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Observations on inshore ocean waves with an analytical stereoplotter
Photogrammetria, 40 (1986) 327-342 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
327
OBSERVATIONS ON INSHORE O C E A N W A V E S WITH AN ANALYTICAL STEREOPLOTTER
H.L. MITCHELL Department of Civil Engineering and Surveying, University of Newcastle, Newcastle, N.S. W. 2308, Australia (Received June 6, 1984; revised and accepted November 18, 1985)
ABSTRACT
Mitchell, H.L., 1986. Observations on inshore ocean waves with an analytical stereoplotter. Photogrammetria, 40: 327--342. The software of an analytical stereoplotter was modified to enable ocean wave parameters to be measured from stereophotographs taken from shore. The separate relative and absolute orientation stages were replaced by a single-stage orientation, which uses the horizon as the only visible control. Non-metric cameras, located close to shore and up to 22 m apart have been used to measure height differences on the water surface with a precision of 0.3 m. The results suggest that the analytical plotter technique could contribute to the collection of inshore wave data.
INTRODUCTION Terrestrial p h o t o g r a m m e t r y is such a useful m e n s u r a t i o n t e c h n i q u e f o r r e m o t e l y sensing objects, t h a t it suggests itself as having a role in the r e c o r d i n g o f t h e shape o f the o c e a n surface. Surf wave heights have been m e a s u r e d previously b y the writer, using an analytical s o l u t i o n o f m o n o c u l a r l y observed plate c o - o r d i n a t e s o n p h o t o g r a p h s t a k e n f r o m shore (Mitchell, 1983). T h e results o b t a i n e d in t h a t m a n n e r suggested t h a t it w o u l d also be feasible to use a s t e r e o p l o t t e r instead o f m o n o c u l a r observations to collect d a t a o n inshore wave processes. This p a p e r outlines the m o d i f i c a t i o n s w h i c h were s u b s e q u e n t l y m a d e t o a Qasco SD-4 analytical s t e r e o p l o t t e r , for t h a t purpose. S o m e results have been o b t a i n e d in the o b s e r v a t i o n o f inshore waves with the s t e r e o p l o t t e r , a n d assessments o f precision have been made. The n e e d for ocean wave data Waves o f various heights, lengths and periods can arrive f r o m d i f f e r e n t directions at a coastal or o f f s h o r e site w h i c h is u n d e r investigation b y a coastal engineer. W h e t h e r t h e engineer's p r o b l e m lies in the design o f c o m mercial or recreational facilities or w h e t h e r it lies in beach a n d shoreline
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© 1986 Elsevier Science Publishers B.V.
328 protection and maintenance, a knowledge of the waves which may be encountered is an important component of the input needed for the design work. The wave data may be used to indicate the average and extreme values of both wave heights and wave energy levels. These suggest the forces which a structure must be expected to face or which will influence the transport of suspended sediment or the erosion of a beach. Estimates of the wave climate may be developed by the engineer from a knowledge of the meteorological conditions which give rise to the sea or swell arriving at the site in question. Alternatively, a description of the wave conditions may be derived from direct measurements made at, or in the vicinity of, the site. If satisfactory observations are n o t already available to the engineer, efforts must be made to record data in whatever time is available. According to Silvester (1974), an engineer may well be fortunate if a year's records can be provided. Various wave recording techniques are presently available (see Silvester, 1974, p. 274 et seq. for a summary of wave recording methods), but automatically recording instruments are widely used. The most c o m m o n instruments would appear to be those which involve accelerometers housed in buoys floating on the water surface. The accelerometers respond to the rise and fall of the water surface as sea and swell pass; double integration of the recorded accelerations with time provides a record of the vertical displacement of the ocean surface. Buoys equipped with accelerometers are widely used and give satisfactory results in m a n y cases. However, they are not w i t h o u t their disadvantages, which include the following: (a) the recorder in the open ocean is prone to damage, corrosion and loss; (b) the buoy must be secured in, and recovered from, deep water; (c) the recorded data must be recovered from the offshore site; (d) because buoy recorders are normally located in deep water, seaward of the site in question, it is necessary to m o d i f y the recorded wave profile to account for the refraction and diffraction of the waves as they travel between the recorder site and the shore, where the wave pattern is needed; (e) w i t h o u t the assistance of a synchronized second recorder, the directions of waves can be difficult to deduce (giving rise to a problem in predicting the arrival of waves at the shore). B o t t o m - m o u n t e d recorders may be used as an alternative to floating-buoy recorders. Although the instruments are protected from damage by the atmosphere, by the waves or by surface traffic, recovery of the instrument and data can be even more difficult than for a float recorder.
