AbstractThe basic problem involved in determining where the ship can not go is an attempt to reconstruct the sea bed. The interpolation of points necessary to reconstruct the sea bed was done using a bicubic spline. This method was chosen because of the similarities between the boundary conditions believed to be characteristic of the modeling problem and those of the natural spline. These include the continuity of the first and second derivatives, and the minimum curvature exhibited by the spline method which is characteristic of the sea bottom. The major problem faced in modeling the sea bed was selecting the extra data points needed in order to find a meaningful solution. This selection was done both by intuition and by constructing splines to model the possible behavior along a straight line. The results were two different models: a ridge model, characterized by a single shallow ridge in the center of the region; and a hill model, characterized by two smaller ridges. By varying one of these extra data points (called critical points), several models of both these extremes as well as intermediate models were generated. However, it was found that the number of given points did not permit a definitive model. Data was needed inside the region, especially at the critical points and at the exterior points in order to better define the boundary. The boundary could not be reliably determined since our spline model does not allow for accurate extrapolation. Thus, the model, although close to what is believed to be the correct model, is not good enough to allow for navigation because of the limited number of given data points

Davies, Andrew

Duff, G.F.D.

Kinoshita, Mark

Van de Water, Richard

Elsevier - Publisher Connector 

0270-0255186 S3.W - .OO 
Copyriaht C 1986 Pqamon Journals Ltd. 
SPLINE ANALYSIS OF HYDROGRAPHIC DATA 
TEAM: AVDREW DAVIES, MARK KINOSHITA, E&HARD VAN DE WATER 
FACULTY ADVISOR: G. F. D. DUFF 
Department of Mathematics 
University of Toronto 
Toronto, Ontario, Canada MS IA1 
Abstract-The basic problem involved in determining where the ship can not go is an 
attempt to reconstruct the sea bed. The interpolation of points necessary to reconstruct 
the sea bed was done using a bicubic spline. This method was chosen because of the 
similarities between the boundary conditions believed to be characteristic of the mod- 
eling problem and those of the natural spline. These include the continuity of the first 
and second derivatives, and the minimum curvature exhibited by the spline method 
which is characteristic of the sea bottom. The major problem faced in modeling the sea 
bed was selecting the extra data points needed in order to find a meaningful solution. 
This selection was done both by intuition and by constructing splines to model the 
possible behavior along a straight line. The results were two different models: a ridge 
model, characterized by a single shallow ridge in the center of the region; and a hill 
model, characterized by two smaller ridges. By varying one of these extra data points 
(called critical points), several models of both these extremes as well as intermediate 
models were generated. However, it was found that the number of given points did not 
permit a definitive model. Data was needed inside the region, especially at the critical 
points and at the exterior points in order to better define the boundary. The boundary 
could not be reliably determined since our spline model does not allow for accurate 
extrapolation. Thus, the model, although close to what is believed to be the correct 
model, is not good enough to allow for navigation because of the limited number of 
given data points. 
INTRODUCTION 
The model studied in this paper is the detailed determination of the depth of a body of 
