NASA TECHNICAL
MEMORANDUM
COCN^«CO
NASA TM X-3423
WAVE REFRACTION DIAGRAMS FOR
THE BALTIMORE CANYON REGION OFTHE MID-ATLANTIC CONTINENTAL SHELF
COMPUTED BY USING THREE BOTTOM
TOPOGRAPHY APPROXIMATION TECHNIQUES
Lamont R. Poole
Langley Research Center
Hampton, Va. 23665 \
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, 0. C. • DECEMBER 1976
https://ntrs.nasa.gov/search.jsp?R=19770009733 2020-01-12T10:54:54+00:00Z
1. Report No.
NASA TM X-3^232. Government Accession No. 3. Recipient's Catalog No.
4. Title and Subtitle WAVE REFRACTION DIAGRAMS "FOR THE BALTIMORECANYON REGION OP THE MID-ATLANTIC CONTINENTAL SHELFCOMPUTED BY USING THREE BOTTOM TOPOGRAPHY APPROXIMATIONTECHNIQUES
5. Report Date
December 19766. Performing Organization Code
7. Author(s)
Lament R. Poole8. Performing Organization Report No.
L-1100U
9. Performing Organization Name and Address
NASA Langley Research CenterHampton, VA 23665
10. Work Unit No.
161-07-02-03
J1. Contract or Grant No.
12. Sponsoring Agency Name and Address
National Aeronautics and Space AdministrationWashington, DC 205 6
13. Type of Report and Period Covered
Technical Memorandum
14.'Sponsoring Agency Code
15.. Supplementary Notes
16. Abstract .
The Langley Research Center and Virginia Institute of Marine Science'waverefraction computer model was applied- to the Baltimore Canyon region of themid-Atlantic continental shelf. -Wave refraction diagrams for a wide range ofnormally expected wave periods and directions were computed by using' threebottom topography approximation techniques: quadratic least squares, cubicleast squares, and constrained bicubic interpolation. Mathematical or physi-cal interpretation of certain features appearing in the computed diagrams isdiscussed.
17. Key Words (Suggested by Author(s))
Wave refractionBottom topography approximationBaltimore Canyon region
18. Distribution Statement
Unclassified - Unlimited
Subject Category k8
19. Security Oassif. (of this report)
Unclassified
20. Security Classif. (of this page)
Unclassified
21. No. of. Pages
155
22. Price*
. $6.25
* For sale by the National Technical Information Service, Springfield. Virginia 22161
WAVE REFRACTION DIAGRAMS FOR THE BALTIMORE CANYON REGION
OF THE MID-ATLANTIC CONTINENTAL SHELF COMPUTED
BY USING THREE BOTTOM TOPOGRAPHY
APPROXIMATION TECHNIQUES
Lament R. Poole
Langley Research Center
SUMMARYV •
• r •Wave re-fraction diagrams were computed by applying the
Langley Research Center and Virginia Institute of Marine Science
wave refraction computer model to the Baltimore Canyon region of
the mid-Atlantic continental shelf. Diagrams were presented for a
wide range of normally expected wave periods and directions. For
each combination of period and direction, diagrams were computed
by using each of three commonly used bottom topography approxima-
tion techniques (quadratic least squares, cubic least squares, and
constrained bicubic interpolation), and all these diagrams are
presented with the exception of cases in which essentially identi-
cal diagrams resulted from the three techniques. Certain features
of the computed diagrams were discussed with respect to physical
or mathematical interpretation, but no conclusions were drawn.
INTRODUCTION
Man's increasing use of continental shelf regions of the
world has stimulated the development of analytical models to aid
in the study of the physical processes affecting offshore and
coastal activities. One such analytical tool is the wave refrac-
tion model, which describes the behavior of surface waves propa-
gating through areas of irregular bottom topography. The Langley
Research ;Center; (LaRC) and .the Virginia Institute o;f Marine'
Science. (VIMS) have, in a cooperative effort, developed a large-
scale -refraction model (ref. 1) for application.to the Virginian
Sea,.a region -of' th.e mid-Atlantic continenta'l-shelf from. Cape
Hatteras.to Cape Henlopen. Computer resource .requirements for
applying the;model to a large geographical region.were signifi-
cantly, reduced by the subsequent incorporation of a random-access
technique for modular storage and retrieval of bathymetry, d.ata
(re,f. -2). '• Recently, the model has been'-modified .to .employ,. ori
,'Optd.pn,. three commonly used techniques for approximating local .
ocean •bottom;topography. Two of these techniques.are-quadratic
least ;squares.and cubic least squares, both of whiph provide some .
smoothing of the discrete bathymetry data but do not 'insure con-
tinuity in the depth or its derivatives. The third .technique, .
constrained; bicubic interpolation, provides continuity without
smoothing.;:: Incorporation of all three techniques stemmed from a
small-scale refraction study (ref, 3) in :which it was concluded
that;, the best approximation technique for use in future .studies
could -.riot ;b'e determined without extensive comparison of, computed
results with; quantitative experimental data. In lieu of such a
comparison,; refraction diagrams can. be computed by using each of
the: techniques. Then more confidence can be placed in .the:cdm-
pute.d results' if no significant discrepancies exist among the
three diagrams. ,
. , The purpose of this paper -is to present wave refraction
diagrams computed by applying the LaRC-VIMS' model to an area of
the mid.-Atlantic shelf called the Baltimore Canyon region :after a
prominent regional'submarine geological feature. Extending from
near.Wachapreague Inlet, Virginia, to near Manasquan Inlet, New
Jersey, the region is of particular current interest with regard
to offshore petroleum deposits and is one of a series of model
regions which to'gether provide broadrscale refraction modeling
capability from Cape Hatteras to Montauk Point", Ne.w .York (ref. 4).