A role for photogrammetry The apparent benefits of a photogrammetric m e t h o d of wave recording when compared with other wave recording methods have been suggested by Mitchell (1983). The advantages centre on the ability of photogrammetry to keep the recording equipment free from contact with the ocean and its
329
users, whilst also enabling easy retrieval of the instrumentation and the recorded information. Furthermore, the collected data refer to occurrences at the shoreline, n o t in deeper offshore water away from the site under investigation. The photogrammetric m e t h o d may also enable the collection of wave parameters other than only wave heights (for example, wave length and wave directions). The automatic wave-recording instruments, whether floating or bottomm o u n t e d , have the advantage of enabling wave profiles to be abstracted from the recorded data by an automated process. Such automation is a distinct advantage when considerable quantities of data must be processed, as is the case when wave statistics are being compiled. It is against this advantage that the apparently labour-intensive photogrammetric methods must be assessed. Photogrammetry is therefore likely to be best suited to situations when the conventional instrumentation is unsuitable: to observe wave action close to shore, or when only a short period of observations is needed or is possible (for example, to record special events, such as storms, to record whilst alternative instruments are inoperative or to check other recorders). An experienced photogrammetrist may have some doubts about a photogrammetric method's usefulness, considering the difficulty of providing control points on the sea surface. With terrestrial photogrammetry, this is not restrictive if an analytical stereoplotter can be used and if its software can be modified. The horizon can be used to indicate camera orientations. Furthermore, the analytical plotter, if its software can be altered, can facilitate data abstraction. It can provide prompts for non-photogrammetrist operators; it can direct the movement of the floating marks along appropriate pre-determined paths when wave data are being abstracted, and it can utilize fiducial marks to orient successive photographic pairs which have the same camera positions. The photogrammetrist must also remember that precision requirements for wave data are n o t severe. Not only is it unnecessary to precisely delineate a moving ocean wave, but also, it is normal to base wave data on a series of observations over time. A single stereopair would provide only part of the data. WAVE HEIGHTS FROM AN ANALYTICAL SOLUTION
Earlier work in this study of photogrammetric wave measurement centred on computer programs providing a least-squares solution of the collinearity equations (Mitchell, 1983). The co-ordinates of both cameras and the orientations of the cameras about two axes were held fixed, the two known camera orientations being determined from the appearance of the horizon on high-oblique photographs. The unknowns in the solution consisted of the remaining orientation parameter for each camera, as well as the ground coordinates of the object points. The computer program also formed the variance-covariance matrix of the determined unknowns, and this enabled
330 error simulation studies to be undertaken to assess the relative accuracies afforded by various camera configurations. A pair of non-metric 3 5 m m Nikon cameras set up on a beach in Newcastle, New South Wales (latitude 32°50 ' south, longitude 151°40 ' east) provided data which were used to co-ordinate points in a breaking wave area. Monocularly observed plate co-ordinates were used to deduce co-ordinates of 12 object points in the ground co-ordinate system. By monocularly reading the plate co-ordinates, any possible errors or difficulties caused by stereoscopic viewing of the high oblique photographs were avoided. Object points which were chosen using an enlargement of one of the photographs of the pair, were used if they could be confidently located on both negatives of the pair. The variancecovariance matrix of the co-ordinates suggested that the standard deviations of height differences were no more than +50 mm for distances up to 75 m from the camera, a result which was considered to be satisfactory for wave height measurement. The effects of systematic errors (in the focal length of the cameras, for example) were examined by obtaining further solutions in which realistic values of the systematic errors were intentionally included. The effects of the errors were not found to be significant (see Mitchell, 1983, table II). Maresca and Seibel (1976) have also used terrestrial photogrammetry to measure wave heights (and to observe currents). Their m e t h o d is fairly simple, and it does not appear to be amenable to the collection of considerable quantities of data. A pair of cameras was oriented so that the two cameras' axes were estimated to be level in two directions, and the direction of pointing of the axes was determined from control markers in the foreground. Co-ordinates were measured off the photographs after corresponding images had been identified. APPLICATION TO THE ANALYTICAL STEREOPLOTTER The analytical solution of monocularly read plate co-ordinates is n o t a satisfactory means of co-ordinating a large number of points on a single stereopair or for collecting data from a number of stereopairs. The approach used in the analytical solution was therefore adapted to an analytical stereoplotter. In this case, modifications were made to the software of a Qasco SD-4 analytical stereoplotter, a device which is described by Elfick and Fletcher (1982). If the apparent horizon is used to determine the camera orientations and if the ground co-ordinates of the cameras are known, the absolute orientation procedure, which requires the use of control points, cannot be followed with wave photographs. Two alternatives were considered: (a) the relative orientation could be followed by a special case of the absolute orientation; or (b) a solution, equivalent to combining the relative and absolute orientation stages, could be adopted. Maintaining separate relative and absolute orientations had two disadvan-
331
tages. Firstly, the relative orientation of the high oblique photographs is weak, presumably because it is not possible to select object points at the optimally located Von Gruber positions. (The photographic overlap varies with the distance of the waves from the cameras, and there is always a loss of usable area above the horizon on high oblique photographs.) Some attempts to obtain reltive orientations of high-oblique surf photographs led to results which were clearly unsatisfactory. Secondly, it is mathematically complex to deduce the absolute orientation parameters given relative orientation results (in which the right-hand camera is positioned with respect to the left-hand camera), given the ground co-ordinates of the two cameras and given two orientations of each camera. The weakness of the relative orientation is not apparent in the combined solution, in which the number of unknowns is reduced. Consequently, it was decided that the software of the Qasco SD-4 analytical plotter would be modified to enable a single-stage orientation to be carried out on the wave photographs. Description o f the revised software