water. Initial data consists of 14 (x, y) water surface coordinates and respective z coor- 
dinate depths. Applying spline interpolation techniques to this data, a detailed three- 
dimensional construction of the sea bed is obtained. The construction is obtained in the 
form of contour maps which can be used as depth navigational charts. We will focus our 
attention on a particular boat with a 5 ft draft. 
ASSUMPTIONS 
In constructing the contour map, an interpolation, based on 14 given depths must be 
made to other regions within the (x, y) domain of the initial data. Thus we are faced with 
the problem of finding a surface z = F(x, y) which represents the sea bed. This function 
does not have to be smooth or continuous (i.e. there can exist sharp peaks, rock for- 
mations, coral reefs, etc). These possibilities will be excluded and it will be assumed that 
the sea bed is smooth and continuous. To be more rigorous, the first and second derivatives 
of F(x, y) are continuous with no jump discontinuities. Physically, this would correspond 
to a fine gravel or sand bottom, which is common in shallow water. It now remains to 
586 ANDREW DAVIES CI (I/. 
find a physical principle that gives us insight into the general shape of a surface acted 
upon by water. Since the region being considered is shallow. it will be assumed that the 
water is flowing over the sea bed. Therefore the water will erode any structures which 
perturb the flow from its minimum lateral kinetic energy (i.e. the kinetic energy of motion 
perpendicular to the direction of flow). This implies that the most favorable sea bed that 
minimizes this function is a plane surface. The direction of motion is not perturbed except 
by otherwise negligible friction effects between water and the sea bed. Therefore any sea 
bed which is not a plane surface will tend towards one through erosion effects. This model 
is of course idealized and does not take into account such factors as sedimentation or 
large unerodible structures. Thus, there are two conditions imposed on the sea bed: 
(1) At least the first two derivatives are continuous. 
(2) Given the initial structure, the surface constructed is of minimum curvature. 
CHOICE OF INTERPOLATING SCHEME 
Interpolation is the fitting of a curve through a given set of data points: in this instance, 
by the use of polynomial approximations. Such a polynomial would have to be of fairly 
low degree to reflect the fact that the sea bed is flat, and the method of cubic splines does 
this well. The idea of a spline through a set of data points is best illustrated by imagining 
a straight flexible rod being placed over a set of points and putting weights on the rod to 
force it to pass through all the points. It is the straightest line through the points. This is 
precisely one of the conditions of the model. Cubic splines are differentiable piecewise 
polynomial functions on an interval [.U,,, X,,] where X,, < X, < ... < X,, are called nodes. 
The function is obtained by fitting a cubic polynomial between each successive pair of 
nodes. This is accomplished by fitting a cubic on [X,,. X,] agreeing with the function at 
X,, and X,. another cubic on [X,, A’,] agreeing with the function at XI and X2, etc. A 
x Crnh4IE (ME) 
Dci%lNfw o UCMlEDFUNT x GTlWFUrir 
DEPTH IN FEET 
Fig. I. Initial data coordinates for spline analysis. 
Spline analysis of hydrographic data 557 
general cubic polynomial involves four constants. Thus there is sufficient flexibility to 
ensure that not only the interpolant is continuously differentiable on the interval, but also 
that it has continuous second derivatives on the interval, even at the nodes. Thus, cubic 
splines satisfy the two conditions of the model; they have continuous first and second 