An,, immediate application of the computed diagrams is planned by
the Philadelphia District and the Coastal Engineering' Research .
2 : . - ; " - . • . - - " . - . - ' . . ; • . ; . • . ,
Center of the U.S. Army Corps of Engineers; these results will-be
used to furnish initial conditions for a fine-scale refractionstudy in the nearshore region off Ocean City, New Jersey.••'•-'•' -
, . . " " ' • »
: DATA INPUT . " - -'
The bathymetry data array for the Baltimore Canyon- regio.n1' was :
developed by first constructing a special transverse-.Mercator map
projection with a central meridian at 75° west, similar, to/the one
constructed for the Virginian Sea region. (See ref v:_1.pfo.r .;details.)The projection was then overlaid with a 0.5-nauti.cal.^raile ;:'sjquare '
grid pattern extending for 150 nautical miles from .north:'td south
and for 127 nautical 'miles from west to east. Depth values at the;r' • • • • *'•'<
76 200 nodes (300 rows by 25^ columns) of the square grid.pattern
were determined by using sounding charts and;'other data' i'h?'a .man- -
ner analogous to that used in developing :the. Virginian Seavvbathyin-
etry data grid (ref. 1). A three-dimensional computer-drawn-plot 'of the Baltimore Canyon region bathymetry data grid (with a verti-cal exaggeration of approximately 300~to 1) is shown in figure 1.:
The regional array was then divided into a series of smaller mod- •'-'•ules for use with the random-access data storage and retrieval- ' "technique described in reference 2. The module size.was selected ' •
as M2 rows by 32 columns to minimize computer cost,, resulting in : ;
a series of 72 overlapped bathymetry data modules. (See ref.- 2'
for sequencing and overlapping details.) As a final adjustment to
the input bathymetry data, the "-program input parameter for. tide
was set so that the input depth values would correspond to mean '
low water. • • ; - : - -
A range of wave direction and period was desired which would"5
provide meaningful refraction diagrams while being realistic from:
the standpoint of normally-occurring real-life wave conditions.
With these requirements in mind, the selected input wave direc-tions ranged from 0° (from due north) through 90° (from due east)to 160° (from due south). Input directions from the west were notdeemed significant for this study because of the limiting fetch for
3
'Cape May.
Delaware
Figure 1.- Three-dimensional computer-drawn plot of the
Baltimore Canyon region bathymetry data grid. Verti-cal exaggeration, 300 to 1.
physical wave generation. Within the selected direction range, '
input .directions were chosen at 22.5° increments (i.e., N, NNE,
NE, etc. ) as was done in the study presented in reference 1..
Input wave periods were selected in 2-second increments from 6 sec-onds to;' 16 seconds. On-the basis of results of'reference 1, wavesof•periods.less than 6 seconds were expected to undergo very little,if-any, refraction; thus, rejfrac-t-ion-:diagrams"~for these periods
would be of little meaning. It was also felt that waves of peri-
ods greater than- 16 seconds would occur so infrequently in nature
that their inclusion in this study was not justified. As a final
input, the initial wave height was fixed arbitrarily at 0.3 meter
for all cases since, according to linear wave theory, wave'height
does not affect the refraction pattern for a given combination of
wave period and direction.
; ; • , . WAVE.REFRACTION DIAGRAMS ' . ' ; . -
i ' ' r ,' a*
Wave refraction diagrams'for'the Baltimore .Canyon region were
computed for the input range of wave period and direction- by using
quadratic least squares, cubic least squares, and constrained
bicubic, interpolation techniques for approximating local bottom
topography. The initial separation distance between wave rays was
selected as 3 nautical miles to provide adequate overall resolu-
tion without cluttering the diagrams. Computations were required
to begin in deep water, which, for the purposes of this paper, was
considered 'to be a water depth greater than one-fourth the initial
wavelength for the particular wave period:being considered. (See
ref. 1 for discussion.) Tick marks were drawn along each computed
ray at 'points equally spaced in time, with the time increment
varied with .wave period to provide relatively uniform spatial reso-
lution among- cases of different periods. By connecting correspond-
ing tick marks o.n adjacent rays so that the connecting curve is
orthogonal to each.ray, wave crest patterns can be derived.