The alteration of the stereoplotter software involved the modification of the relative orientation procedure to account for the change in the set of known and u n k n o w n parameters. The modified relative orientation involves the least-squares solution of the collinearity equations as used for the fully analytical solution. The standard absolute orientation routine normally provided with the plotter becomes redundant when the altered routines are being used. It should be made clear that the operator can still call up the original software, when the stereoplotter is needed for conventional work. In the standard SD-4 software, the relative orientation may be based on a minimum of six and a maximum of 10 points. The least-squares solution of the relative orientation is based on linearized collinearity equations, written in the form of the conventional observation equations, that is, each equation is in terms of only one observed plate co-ordinate. To minimize the a m o u n t of alteration needed, the revised software which undertakes the "combined relative and absolute orientations" is also based on observation equations. Accordingly, the values of w and K, which are determined for each camera from the horizon, are held fixed in the solution. According to a rigorous approach, they should be regarded as observations, but this requires a more complex solution, as the following explanation suggests. The set of linearized collinearity equations is written in matrix form as: BV+AX+C
= 0
(1)
where V is the vector of corrections to the observations; X is the vector of corrections to the a priori estimates of the required parameters; B and A are coefficient matrices; and C is the vector of constants. The solution in the Qasco SD-4 requires that:
332 B=I
The solution for X is then given by: (2)
NX + A Tc = 0
where N
(3)
= ATA
If, on the other hand, w and K are observations, then: B =/= I
and the solution for X is given by: X
= -N -1AT(BGBT)
-1 C
(4)
where N
= A T ( B G B T ) -1 A
(5)
and where G is the variance-covariance matrix for the observations. The stereoplotter software was modified to provide the simpler of the two solutions. The deficiency of this solution, in comparison to a solution in which co and K are included as observations, was tested; see below. Because the set of unknowns in the stereoplotter solution comprises the X, Y and Z co-ordinates of the object points and the ~ orientations of both cameras, the minimum number of object points needed for a solution without redundant observations is two. A solution with ten object points, for example, would provide eight redundancies. The results of orientations reported in this paper were checked using the computer program which was written for the fully analytical solutions reported earlier (Mitchell, 1983). The program was based on the same equations as those used in the SD-4 plotter, but the program used a different solution algorithm to that of the stereoplotter. This program also has the facility to incorporate ¢o and K as observations; its use in this manner is mentioned below. In addition, the program formed Z XX, the variancecovariance matrix for the parameters: ZXX = ao N-1
(6)
where Oo is a variance factor estimated by: ao = v T G - 1 V / r
(7)
and r is the number of redundancies. In the standard SD-4 software, the least-squares solution is obtained by an elimination m e t h o d applied to eqn. (2), and n o t by inversion of the matrix N, as used in eqn. (4). Consequently, the matrix ZXX is not presently formed by the stereoplotter software.
333
The use of the horizon Mitchell (1983) explained t ha t the horizon on high oblique photographs does n o t enable camera orientations to be deduced immediately if the (x,y) plate co-ordinates are defined in the conventional manner (x across the p h o to g r ap h to the right, y towards the top of the photograph). This was overcome in the analytical solution of the monocularly observed points by rotating the x and y plate co-ordinate systems on the photographs and making corresponding changes to ground co-ordinate system. For example, heights were given by the Y co-ordinates. To avoid difficulties caused by redefining the plate co-ordinate system in the SD-4 software, an alternative solution has been a dopt e d for the stereoplotter routines. The rotations ¢o, ~, K ab o u t the cameras' x, y, z axes are assumed to have occurred in the order (¢, co, K), instead of (co, ~, K). The r o t at i on matrix M ~ relating the ground and camera co-ordinate systems for the SD-4, has elements as follows: m,, = cos ~ cos K + sin co sin ~ sin m,2 = c o s w sin m13 = s i n ~ c o s ¢ s i n ~ - s i n ~ c o s K rn2, = sin co s i n O c o s ~ - c o s @ s i n K m22 = cos co cos K m23 = s i n ~ s i n ~ + s i n c o c o s ¢ cosK m3, = cos co sin
m32 = - sin m33
=
cos
¢o c o s
The relevant co-efficients of the corrections to the parameters in the linearized collinearity equations are as in Table 1, rather than as given in most t e x t and reference books for M~¢~ (e.g. Wolf, 1974, p. 533; Wong, 1980, p. 57). The orientations w and K for any camera can now be calculated from the apparent 'dip' and 'slope' of the horizon as seen on a photograph: = ~' + ~
(8)
where co' is the dip of the apparent (rather than true) horizon, and is given by: tan co' = - p / f
(9)
Aw is the correction for refraction and for the difference between the true and apparent horizons; p is the perpendicular distance from the horizon to the principal point, measured positive upward from the principal point, f is the camera focal length, and
334 TABLE 1 C o e f f i c i e n t s o f c o r r e c t i o n s t o t h e a priori e s t i m a t e s o f t h e p a r a m e t e r s in t h e linearized c o l l i n e a r i t y e q u a t i o n s f o r ith c a m e r a a n d for o b j e c t p o i n t A Parameter
C o e f f i c i e n t o f c o r r e c t i o n to e s t i m a t e s o f p a r a m e t e r in e q u a t i o n in x
in e q u a t i o n in y
f ( A X m~ : sin~ - A Y sinco sink
f ( A X m22 s i n ¢ - A Y s i n ¢ o cosK
r
r
+ A Z m ~ cos¢) X +--
( A X m 3 2 sinq~ r + AZ m ~ cos40
+ AZ m ~ cos~)
f ( ~ X m,3 - a Z m,~) r X
+-
+ Y ( A X m~2 sin~ - A y cos¢o
~Ycos~0
r
+ ~ Z m32 c o ~ ) f ( A X m23 r
( ~ X m3~ - ~ Z m ~ , )
+ y ( ~ X m ~ - ~ Z m~,)
r
K
X Aand
Z i
r
fq
fP
r
r
f --m~l r
x + -- m3x r
YA a n d - Y i
f x - m~2 + - r r
ZA and-Z i
--
f
m32
x m~3
AZ ms, )
+
r
--
f y --rn21 + - - m31 r r
f y m22 + -- m3~ r r
--
f m3a
r
--
r
y m:~
+
-
-
ma3
r
p = r n l l ~ X + m l ~ A Y + m 1 3 A Z ; q = m 2 1 A X + m ~ A Y + m2~,~Z; r = m 3 1 A X + m 3 2 A Y + m 3 3 ~ Z ; A X = X A - Xi; A Y = Y A - - Yi; a n d A Z = Z A - Zi.