derivatives and can furnish the straightest curve fittin g the given data points. For this 
model the natural spline boundary condition is chosen. This corresponds to the second 
derivatives being zero at the end points X0 and X,. Such a condition allows for the max- 
imum flexibility in determining the straightest curve through the given nodes. 
BICUBIC SPLINE PROCEDURE 
Since the sea bed has two independent variables, it requires generalization of the one- 
dimensional spline method to two dimensions. This is called a bicubic spline. It is nec- 
essary to arrange the data points in an m x n grid formation = {(Xi, Yj) / 0 < i < n, 0 <j 
< m}. From this it is possible to find the value of the interpolating surface F(X, Y) at (X’. 
Y’) by running one-dimensional cubic spline interpolations along the grid lines X = Xi 
Table I. Initial grid data (nodes) 
(a) The data are given as (.v coordinate. depth). 
The points A. B and C are critical points which determine the form of the model. A = CR (145.50) is the 
dominant critical point. with 3 < Z < 7 the depth taken at the critical point. 
The points B = CR (125.115) and C = CR (I85.-20) are milder constraints on the model. BZ may var? 
between 5 and 7.5, and C is generally taken as shown (although it can be moved and have its : coordinate 
changed). 
The columns correspond to the x coordinates: Col. I: X = 85: Col. 2: X = 105: Col. 3: X = 125; Col. 4: X = 
145; Col. 5: X = 165: Col. 6 X = I85 
No\Col no I 2 3 4 
I ( - 80.9) ( - 8 I .9) (-75.9) ( - 70,9) 
2 (3.7.9) (28.6) ( - 40.9) ( - 25.9) 
3 (56.5.8) (85.5.8) (7.5,4) 
4 (147.8) (113.8) B (11:5,8, 
5 (145,8) 
(b) Example: of spline calculations for determination of depth of A(50.Z) 
5 6 
(-65.9) (-65,9) 
( - 6.5.9) c 
(84.4) (22.5,6) 
(146.8) (140.8) 
The three one-dimensional spline calculations: 
I. Line A: Ridge model result CR (l-15.50). Z = 3: 2. Line B: Hill model result CR (14550). Z = 6.9; 3. Line 
C: result CR (145,60). Z = 4. 3 or CR (l-15.70) = Z = 4 
Coordinates 
Line X Y Depth Description 
A 117.5 -38.5 9 
I29 7.5 4 
162 84 4 
I85 l-10 8 
I45 50 3.02 
B 185.5 22.5 6 
IO5 85.5 8 
l-t5 50 6.87 
C 75 -5 8 
108.5 28 6 
I62 8-t 4 
215 i-to 8 
Given value 
Given value 
Given value 
Modified value 
Spline result 
Given value 
Given value 
Spline result 
Modified value 
Given value 
Given value 
Modified value 
I-t5 40 4.01 Spline result 
588 ANDREW D.-\CIES et al. 
through the data F(Xi, rj), 0 <j < m and evaluating each spline at Y = Y’. Then a cubic 
spline interpolation is performed along Y = Y’, through the values obtained above, and 
then this is evaluated at X = X’. 
A one-dimensional cubic spline program[l] was modified so as to perform the bicubic 
spline interpolation as outlined above. The program listing is given in the Appendix. The 
region (75,200) x (-50, 150) was divided into six columns parallel to the y-axis. This 
orientation was chosen because the maximum number of nodes (given data points) could 
be obtained and they were better distributed along this direction [refer to Fig. 1 and Table 
I(a)]. The node grid input into the program consists of initial data points along the ap- 
propriate columns. The position values assigned to the columns are given in Table 1, along 
with the 4’ and z coordinates of the nodes and their grouping into columns. &lost of the 
nodes used are given data points, or given data points slightly extrapolated to correspond 
to the center of the column. The program then performed a spline analysis down the I 
columns and interpolated 41 equally spaced points per column. These points were then 
used to construct a spline along y = constant with 20 equally spaced points interpolated. 
The output was in the form of a contour map with a resolution of 70 x 41 extrapolated 