Computed- refraction diagrams are presented in figures 2 to 52
in the sequence given in table I. Diagrams computed for a wave
direction'of 0° and periods of 12, 14, and 16 seconds have been
omitted because only one or two rays met the deep-water condition
for initiating computations. Unless noted otherwise, each figure;,
consists of three parts: part (a) presents the diagram computed
by using the quadratic least squares technique for approximating .
bottom topography; part (b) presents the diagram computed by using
the cubic least squares technique; and part (c) presents the dia-
5
gram computed by using the constrained bicubic interpolation tech-
nique. When essentially identical diagrams resulted from the
three techniques, only the one computed by using the quadratic
least squares technique is presented. :
Some general comments are in order as an aid to potential
users in interpreting the various diagrams. Since the tick marks
along each ray denote equal time spacing, an increase in the spa-
tial density of tick marks as a ray approaches the shore indicates
decreasing wave phase speed. Convergence of a group of :neigh-
boring rays indicates an area of relatively high wave energy,;
whereas a sparsity of rays indicates relatively low energy. On
occasion, individual rays stop abruptly in an area of relatively
deep water (ray 23 in fig. 20(a), for example). Such ray; termi-
nations indicate failure to converge on a proper value fo;r local
ray curvature or calculation of a physically unrealistic value
for wave height and, as such, should be considered of no physical
significance. In several instances, individual rays undergo
abrupt changes in direction relative to the mean direction of
neighboring rays (ray 40 in fig. 32(c), for example). These
abrupt changes are induced by sharp gradients in the local bottom
topography, which are influenced, in turn, by the technique
selected for approximating the topography within the finite-mesh
bathymetry grid. Since use of the constrained bicubic interpola-
tion technique results in a surface which passes directly through
the bathymetry data points, sharp topography gradients tend to
occur more often with this technique than with the two least
squares techniques, both of which provide some smoothing .of the
data.
TABLE I.- SEQUENCE OF WAVE REFRACTION DIAGRAMS
Figure
2 ;3 .4
5 .6789101112
1314
1516
1718
19.20
21
22
2324
2526
27
Wave direction,
a, degrees
00
0
22 . 5 '
22.5
• ' • ' .22.5
22.522, 5
22.5
45
45
.45
, ' 45
'45 '
45
'.67 ..5
'.-'"•. ,67.5 .. ..67.5
67.5
'•;" 67.5
67.5
:• ; 90
90
, ' , 90
":;•: 90"
• ' ' 90
Wave period,
T, secondsa6a810a68
10
12
14
16
6
8
10
12
14
16
68
10
12
14
16
6
8
10
12
14
Figure
28
2930
. 3132
3334
-: 3536
3738
3940
41
42
4344
4546
4748
4950
5152
Wave direction,
a , degrees
90
112.5
112.5
112.5112.51 12.5
112. 5
135
135
135
135
135
135
157.5
157.5
157.5
157.5
157.5
157.5180
180
180
180,
180
180
Wave period,
T, seconds
16
6
8
1012- •
14 -.-
16 ,
6
,.(. 8
10: 3 1 •12
14
16
6
8'
10,1214
16
6
8
10
12
14 '
16
aOnly the diagram computed by using the quadratic least squarestechnique is given because no resolvable difference existed among dia-
grams computed by using the three approximation techniques.
75°|00' 74°|00'Y-axis, nautical miles
73°|00'
+•1 i imiiM1111ii i 111iii 11111111in11111111i i11 i
Figure 2.-- Wave refraction diagram. a = 0°; T = 6 seconds.
Bottom topography approximated by quadratic least squares
technique. • . : . : .
75°|00' 74°|00'Y-axis, nautical miles
73°|00'
Figure 3-- Wave refraction diagram. a = o°; T = 8 seconds.
Bottom topography approximated by quadratic least squares
technique.
-. .:75°|00' 74°|00'
Y-axis, nautical.miles
73°|00'
•' . (a) Bottom topography approximated by quadratic
' ' • .•' least squares technique.
Figure-4.- Wave refraction diagrams, a = 0°; T = 10 seconds
10
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40°
(b) Bottom topography approximated by cubic
least squares technique.
Figure 4.- Continued. .
11
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
ra uI I I II I I I I II 11 II 11 II II I 11 I I MM I I I I I I I I I I II I 11 IMI
12
,(G) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 4.- Concluded.
74°|00'Y-axis, nautical miles
Figure 5.- Wave refraction diagram. a = 22.5°; T = 6 seconds.
Bottom topography approximated by quadratic least squares
technique.
13
75°00' 74°|00'Y-axis, nautical miles
73°00'
40° °g
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 6.- Wave refraction diagrams, a = 22.5°; T = 8 seconds
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 6.- Continued.
15
75°00' 74°|00'Y-axis, .nautical miles
73°|00'
,40°
- DELAWAREE, ^ BAY':'
16
(c) Bottom topography approximated by constrained
• bicubic interpolation, technique.
. , • Figure. 6.-, .Conclude.^.
75°00' 74°|00'Y-axis, nautical miles'
73°00'
40° °g
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 7.- Wave refraction diagrams, a = 22.5°; T = 10 seconds.
