N ational Science F oun dation u nd er
N S F G ran t — G 15310 and b y the O ffice of N a v a l R esearch under Contract
N o n r-1823(00).
References
[1 ]
F . G. S h u m a n : N u m e ric a l m ethods in w eath er prediction : II. Sm ooth­
ing and filtering. M o n t h l y W e a t h e r R e v i e w , Vol. 85, Novem ber
1957, 357-361