points. The contour map resolution 20 x 41 was selected because it is the compromise 
between tolerable computation time and maximum resolution. 
MODEL PARAMETERS 
The data given is not sufficient. Extra points must be given in order to clarify the 
possible nature of the sea bed surface. Observing the general form of the surface based 
of the given data. it becomes apparent that two possible structures exist. These structures 
are centered about the two given data points with depths z = 4. These structures can be 
either a long ridge or two smaller hills each centered about the points (129, 75) and (162, 
8-t). The point that discriminates between these two models is the critical point CR(l45, 
50). A variation in depth of this point will decisively alter the results. To determine the 
Table 2. Impassable area sensitivity analysis [for depths (Sft)] 
(Total grid area = 820 sq. units) 
&lode1 
Depth Central 
5 peak Other 
Total 
area 
Percent 
of 820 
Ridge 3.0 
CRI 185. - 20) -8.0 I40 6 I16 17.80 
CR1185.--20) 9.0 141 I4 155 18.90 
CR(l85.-85) 8.0 
CR( 192.81) 6.4 I37 22 1’7 19.15 
CR( 192.81) 6.0 II3 9 122 11.88 
CRI 192.81) 7.0 108 I-I I22 Il.88 
CRIl85.-‘0, 7.5 0.00 
CRf 192.85) 6.0 113 7 I20 I-1.63 
CRC 125.5.5) 5.5 Ii7 5 163 19.88 
Intermediate 
CRI I15 50) 
CRi 145 50) 
CRf I15 501 
4.0 130 1 134 16.3-I 
5.0 II3 3 II6 II.15 
6.0 85 0 85 IO.37 
Hill 
CR( I45 50) 7.0 86 0 86 IO.19 
“CR(X. )‘I = critical point at position .r..v Other peak centralized at 1X0.24) ‘Standard model CRC 185. -20) 
R = 8. CR1125.1 IS, R = 7.5 
Spline analysis of hydrographic data 589 
possible range of depths for this point a one-dimensional spline analysis was carried out 
along the major trend lines of the sea bed. Trend lines for the critical point CR(I-tS. 50) 
are the lines between the two points where ; = 4 (trend A) and the line that is perpendicular 
to this direction and passes through (185.5. __._ ?’ 5) and (103, 85.5) (trend B, see Fig. I). X 
spline analysis is performed along these two lines to determine the range of variation of 
depth at the critical point [see Table I(b)]. Two other milder critical points exist at CR(l3. 
115) and CR( 185. - 20). Analysis similar to the one performed on CR(145, 50) gave depth 
ranges as indicated in Table I(a). These two critical points are less sensitive to variations. 
SPLISC DiIPTH ASALYSIS 
:2!:TC'JR RAF 
RID:E rlclOCL CAii4S 501 ze3 
Y 
I50 
14s 
140 
13s 
130 
12s 
120 
IIS 
I10 
IOJ 
LOO 
95 
90 
BS 
80 
7S 
70 
bS 
60 
55 
50 
45 
40 
3s 
30 
2S 
20 
1s 
10 
5 
0 
-s 
-10 
-1) 
-20 
-25 
-30 
-3s 
-40 
-4s 
-SO 
Y 
X- COOROlNATE 
7s 88 101 IIS 128 I41 IS4 lb7 180 LPI Y 
10 
80 
80 
79 
79 
79 
79 
7) 
78 
78 
?I 
78 
78 
79 
110 IO 80 80 79 79 80 1 84 87 89 90 PI 90 88 86 84 I I IS0 
14s 
140 
a2 9s 108 121 114 147 161 174 ia7 200 Y 
X- COORDINATE 
"OEPTHS IN II10 FEET 
X.Y COORDINATES IN YARDS 
(a) 
Fig. 2. (a) Spline depth analysis contour map. Depths in Tt, ft: A’. Y coordinates in yards. Ridge klodsl: CRI II!. 
50) Z = 3; (b) Intermediate Model: CR(1-U. 50) Z = 5: (c) Hill Model: CR( I-15. 50) Z = 7. 
590 ANDREW D.*VIES zf ul. 
Thus, for the analysis of the main critical point variations, they are assigned values: for 
CR(125, 115), z = 7.5 and for CR(185, -20). ; = 8. 
ANALYSIS 
Thirteen bicubic spline analyses were run. The results are summarized in Table 2. Most 
runs were performed with the aim of determining the effects from the variation of the 
depths of the mild critical points. These effects were not the main purpose of the analysis 
but were used to determine sensible values of the depths of the milder critical points. The 
SPLINE DEPTH ANALY:!j 
CONTOUR KAP 
INTERHEOIATE KOOEL CR(l4S 551 zr5 
Y 
IJO 
14s 
140 
13S 