17
75°00' 74°|00'Y-axis, nautical miles
73°00'
111 111 I M I II 111 1 1111 111 I 11 II II I I I 1111 I M I III 111 I I I I 11 I I [ I I 11 I I I II I M I I I M 11 I I I I II I I I I 111 I II I 11 I [ I II I 111 M III I 11I . I - I I ~T~1* I ,1 - v r* *. ' » " l ~ *:' * \ L f. ^^—»^^^— . I ' H I >./ ^ 7"~* / j ^ * t S.K f
(b) Bottom topography approximated by -c.ubic
least .squares technique.
Figure 7.- Continued.. .
18
75°00' 74°|00'Y-axis, nautical miles
JTT
73°|00'
(c) Bottom topography approximated by constrained ...'
bicubic interpolation technique. -"..-'-' • . '.** .-i
Figure 7.- Concluded. • . _ '.-'' .;19
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40°
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 8.- Wave refraction diagrams. a = 22.5°; T = 12."seconds
20
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40° °G?
(b) Bottom topography approximated by cubic
least squares technique.
Figure 8.- Continued.
21
75°00' 74°|00'
Y-axis, nautical miles
73° 00'
(c) Bottom topography approximated by constrained.
bicubic interpolation technique.
. ; . : • • . . - • • Figure 8.- Concluded. ;
22
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(a) Bottom topography approximated by quadratic
least squares technique. .
Figure 9-- Wave, refraction diagrams, a = 22.5°; T = 1 *J seconds
23
75° 00' 74°|00'
Y-axis, nautical miles
73°|00'
40°
24.
(b)' Bottom topography approximated by cubic
least squares technique.f
Figure 9.- Con t inued .
750100' 74°[00'Y-axis, nautical miles
73°|00'
•(c) Bottom topography approximated by constrained
• ;. bicubic interpolation technique.
. . - • ' • Figure 9 . - Concluded.
25
•>-75 000' . - - 74°|00'Y-axis, nautical niiles
73° 00'
r '* . . . -- i - •/••= . . ' - / - . • : - : - - . • •
i- ••»;*C£;)\Bottom;:tpp'O"graphy; approximated by quadratic
F r: -?'::-i->"V:;";. -:''i\";i s-t ;square,s technique.
Figure;^^^l^aV^^e^aGtio^rdiagrams. a =22.5°; T = 16 seconds
26 /|S|:>::;S.:.;K • . :-
75°|00' 74°|00' '"Y-axis; nautical miles
73°|00' '
(:b) Bottom topography approximated by cubicleast squares technique.Figure 10.- Continued. "•/'•
27
75°00' 74°|00'Y-axis, nautical miles
73°|00'
28
(c) Bottom topography app>oximated by constrained
bicubic interpolation technique.
Figure 10.-' Concluded.
75c|00r ' . 74°|00' .Y-axis, nautical miles
73°|00'
(a-) Bottom topography approximated by quadratic - '
. '-'-:•.-':.'lea"s.t'.squares technique. ; • •
Figure 1 iv-^Wave^refraetion diagrams, -a = 45°; "T" =• 6'seconds.
29-
-- 75°|00' 74°|00'Y-axis, nautical miles
73°001
40°
'S,
30
:•' (b) Bottom topography approximated by cubic
:'.;-: . least squares technique.
: • : . . ; • Figure 11.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°00'
ra ut
'S,
(c) Bottom topography approximated by constrained
"-.•".:.•/••• .-.bicubic interpolation technique.
••" V;f: .. '" Figure 11.- Concluded.
31
75°00' 74°|00' '
Y-axis, nautical miles
(a) Bottom topography approximated .by quadratic,^-. .- . . • ' " ' * : .''•"' ', ' ^f-
. -.-; ••---•-'.- least squares technique.; -.^/i "•.'•"-'.;;V:
Figure 12.-.'Wave refraction diagrams. a -= 45°; . T:,-='X8' "seconds
32 : -•-'•.-••"•• - - . ' - ' ' ' :'>>""' ' r.
75°00' 74°|00'
Y-axis, nautical miles
73°|00'
40° °E
(b) Bottom topography approximated by cubic
least squares technique.
Figure 12.- Continued.
33
••• 75° 00' 74°|00' v•Y-axis, nautical miles
73° 00'
T (c-). Bottom topography approximated by constrained- c';.:.. :;, ;-'. . bicubic interpolation technique. •[ • . ..
,- •'~y?:;: . .. Figure 12..- Concluded. ' • • ' " ' .
75°00' ' 74°|00'Y-axis, nautical miles
73°|00'
(a) Bottom topography approximated by quadratic •• '•'
. • ' ' least squares technique. . • " '.'"'.
Figure 13'-- Wave refraction diagrams. <* = 45°;. T .= 10 seconds.
35.
75°00' 74°|00'Y-axis, nautical miles
73° 00'
+i 1 1 1 1 1 1 1 1 1 1 1 1 M 11 |i 11 ii 1111| 111 M M 11 11 M M 111| i M 11 up i
36
(b) Bottom topography approximated by cubic
least .squares, technique.
Figure .13.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40° fl I MM 11111111111111111111111 I— 1 - M ' I I
: (c) Bottom-topography approximated by constrained *. - • . . , , «• * — , <f
.' " - " ' " '"" .bicubic interpolation technique,
i .'• ;' '. Figure 13-- Concluded.