tan
K = -(Y2
- yl)/(x2
- xl)
(10)
where (Xl, Yl) and (x~, Y2) are points on the apparent horizon. When placed in the plate-carriers of the SD-4 plotter, the photographs may have any orientation relative to the stereoplotter's ( x , y ) plate coordinate system. Therefore, the value calculated for K is not necessarily that of the camera in the field, unless fiducial marks are utilized. The values of w and ~ obtained in the orientation procedure are not affected by the value of K. To improve the accuracy estimates of K, three points spread along the horizon are used for each photograph. The values of K are then calculated by:
335
Y2-Yl)
1(y3-Y2 + 2 x3 -- X 2
tan K =
X2
(11)
Xl
where points ( x l , y ~ ) and ( x 3 , y 3 ) a r e near the edges of the frame and (x2, Y2) is close to the centre. The value of co' is calculated from not one but five points, spread evenly along the horizon: tan
co'
1 =
-p/f
-
~
(YiCOS~-xisin~)
(12)
~
f
i=1
5
PRACTICAL TESTS WITH THE ANALYTICAL PLOTTER
A series of tests was undertaken on the stereoplotter, using the modified software, to estimate the accuracies and precisions which could be achieved in realistic situations. Details of four stereopairs which were used in the tests are given in Table 2. Each pair of photographs was taken in the vicinity of Newcastle in New South Wales (150 km north of Sydney). Stereopair number 1 did n o t involve ocean waves, and the photographs were not taken synchronously. The surf photographs of pair number 2 provided a clear object with plenty of detail, and were taken from camera positions optimized for accurate results; on the other hand, pairs 3 and 4 were taken in poor weather and in less-than-optimum situations. TABLE
2
S t e r e o p a i r s u s e d i n t e s t s o f m o d i f i e d SD-4 s o f t w a r e Pair n u m b e r :
1
2
3
4
Date:
January 1982
March 1982
August 1983
August 1983
Cameras:
Wild P-32, 64 r a m f o c a l length
N i k o n 3 5 m m n o n - m e t r i c c a m e r a s w i t h N i k k o n u s lens, n o m i n a l 50 m m f o c a l l e n g t h , c a l i b r a t e d t o be 51.5 m m
Object:
Test fence with co-ordinated object points. Ocean and h o r i z o n visible
Breaking surf waves, taken from beach (see Fig. 1)
Film:
120 ram, black and white, nford Pan F
35 m m , b l a c k a n d w h i t e , K o d a k A S A 1 0 0
Inshore waves, some breaking, taken from point close to s h o r e
Inshore waves, mostly unbroken, taken f r o m headland
Weather:
Clear a n d s u n n y
Clear a n d s u n n y
Dull, cloudy, rain in vicinity
Distance between cameras:
4.4 m
9.2 m
7.1 m
22.4 m
Height of cameras a b o v e sea-level:
Not applicable
2 m
12 m
45 m approx.
Synchronization method:
Not applicable
C a m e r a s e x p o s e d b y air h o s e s f r o m a single s o u r c e
336 C a l i b r a t i o n o f t h e t w o n o n - m e t r i c c a m e r a s led t o t h e use o f a focal length o f 51.5 m m in all c a l c u l a t i o n s involving these c a m e r a s . This f o c a l length was c o n s i d e r e d t o be c o r r e c t t o b e t t e r t h a n 0.5 m m . S y n c h r o n i s e d e x p o s u r e o f t h e n o n - m e t r i c c a m e r a s was a c h i e v e d b y t h e o p e r a t o r , w h o s q u e e z e d a r u b b e r b u l b w h i c h was c o n n e c t e d b y t h i n airhoses to each c a m e r a . T h e s i m u l t a n e i t y o f this p r o c e d u r e h a d b e e n e s t i m a t e d p r e v i o u s l y t o b e q u i t e s a t i s f a c t o r y (Mitchell, 1983). Figure 1 s h o w s b o t h c a m e r a s o n t h e b e a c h at Newcastle, in t h e configurat i o n u s e d f o r p h o t o g r a p h i n g pair n u m b e r 2.
Fig. 1. Non-metric cameras positioned for photography of surf waves referred to as stereopair number 2.