130 
125 
120 
11s 
II0 
IOS 
100 
95 
90 
as 
80 
7S 
70 
65 
60 
ss 
so 
4s 
40 
3S 
30 
ZS 
20 
1s 
IO 
S 
0 
-S 
-10 
-1s 
-20 
-25 
-JO 
-35 
-40 
-4s 
-so 
Y 
X- COORDINATE 
7s aa 101 IIS 128 I41 IS4 167 1110 IV3 Y 
a0 a0 80 80 79 ?9 79 a 81 82 a4 8b a 
BG 79 78 77 7 
4a 44 41 41 4 
2 91 94 9s 94 9 
3 93 93 93 94 94 9 
a2 9s 108 121 134 147 lb1 174 187 200 Y 
X- COOROINATE 
(b) 
**DEPTHS IN I110 FEET 
X.Y COOROINATCS IN YAROS 
Fig. 2. (continued) 
Spline analysis of hydrographic data 591 
primary analysis was the variation of the depth of the main critical point. Figures Z(a). 
(b), (c) are contour maps that represent the topological features of the sea bed. Fig. I(a) 
is the ridge model (i.e. : = 3) in which the main feature is a large central ridge. The area 
within the dashed contour line represents the area impassable to the boat. This area 
represents about 17.8% of the total area in consideration. Figure 3_(c) is the hill model 
with z = 7. It is observed that there are two distinct impassable shallow regions with a 
small passable ridge between them. Figure 3 is a plot of the percent of the area impassable 
to the ship as a function of depth of the main critical point CR(145, 50). Note that the 
contour map for the ridge model shows a second shallow region at (200, 30). This result 
was totally unexpected and further analysis [variation of critical point CR( 185, - 20) and 
SPLINL DEPTH ANALYS:S 
CONTOUR XAP 
HILL IlODiL CR(I45 SO) 2=7 
Y 
1sa 
195 
140 
13s 
130 
12S 
120 
115 
110 
LOS 
100 
9s 
90 
8s 
80 
7S 
70 
65 
60 
5s 
so 
IS 
40 
3s 
30 
25 
20 
IS 
10 
S 
0 
-5 
-10 
-1s 
-10 
-2s 
-10 
-IS 
-40 
-4s 
-50 
Y 
X- COORDINATE 
7s a11 101 11s 128 IPI IS4 167 180 I91 
0 81 81 81 0 00 79 79 ,9 7 
* 81 82 at 0 79 78 fa 78 7 
1 81 al 82 0 78 7a 7a 77 7 
0 77 76 77 76 7 
I 83 81 82 9 76 7S 76 7S 7 
a3 a4 at 8 74 73 74 74 7 
1 94 9s 94 
82 9s 108 121 134 147 ii: 174 187 200 
I- COORDINATE 
Y 
IS0 
145 
I40 
135 
130 
12s 
120 
115 
110 
10s 
100 
9S 
90 
8s 
80 
7s 
70 
6S 
60 
SS 
so 
4S 
4c 
3s 
30 
25 
20 
IS 
IO 
5 
0 
-5 
-10 
-IS 
-20 
-25 
-30 
-IS 
-40 
-45 
-sa 
(cl 
Fig. 2. (continued) 
592 ANDREW DAVIES et al. 
40 
( Standard model : CR (185,- 20) z = 8 ) 
5- 
Depth of CR (145,50) (ft) 
Fig. 3. Impassable area vs. depth of CR(145, 50). 
other points in column 61 showed it to be stable to variations. Thus this must be a real 
result. 
CONCLUSION 
The single largest source of error in the analysis is the lack of sufficient data. If a larger 
set of depth measurements are systematically made, then we would feel more confident 
telling the captain of the boat to use our contour maps. As it stands, a decision as to 
which model is correct, the ridge or hill model, could be obtained by making a depth 
measurement at the critical point CR(145, 50). Further analysis with larger sets of data 
would determine whether the overall assumptions made are valid. We were in the process 
of running the spline program on data taken from real contour maps and making further 
comparisons before time ran out. 
Acknowledgements-We would like to thank Research Services, Great Lakes Region, Canadian National Rail- 
ways for the use of their computer systems during the weekend of February 8, 1986. 
1. 
. 2. 
3. 
REFERENCES 
W. H. Press et al., Numerical Recipes. Cambridge U. P., Cambridge. 
R. L. Burden, J. D. Faires and A. C. Reynolds, Numerical Analysis, 2nd Edn. Prindle, Weber & Schmidt, 
Boston, MA (1981). 
J. F. A. Sleath, Sea Bed Mechanics (1984). 
Spline analysis of hydrographic data 
APPENDIX 
Program listing 
593 


Spline analysis of Hydrographic data 

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Spline analysis of Hydrographic data

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