37
75°00' 74°|00'
Y-axis, nautical miles73°|00'
40°
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 14.- Wave refraction diagrams, a = 45°; T = 12 seconds
38 "
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40°00'
(b) Bottom topography approximated by cubic
least squares technique.Figure 1M.- Continued.
39
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40° °g
40
(c). Bottom topography approximated by constrained
bicubic interpolation technique.
. . . Figure 14.- Concluded.
75°00'
40°°El
74°|00'Y-axis, nautical miles
en o> -jm
73°|00'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 15.- Wave'refraction diagrams. a = 45°; T = 14 seconds.4-1
74°|00Y-axis, nautical miles
I
•» . (b.) Bottom topography approximated by cubic.;'"_...;-.••. •. •-•. least squares .technique.;* '•.•'. :-,'. Figure 15.- Continued.
. 75^00' 74°|00'Y-axis, nautical miles
73°|00'
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 15.- Concluded.
75° 00' 74°|00'
Y-axis, nautical miles73°|00'
40°
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 16.- Wave refraction diagrams. <*j= 450. T = 16 seconds
44
75°00' 74°|00'Y-axis, nautical miles
730|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 16.- Continued.
75°|00' 740|00' '
Y-axis, nautical'miles
.73°|00'
(c) Bottom topography approximated by constrained,
bicubic interpolation technique. . -.
Figure 16.- Concluded.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(a) Bottom topography approximated by quadratic " ."
least squares technique.- > . - ' - " • ; - - - •
Figure 1?.- Wave refraction diagrams, a = 67-5°; T =.6 seconds
,• " 75° 00' 74°|oo' : :Y-axis, nautical miles
73° 00'
,, (b) Bottom top.ography approximated ;by cubic
. . .least squares .technique. " -.
Figure 17..- Continued.
"48
75°l00' 74°|00'Y-axis, nautical miles
73°|00'
40°°gl
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 17.- Concluded.
-75° 00' \ 74°|00'Y-axis, nautical miles
73°|00'
(a) Bottom,topography approximated by quadratic.- ;" - - - ' , . * . . . • - " '
•': " -.'.r- least squares technique. ' :
Figure 18.--Wave.refraction diagrams. a = 67-5°; "T.= 8 seconds,
50 > ''!-'••> " :
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40° °g
(b) Bottom topography approximated ,,by cubic
least squares technique.
Figure 18.- Continued. -
51
75°00' 74°|00'Y-axis, nautical miles
73° 00'
40°°gl
52
(c)'Bottom .topography approximated by constrained
bicubic interpolation technique.
Figure 18.- Concluded.
75°00' 74°|00'Y-axis, nautical miles
73° 00'
m cn
(a) Bet-torn topography approximated by quadraticleast squares technique.
Figure 19. - Wave' refraction diagrams.' a = 67.5°; T = 10 seconds.
• - • • ' • • - . , ' 5 3
t.
'»- .Contlnu.,
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
' Figure 19.- Concluded.
55
75°|00' 74°|00'Y-axis, nautical miles
01nr
73°|00'
(a) Bottom topography approximated by quadratic;-.-"-.
. least squares technique. ' .' -• : .
Figure 20.- Wave refraction diagrams, o = 67.5°.;. T =. 12.'.-seconds
5 6 , . " ' • • ' " ' ' , . - • - . • • • • . • , ; . ; . .
75° 00' '
40°°E
74°|00'Y-axis, nautical miles
O!rn
73°00'
.. (b) Bottom topography approximated by .cubic'
least squares technique. ; .
Figure 20.- Continued. ' " '.'- .-'••-
57
58
iiiiSS-----,'•* ;?§::S1 ;§ ^ tec;
: -&&&&s&^:&$*;-i-',f'.,: "~ Concluded
r-;\'>:\%x^;^*.;rM:^^y .^:^§>^^^%::<^-:w>/;''.-,
^^
75°|00' 74°|00'Y-axis, nautical miles'
73°|00'
(a) Bottom topography approximated by"quadratic
. •;. •'• least squares 'technique. - . • :
Figure 21.- Wave refraction diagrams. a =• 67.5°; T .'= I1*' seconds.
. - . 59
• 73° 00'Y-axis, nautical''miles
60
(b) Bottom topography •approximated by cubicleast squares technique.
Figure 21 .'-.Continued.
75°00' 74°|00' . • ".Y-axis, nautical miles
73°|00'
(c) Bottom topography approximated By "constrained
bicubic interpolation technique.- - . .-r-:
• .Figure: 21 Concluded. ...
61
75900'" ' 74°|00'Y-axis, nautical miles
73°00'
Figur,e
62 "•
.'(a). Bottom topography approximated by quadratic •:•;.".'..'>".'>::S' ' least-squares technique. [
22;.'r':Wa.v;e 'refraction diagrams, a = 67.5°;- T = 16 seconds
75°00' 74C|00'Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 22.- Continued.
63
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40°
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 22.- Concluded.
75°00' 74°|QO'Y-axis, nautical miles
73°00'
40°°FF
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 23.- Wave refraction diagrams, a = 90°; T = 6 seconds.