(a) Testing the software In t h e first test, w h i c h was a i m e d at v e r i f y i n g t h e s o f t w a r e m o d i f i c a t i o n s , a pair o f p h o t o g r a p h s t a k e n w i t h a Wild P-32 m e t r i c c a m e r a was used. A fence, o n w h i c h t h e r e w e r e 12 c o - o r d i n a t e d p o i n t s , a n d b e y o n d w h i c h the o c e a n and h o r i z o n w e r e visible, was p h o t o g r a p h e d f r o m t w o c a m e r a s t a t i o n s 15 m f r o m t h e fence. T h e c a m e r a s t a t i o n s w e r e 4.4 m a p a r t . A n o r i e n t a t i o n was u n d e r t a k e n in t h e s t e r e o p l o t t e r using t e n p o i n t s o n t h e fence. T h e f l o a t i n g m a r k was t h e n driven t o t h e t e s t p o i n t s . T h e m e a n o f t h e a b s o l u t e values o f t h e errors in height d i f f e r e n c e s b e t w e e n eight pairs o f t e s t p o i n t s was 7.5 m m . T h e test, t h e r e f o r e , did n o t reveal a n y flaws in t h e software.
337
(b) Precision of camera orientation deduced from the horizon The precisions of the orientations w and K were estimated by orienting the stereopair n u m b e r 3 ten times in the stereoplotter. The precision {one standard deviation) of ¢z was 0.0001 radians (+ 30 seconds of arc) and for K, +0.001 radians (+4 minutes of arc). It is w ort h noting t hat the focal length of a camera must be measured to 0.1% accuracy if a 0.1% accuracy is to be ensured for w. See eqn. (9).
(c) Precision of the orientation of the stereoscopic model The reliability of results obtained from the ocean photographs could only be estimated from the internal consistency of the results, since no independent means o f measuring the water surface at the instant of p h o t o g r a p h y , was available. The precision of the combined relative and absolute orientation procedure was tested in the modified SD-4 pl ot t er using stereopairs numbers 2 and 3. The precision was assessed by the error which the orientation procedure was likely to introduce into height differences measured on the stereoscopic model. The precision of the orientation of the model was also indicated by the variances of the cameras' ¢ orientations, which are derived in the model orientations. On stereopair n u m b e r 2, ten object points, comprising five pairs of nearby points, were selected as suitable for the test. The points were, for example, droplets of spray and discernible particles of foam; t h e y were between 30 m and 90 m f r o m the cameras. An orientation of the stereopair was then effected in the plotter, and the co-ordinates of the ten points were read, three times each. A mean height difference was then calculated for each of the five pairs o f points. This procedure was t hen repeated with t w o further orientations, the same ten points being observed. The three height differences for each of the five pairs were used t o estimate t hat the standard deviation o f a height difference was appr ox i m at el y + 20 mm. However, the mean height differences suffer from an error due to locating an object point and setting the floating mark on it, so the error from this source had to be isolated. All 45 height differences were used to estimate that, for any height difference, the standard deviation representing the latter error source was + 10 mm. It was concluded therefore t hat the standard deviation of a height difference arising f r om the orientation procedure was only 15 mm. Using the same logic for height differences measured between all points (not only between close points), the standard deviation for a height difference was calculated to be + 40 mm. The three orientations also provided six estimates o f the ¢ orientation of the left and right cameras, f r om which a standard deviation of -+0.03 radians {+ 2 degrees) for ¢ was calculated. One o f the three orientations was t hen checked using the observed plate co-ordinates as data in the independent c o m p u t e r program. The variance-
338 covariance m a t r i x c o m p i l e d in this m a n n e r gave s t a n d a r d deviations for ¢ o f + 0 . 0 0 6 radians (left) and + 0 . 0 0 8 radians (right). These values are smaller t h a n the practically e s t i m a t e d values b y a f a c t o r o f a b o u t f o u r t o five. The variance-covariance m a t r i x also enabled standard deviations t o be calculated for t h e height differences b e t w e e n the ten points used for the s t e r e o - m o d e l o r i e n t a t i o n . T h e m e a n o f these 45 standard deviations was + 14 m m , which is a b o u t one-third the standard deviation d e d u c e d f r o m the t h r e e o r i e n t a t i o n s themselves. S t e r e o p a i r n u m b e r 3 (see Fig. 2) was used in a separate test. T h e o p e r a t o r used, as closely as c o u l d be judged, the same ten points on t h e w a t e r surface to carry o u t five separate o r i e n t a t i o n s o f the model. The p o i n t f u r t h e s t to sea was 60 m f r o m the cameras. T h e deviations o f all 45 height differences f r o m their means given b y the five orientations, were used t o estimate t h a t the standard deviation for a height d i f f e r e n c e was + 0.2 m. P r e s u m a b l y , this figure includes an error d u e to re-identifying the points.
Fig. 2. Stereopair number 3. See Table 2 for details. Again, the observations f r o m one o f the o r i e n t a t i o n s were recalculated using the c o m p u t e r program. T h e EXX m a t r i x f o r m e d b y this p r o g r a m enabled standard deviations for all height differences t o be calculated; their m e a n value was + 0.04 m or a b o u t one-fifth the s t a n d a r d deviation d e d u c e d f r o m the r e p e a t e d orientations. T h e standard deviation for ¢ (which is free f r o m the p o i n t r e l o c a t i o n error) was calculated f r o m the five o r i e n t a t i o n s to be + 0 . 0 0 8 radians (+ 30 minutes); the E x x m a t r i x gave a value o f + 0 . 0 0 1 6 radians for b o t h l e f t - a n d right-hand cameras. Thus, the actual standard deviations are five times the p r e d i c t e d size, in this case. T h e possibility t h a t the size o f the practically d e t e r m i n e d standard deviations were larger t h a n the p r e d i c t e d value because ¢o and ~ were being held fixed in the solutions, has been e x a m i n e d . It was suspected t h a t holding these o r i e n t a t i o n s fixed (rather t h a n treating t h e m as observed values) c o u l d result in an artificially w e l l - c o n d i t i o n e d N matrix. This, in t u r n , c o u l d produce unrealistically small e l e m e n t s in the inverse o f N, and h e n c e in the E x x m a t r i x also. T o test this hypothesis, the w and K values were t r e a t e d as
339
observations, with standard deviations of +0.0001 and +0.001 radians, respectively. The final co-ordinates and estimated variances were n o t significantly different f r om those obtained when the orientations were fixed.