65
.75°|00' 74°|00'Y-axis, nautical miles
73°|00'
\* > 1 4 1 1- *V. i *—J,—4 1 I \ t-J
-' .(r ? \ ' \ 7. "r'. \ '.
\ f i , /t—i—i—*•)* j/ »—i—i—» •»- *,-*—i
66
(b) Bottom topography approximated by cubic
least squares technique.
Figure 23-.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°00'
V ' V ^ - • • .
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 23.- Concluded.
67
75° 00' 74°|00'
Y-axis, nautical miles
73° 00'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 24.- Wave refraction diagrams. a = 90°; T = 8 seconds
68
75°00' 74°|00'Y-axis, nautical miles
73°|00'
tn atTII ii i imp 11 mi 111 in i M ii i ii I M I I I 111 ii i MM 111 M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mi n 11 [ 111 m i n 11 n n »
(b) Bottom topography approximated by cubic
least squares technique.
Figure 24.- Continued.
69
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40° °£
70
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 24.- Concluded.
75°00' 740|00'Y-axis, nautical miles
73°|00'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 25.- Wave refraction diagrams. a = 90°; T = 10 seconds.
71
75° 00' 740|00'
Y-axis, nautical miles
73°|00'
40°.
72
(b) Bottom topography approximated by cubic
least squares technique.
Figure 25.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°00'
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 25.- Concluded.
7-3
75°00' 74°|00'Y-axis, nautical miles
73° 00'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 26.- Wave refraction diagrams, a = 90°; T = 12 seconds
74
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 26.- Continued.
75
75° 00' 74°|00'
Y-axis, nautical miles
73°|00'
76
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 26.- Concluded.
75POO' 74°|00'Y-axis, nautical"mifes
73°00'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 27.- Wave refraction diagrams. a = 90°; T = 14 seconds.
77
75° 00' .• ,74°|00' •L"Y-axi's,; nautical miles
73° 00'
en * ,o>
78
(b) Bottom topography approximated by cubic
least squares technique.
Figure 27.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40° E
(c) Bottom topography approximated by constrained'
bicubic interpolation technique.
Figure 27.- Concluded.
79
75° 00' 74°|00'
Y-axis, nautical miles73°|00'
,4(0:
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 28.- Wave refraction diagrams, a = 90°; T = 16 seconds
•80
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 28.- Continued.
81
75°00' 74°|00'Y-axis, nautical miles
73°00'
40° °g00' E
iiiiiiiiiimniiiiiiiiiiiim-N
54.9m\V
82
(c) Bottom topography approximated by constrained
• .-' bicubic interpolation technique.
Figure 28.- Concluded.
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40°°B?
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 29.- Wave refraction diagrams. a = 112.5°; T = 6 seconds.
83
75° 00' 74°|00'
Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 29.- Continued.
75°log' 74°|00'Y-axis, nautical miles
73°|00'
en 01
j 1 1 1 1 1 n •
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 29.- Concluded.
85
74°00'Y-axis, nautical miles
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 30.- Wave refraction diagrams. a = 112.5°; T = 8 seconds
86
75°00' 740[00'Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 30.- Continued.
87
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40° °E
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 30.- Concluded.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40°°E
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 31.- Wave refraction diagrams, a. = 112.5°; T = 10 seconds,
89
75? 00' 74°|00'
Y-axis, nautical miles
73°|00'
90
(b) Bottom topography approximated by cubic
least squares technique.
Figure 31-.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
re u> en o»
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 31.- Concluded.
91
75° 00' 74°|OQ'
Y-axis, nautical miles
73°00'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 32.- Wave refraction diagrams. a = 112.5°; T = 12 seconds
92
75° 00' 74°|00' .Y-axis, nautical miles
73°|00'
40°
( b ) . Bottom topography approximated by cubicleast squares- techri ique.Figure 32.- Con t inued .
93
74°iQQ'Y-axis, nautical miles
73°l00'
Cc) Bottom topography approximated by constrained
•'.','• bicubic interpolation technique,
•'„;. Figure 32.- Concluded.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
•(a) Bottom topography approximated by quadratic
.,',.' least squares technique.
Figure 3 3...' .-Wave refraction diagrams. a = 112.5°; T = 14 seconds
:.';••?'•• • 95
75°00' 74°|OQ'
Y-axis, nautical miles
73°|00'
96
(b) Bottom topography approximated by cubic
least squares technique.
Figure 33-- Continued.
75°00' 74"|00'Y-axis, nautical miles
73°00'
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 33-- Concluded.97
74°100'Y-axis, nautical miles
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 3U.- Wave refraction diagrams, a = 142.5°; T = 16 second
98
75°|001 . 74°|00'Y-axis, nautical miles
mimiinii
73°|00'
(b) Bottom topography approximated by cubicleast squares technique.
:Figure 3.4 •- Continued.99
75°|00' 74°|00'Y-axis, nautical miles
73°|00'
100
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 3^.- Concluded.
.75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40°°E
_(a) Bottom topography approximated- by quadratic
least squares technique. .
Figure 35'.-"Wave, refraction diagrams, a = 135°; T = 6" "seconds.
101
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40° °E
(b) Bottom topography approximated by cubic
least squares technique.
. Figure. 35.- Continued.
102
75°00' 74°|00'Y-axis, nautical miles
73°00'
(c) Bottom topography approximated by constrained
bicubic interpolation technique.Figure 35-- Concluded.
103
75° 00' 74°|00'
Y-axis, nautical miles
73°|00'
ro to) en tn
(a) Bottom topography approximated by quadratic
least squares technique.
Figure^36.- Wave refraction diagrams, a = 135°; T = 8 seconds
104
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40" "gl
(b) Bottom topography approximated by cubic
least squares technique.
Figure 36.- Continued.
105
75°00' 74°|00'Y-axis, nautical miles
73°|00' ' .
(c) Bottom topography approximated by constrained-
bicubic interpolation technique.
Figure 36.- Concluded.
106
75°JOO' 74°|00'Y-axis, nautical miles
73°00'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 37.- Wave refraction diagrams. a = 135°; T = 10 seconds.
107
75°l00' 74°|00'Y-axis, nautical miles
73° 00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 37.- Continued.
.108
75°l00' . 74°|00'Y-axis, nautical miles
73°|00'
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 37.- Concluded.
109
75° 00' 74°|00'Y-axis, nautical miles
73° 00'
(a) Bottom topography approximated by quadratic
least squares technique. '
Figure 38.- Wave refraction diagrams, a = 135°; T = 12 seconds
110 :
75°|00' 74°|00'Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 38.- Continued.
1 1 1
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40°
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 38.- Concluded.
.112
75°|00' 74°[00'
Y-axis, nautical miles
73°|00'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 39.- Wave refraction diagrams, a = 135°; T = 14 seconds.
113
75°00' 74°|00'
Y-axis, nautical miles
73°|00'
114
(b) Bottom topography approximated by cubic
least squares technique.
Figure 39.- Continued.
74°|00'Y-axis, nautical miles
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 39.- Concluded.
115
75° 00' 74°|00'
Y-axis, nautical miles
73° 00'
40°°gTTTTTTTT
00' E
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 40.- Wave refraction diagrams. a = 135°; T = 16 seconds
116
75°|00' 74°|00'
Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 40.- Continued.
117
75°|00' 74°|00'Y-axis, nautical miles
73°|00'
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 40.- Concluded. • .
118
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(a) Bottom topography approximated by quadratic
• . least squares technique.
Figure 41.- Wave refraction diagrams. a = 157.5°; T = 6 seconds.
••'•• 119
75° 00' 74°|00'Y-axis, nautical miles
73° 00'
40°°g1
120
(.b) Bottom topography approximated by cubic
least squares technique.
Figure 41.- Continued.
75°00'
40°°E
74°|00'
Y-axis, nautical milesJim mm it
73° 00'
(c) Bottom topography approximated by constrained'
bicubic interpolation technique.:- .• • •'
Figure H'\.- Concluded. ' ' . '
121
'75° 00' 74°|00'
Y-axis, nautical miles
73100'
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 42.- Wave refraction diagrams, <* = 157.5°; T = 8 seconds
122 .
75° 00' 74°|00'Y-axis, nautical miles
73°00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 42.- Continued.
123
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40° °pra u>
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 42.- Concluded.
124
75°|00' 74°|00'
Y-axis, nautical miles
73° 00'
(a) Bottom topography approximated by quadratic
least' squares technique.
Figure 43.- Wave refraction diagrams, a = 157.5°;- T =. 10 seconds,
125
75° 00'- 74°|00'
Y-axis, nautical miles
73°|00'
126
-.'. :(.b) Bottom topography approximated by cubic
'..',-;..;.,'; least squares technique.
:".'.-•;. Figure 43.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40°°g
(c)-Bottom topography approximated by constrained
• .'•, bicubic interpolation technique.
;. ••• • ' Figure 43.- Concluded.
127
•75°00' 74°|00'Y-axis, nautical miles
73°|00'
(a) Bottom
Figure -44.-
128 .
topography approximated by quadratic ,
least.-squares technique. - -
Wave, refraction diagrams. a = 157.5°; T.-= 12 seconds
75°00' 74°|00'
Y-axis, nautical miles
73°00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure 44.- Continued.
129
75° 00' 74°|00'Y-axis, nautical miles
73°00'
I'3'O
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 44.- Concluded.
75°00' 74°.|00' :Y-axis, nautical miles
73°l00'
; . .(a) Bottom topography approximated by quadratic
• •;' . " • least squares technique.
Figure--45VT Wave refraction diagrams. o = 157.5°; T = 14 seconds,
":'•'.' ' 131
132
•75° 00' 74°|00'Y-axis, nautical miles
73°|00'
(c) Bottom topography approximated by constrained
bicubic interpolation technique. -; '
Figure 45.- Concluded.
133
75° 00' 74°|00'Y-axis, nautical miles
73°00'
40°
(a) Bottom topography approximated by quadratic
least squares technique. :
Figure 46.- Wave refraction diagrams. a = 157.5°; T = 16" seconds
134 • .
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(b) Bottom topography approximated by cubic
least squares technique.
Figure ^6.- Continued.