(d) Precision of abstraction of wave detail The precision with which co-ordinates of wave points could be observed on the fully-oriented model was examined using stereopairs numbers 2 and 4. On pair n u m b e r 2, the top edge of a steeply form ed wave, a b o u t 55 m from the cameras, was delineated twice. The floating mark was set on the wave crest at intervals, usually where the shape of the crest was judged by the o p er ato r to change. The two delineations are depicted in Fig. 3. T h e y suggest a precision certainly bet t er than + 0.05 m in this case. -v
kl
o 5
1'0 Co-ordinate
in d i r e c t i o n
2'0
1'5 of camera
base
(metres)
Fig. 3. D e l i n e a t i o n o f c r e s t o f s u r f w a v e o n s t e r e o p a i r n u m b e r 2.
After an orientation of stereopair n u m b e r 4 (which could n o t be oriented rigorously because the camera heights were not measured exactly, because o f the difficulty of levelling the 45 m to the sea surface) a profile across the water surface was abstracted four times. The X co-ordinate of the floating mark was held fixed, so that all profiles followed the same path in a direction away from the camera base. The floating mark was moved an arbitrary distance (averaging a bout 2 m) by the operator, and then set on the water surface. Figure 4 shows the stereopair and Fig. 5 shows the four profiles. A precision of a b o u t + 0.2 m for the delineation seems to have been achievable for this particular pair of photographs, for points up to 150 m from the camera. An apparent slope of a bout six percent on the water surface has been removed f r om the profiles. This grade can be attributed to an error in the camera heights (see Discussion), but its existence should n o t have influenced the precision of the profiling. DISCUSSION
A height difference between two points on the water surface, when measured during wave data abstraction with the stereoplotter, will have
340
Fig. 4. Stereopair number 4. See Table 2 for details.
-r' ¢0
I~-
Points from four profiles shown: ~r
o o
~
Points
in firs! p r o f i l e a r e j o i n e d
× ~lo
o
~_A
v O
110
120
o
~
i
T
°
o
i
150
~o
Distance from cameras (metres) J
Fig. 5. Water surface profiles from stereopair number 4.
errors due to two sources: one, the orientation of the stereoscopic model, and the other, the setting of the floating mark on the water surface. Results of the tests suggest that the magnitude of the first error will vary with the separation between the points. In the models tested, the standard deviation of a measured height difference due to this error was from + 20 m m (for adjacent points) to + 200 mm (for widely separated points). The error due to the setting of the floating mark on the surface was given a standard deviation of +2 0 0 mm. Therefore, the overall precision of a height difference was a b o u t + 300 mm. These results should apply to situations similar to those of the test cases. Error simulations can be used to compare the precisions to be expected in various ot he r camera configurations (Mitchell, 1983). During the orientation procedure, a strong solution should be sought by using points which are located to provide good ray intersection geometry. Normally, such points will be close to shore, provided that subsequent measurements on the model are to be made in this vicinity. Reading coordinates a n u m b e r of times will normally improve p h o t o g r a m m e t r i c results, but when obtaining wave heights, there may be restrictions on the time which the analysis can realistically take. A slope of the sea-surface which has been noticed on some surface profiles has been found to be a useful
341 indicator of a poor orientation solution. The use of the apparent slope of sea-surface to strengthen the solution could be investigated further. It is worth noting that: (a) standard deviations estimated by variancecovariance matrices derived from least squares solutions, appear to be optimistic (by a factor of five in the worst case encountered during the tests); and (b) horizontal co-ordinates were measured to precisions which were inferior to those of the heights (as the cameras were configured for heighting accuracy), but which were adequate for fixing points on the water surface. Although the results of a number of experimental orientations have not been reported here, the tests which have been described are easily reproducible. Indeed, better results could have been achieved with longer camera bases, with cameras closer to sea-level, in better weather conditions, with larger format cameras, and probably with colour film. Useful experience with observations in high-oblique photographs has been obtained during this study. Three-dimensional vision was found to be easily attainable. Nevertheless, further investigations should be undertaken into the tolerance of a photogrammetric operator for sustained viewing of such stereomodels. It is envisaged that a practical wave-measuring procedure would involve: (a) permanent mountings for a pair of cameras so that the cameras' locations and orientations would remain fixed for a series of photographic pairs; (b) with a suitable analytical plotter, fiducial marks on the photographs would enable a number of stereopairs to be oriented by reference to the previous orientation and without the need for repeating the complete orientation procedure; and (c) the plotter would be programmed to help the operator extract the required results (such as profiles across the surface). Further work on other aspects of the m e t h o d is desirable in order to assess its practicality: photographic limitations, the relative merits of metric and non-metric cameras and the scope of data that can be extracted from stereomodels of the ocean surface. CONCLUSION Because a precision of +0.3 m is adequate for many wave-measuring purposes, the use of terrestrial photogrammetric methods to measure the heights of waves in the near-shore region appears to be worth pursuing. The convenience of model orientation and data abstraction which can be achieved using an analytical plotter for which the software can be modified, renders any other plotter too unwieldy. A full assessment of the analytical plotter's role in collecting wave data would require an examination of its convenience, versatility and cost, but it would appear that, with limited further development, it should be possible to use p h o t o g r a m m e t r y to collect wave data over short periods, in special events, or to supplement or check alternative data collecting methods.
342 ACKNOWLEDGMENTS T h e w o r k d e s c r i b e d in this p a p e r was p a r t o f a p r o j e c t w h i c h has received financial s u p p o r t u n d e r t h e Australian R e s e a r c h G r a n t s S c h e m e . T h e Q a s c o SD-4 S t e r e o p l o t t e r in t h e D e p a r t m e n t o f Civil E n g i n e e r i n g and S u r v e y i n g at t h e U n i v e r s i t y o f N e w c a s t l e , Australia, was d o n a t e d b y Q a s c o Pty. L t d . o f S y d n e y . M u c h o f t h e o b s e r v i n g on t h e SD-4 s t e r e o p l o t t e r f o r the p u r p o s e s o f this s t u d y , was u n d e r t a k e n b y Mr. H.T. Kniest.
REFERENCES Elfick, M.H. and Fletcher, M.J., 1982. The Qasco SD-4. Photogramm. Eng. Remote Sensing, 48 (6): 925--930. Maresca, J.W. and Seibel, E., 1976. Terrestrial photogrammetric measurements of breaking wave and longshore currents in the nearshore zone. Proc. Conf. Coastal Eng., 15th, Honolulu, pp. 681--700. Mitchell, H.L., 1983. Wave heights in the surf zone. Photogramm. Rec., 11(62): 183-193. Silvester, R., 1974. Coastal Engineering, 1. Elsevier, Amsterdam, 457 pp. Wolf, P.R., 1974. Elements of Photogrammetry. McGraw Hill Kogakusha, Tokyo, 562 pp. Wong, K.W., 1980. Chapter II. In: C.C. Slama (Editor), Manual of Photogrammetry (4th Edition). ASP, Falls Church, Va. 1056 pp.
327
OBSERVATIONS ON INSHORE O C E A N W A V E S WITH AN ANALYTICAL STEREOPLOTTER
H.L. MITCHELL Department of Civil Engineering and Surveying, University of Newcastle, Newcastle, N.S. W. 2308, Australia (Received June 6, 1984; revised and accepted November 18, 1985)
ABSTRACT
Mitchell, H.L., 1986. Observations on inshore ocean waves with an analytical stereoplotter. Photogrammetria, 40: 327--342. The software of an analytical stereoplotter was modified to enable ocean wave parameters to be measured from stereophotographs taken from shore. The separate relative and absolute orientation stages were replaced by a single-stage orientation, which uses the horizon as the only visible control. Non-metric cameras, located close to shore and up to 22 m apart have been used to measure height differences on the water surface with a precision of 0.3 m. The results suggest that the analytical plotter technique could contribute to the collection of inshore wave data.
INTRODUCTION Terrestrial p h o t o g r a m m e t r y is such a useful m e n s u r a t i o n t e c h n i q u e f o r r e m o t e l y sensing objects, t h a t it suggests itself as having a role in the r e c o r d i n g o f t h e shape o f the o c e a n surface. Surf wave heights have been m e a s u r e d previously b y the writer, using an analytical s o l u t i o n o f m o n o c u l a r l y observed plate c o - o r d i n a t e s o n p h o t o g r a p h s t a k e n f r o m shore (Mitchell, 1983). T h e results o b t a i n e d in t h a t m a n n e r suggested t h a t it w o u l d also be feasible to use a s t e r e o p l o t t e r instead o f m o n o c u l a r observations to collect d a t a o n inshore wave processes. This p a p e r outlines the m o d i f i c a t i o n s w h i c h were s u b s e q u e n t l y m a d e t o a Qasco SD-4 analytical s t e r e o p l o t t e r , for t h a t purpose. S o m e results have been o b t a i n e d in the o b s e r v a t i o n o f inshore waves with the s t e r e o p l o t t e r , a n d assessments o f precision have been made. The n e e d for ocean wave data Waves o f various heights, lengths and periods can arrive f r o m d i f f e r e n t directions at a coastal or o f f s h o r e site w h i c h is u n d e r investigation b y a coastal engineer. W h e t h e r t h e engineer's p r o b l e m lies in the design o f c o m mercial or recreational facilities or w h e t h e r it lies in beach a n d shoreline
0031-8663/86/$03.50
© 1986 Elsevier Science Publishers B.V.