135
75°mO' 74°|00'Y-axis, nautical miles
73° 00'
01 <n
136
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 46.- Concluded.
75° 00' 74°|00'Y-axis, nautical
73°|00'miles
00'
39°00'
•-13mc
38°00'
.6|m
V
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 4?.- Wave refraction diagrams, a = 180°; T = 6 seconds.
137
75° 00' 74°|00'
Y-axis, nautical
73?|00'miles
(b) Bottom topography approximated by cubic
least squares technique. • . -.-
Figure 4?.- Continued. . .
138
75°00' 74°|00'Y-axis, nautical
73°|00'miles
40°c
00'
39°00' :CAPE
•3"roc
t/f
'isX
38°00'
.6m 9n
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 4?.- Concluded.
139
75° 00' 74°|00'Y-axis, nautical miles
73°|00'
40°°gl
(a) Bottom topography approximated by quadratic
least squares technique.
Figure 48.- Wave refraction diagrams. a = 180°; T = 8 seconds
140
75°00' . 74°|00'Y-axis, nautical miles
730|00'
(b)' Bottom topography approximated by cubic
least squares technique.
Figure 48.- Continued.
74°00'Y-axis, nautical miles
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 48.- Concluded-.
142
75°00' 74°|00'Y-axis, nautical miles
730|0d'
*i1111111 111uH11 111111 i i1111111111111n i i i 11 i i i i t 1111
(a) Bottom topography approximated by quadratic- ' . '
least squares technique. . . ; ,
Figure 49.- Wave refraction diagrams. a = 180°; T = 10 seconds.
' 143
75° 00' 74°|00'Y-axis, nautical miles
73°00'
.,(b). .Bottom topography approximated by cubic
.: least squares technique. :•: . '
Figure .49..- Continued. '. ''.'.'
•T44
75°00' 74°|00'Y-axis, nautical miles
40°c
00'
39°00' CAPE MA
•38°00'
E /
9i
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 49.- Concluded.
145
40°'00'
39°00'
38°00'
TTTTTTTTT
= DELAWAREBAY
£APE
74°|00
Y-axis, nautical miles
(a) Bottom topography approximated by quadratic
least squares technique.
Figure. 50.- Wave refraction diagrams. a = 180°; T = 12 se.conds
146 -
74°|00'Y-axis, nautical miles
73°00'
(b) 'Bottom topography approximated by cubic
least squares technique.
Figure 50.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
40° °F
148
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 50.- Concluded.
75°00' 74°|00'Y-axis, nautical miles
73°|00'
(a) Bottom topography approximated by quadraticleast squares technique.
Figure 51.- Wave refraction diagrams. a = 180°; T = 14 seconds.
1,49
74°|00'Y-axis, nautical miles
150
(b) Bottom topography approximated by cubic
least squares technique.
Figure 51.- Continued.
75°00' 74°|00'Y-axis, nautical miles
73°00'
f11 1 1 1 1 i i 1 1 1 1 1 1 1 1 1 1 1 i t 1 1 1 1 1 1 1 1 1 1 1 n 1 1 i i 1 1 1 1 1 1 1 1 1 1 1
(c) Bottom" topography approximated by constrained
'•bicubic interpolation technique.
. ' Figure 51.- Concluded.
151
75° 00' 740|00'
•Y-axis, nautical miles
73°|00'
40°e
00'
£ DtLAWAREBAY
39°00' tCAPE MA
38°00'
(a) Bottom topography approximated by quadratic ';'.
• . '.•'.• .least, squares technique.
Figure 52.- Wave refraction'diagrams. a = 180°; T = '16 seconds
152 : : ; : - " - •' • ' . ' ':'.' ' . : • ' -.
75°00' 74°|00'Y-axis, nautical miles
73°00'
40°°El
(b) Bottom topography approximated by cubic
least squares technique.
Figure 52.- Continued.
153
75° 00' 74°|00'
Y-axis, nautical miles
73°|00'
40°
154
(c) Bottom topography approximated by constrained
bicubic interpolation technique.
Figure 52.- Concluded.
REFERENCES
1. Goldsmith, Victor; Morris, W. Douglas; Byrne, Robert J.; and
Whitlock, Charles H.: Wave Climate Model of the Mid-
Atlantic Shelf and Shoreline (Virginian Sea) - Model Devel-
opment, Shelf Geomorphology, and Preliminary Results. NASA
SP-358, VIMS SRAMSOE No. 38, 1974.
2. Poole, Lament R.: Random-Access Technique for Modular Bathym-
etry Data Storage in a Continental-Shelf Wave-Refraction
Program. NASA TM X-3018, 1974. ' •• .
3. Poole, Lamont R.: Comparison of Techniques for Approximating1 Ocean Bottom Topography in a Wave-Refraction Computer, Model.
NASA TN D-8050, 1975.
4. Goldsmith, Victor: Shoreline Waves, Another Energy Crisis.
VIMS Contrib. No.'734 to First Annual Conference of the
Coastal Society (Arlington, Virginia), Nov. 1975. '
Langley Research Center .
National Aeronautics and Space Administration
Hampton, VA 23665
September 8r 1976 . { ' :
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