328 protection and maintenance, a knowledge of the waves which may be encountered is an important component of the input needed for the design work. The wave data may be used to indicate the average and extreme values of both wave heights and wave energy levels. These suggest the forces which a structure must be expected to face or which will influence the transport of suspended sediment or the erosion of a beach. Estimates of the wave climate may be developed by the engineer from a knowledge of the meteorological conditions which give rise to the sea or swell arriving at the site in question. Alternatively, a description of the wave conditions may be derived from direct measurements made at, or in the vicinity of, the site. If satisfactory observations are n o t already available to the engineer, efforts must be made to record data in whatever time is available. According to Silvester (1974), an engineer may well be fortunate if a year's records can be provided. Various wave recording techniques are presently available (see Silvester, 1974, p. 274 et seq. for a summary of wave recording methods), but automatically recording instruments are widely used. The most c o m m o n instruments would appear to be those which involve accelerometers housed in buoys floating on the water surface. The accelerometers respond to the rise and fall of the water surface as sea and swell pass; double integration of the recorded accelerations with time provides a record of the vertical displacement of the ocean surface. Buoys equipped with accelerometers are widely used and give satisfactory results in m a n y cases. However, they are not w i t h o u t their disadvantages, which include the following: (a) the recorder in the open ocean is prone to damage, corrosion and loss; (b) the buoy must be secured in, and recovered from, deep water; (c) the recorded data must be recovered from the offshore site; (d) because buoy recorders are normally located in deep water, seaward of the site in question, it is necessary to m o d i f y the recorded wave profile to account for the refraction and diffraction of the waves as they travel between the recorder site and the shore, where the wave pattern is needed; (e) w i t h o u t the assistance of a synchronized second recorder, the directions of waves can be difficult to deduce (giving rise to a problem in predicting the arrival of waves at the shore). B o t t o m - m o u n t e d recorders may be used as an alternative to floating-buoy recorders. Although the instruments are protected from damage by the atmosphere, by the waves or by surface traffic, recovery of the instrument and data can be even more difficult than for a float recorder.
A role for photogrammetry The apparent benefits of a photogrammetric m e t h o d of wave recording when compared with other wave recording methods have been suggested by Mitchell (1983). The advantages centre on the ability of photogrammetry to keep the recording equipment free from contact with the ocean and its
329
users, whilst also enabling easy retrieval of the instrumentation and the recorded information. Furthermore, the collected data refer to occurrences at the shoreline, n o t in deeper offshore water away from the site under investigation. The photogrammetric m e t h o d may also enable the collection of wave parameters other than only wave heights (for example, wave length and wave directions). The automatic wave-recording instruments, whether floating or bottomm o u n t e d , have the advantage of enabling wave profiles to be abstracted from the recorded data by an automated process. Such automation is a distinct advantage when considerable quantities of data must be processed, as is the case when wave statistics are being compiled. It is against this advantage that the apparently labour-intensive photogrammetric methods must be assessed. Photogrammetry is therefore likely to be best suited to situations when the conventional instrumentation is unsuitable: to observe wave action close to shore, or when only a short period of observations is needed or is possible (for example, to record special events, such as storms, to record whilst alternative instruments are inoperative or to check other recorders). An experienced photogrammetrist may have some doubts about a photogrammetric method's usefulness, considering the difficulty of providing control points on the sea surface. With terrestrial photogrammetry, this is not restrictive if an analytical stereoplotter can be used and if its software can be modified. The horizon can be used to indicate camera orientations. Furthermore, the analytical plotter, if its software can be altered, can facilitate data abstraction. It can provide prompts for non-photogrammetrist operators; it can direct the movement of the floating marks along appropriate pre-determined paths when wave data are being abstracted, and it can utilize fiducial marks to orient successive photographic pairs which have the same camera positions. The photogrammetrist must also remember that precision requirements for wave data are n o t severe. Not only is it unnecessary to precisely delineate a moving ocean wave, but also, it is normal to base wave data on a series of observations over time. A single stereopair would provide only part of the data. WAVE HEIGHTS FROM AN ANALYTICAL SOLUTION
Earlier work in this study of photogrammetric wave measurement centred on computer programs providing a least-squares solution of the collinearity equations (Mitchell, 1983). The co-ordinates of both cameras and the orientations of the cameras about two axes were held fixed, the two known camera orientations being determined from the appearance of the horizon on high-oblique photographs. The unknowns in the solution consisted of the remaining orientation parameter for each camera, as well as the ground coordinates of the object points. The computer program also formed the variance-covariance matrix of the determined unknowns, and this enabled
330 error simulation studies to be undertaken to assess the relative accuracies afforded by various camera configurations. A pair of non-metric 3 5 m m Nikon cameras set up on a beach in Newcastle, New South Wales (latitude 32°50 ' south, longitude 151°40 ' east) provided data which were used to co-ordinate points in a breaking wave area. Monocularly observed plate co-ordinates were used to deduce co-ordinates of 12 object points in the ground co-ordinate system. By monocularly reading the plate co-ordinates, any possible errors or difficulties caused by stereoscopic viewing of the high oblique photographs were avoided. Object points which were chosen using an enlargement of one of the photographs of the pair, were used if they could be confidently located on both negatives of the pair. The variancecovariance matrix of the co-ordinates suggested that the standard deviations of height differences were no more than +50 mm for distances up to 75 m from the camera, a result which was considered to be satisfactory for wave height measurement. The effects of systematic errors (in the focal length of the cameras, for example) were examined by obtaining further solutions in which realistic values of the systematic errors were intentionally included. The effects of the errors were not found to be significant (see Mitchell, 1983, table II). Maresca and Seibel (1976) have also used terrestrial photogrammetry to measure wave heights (and to observe currents). Their m e t h o d is fairly simple, and it does not appear to be amenable to the collection of considerable quantities of data. A pair of cameras was oriented so that the two cameras' axes were estimated to be level in two directions, and the direction of pointing of the axes was determined from control markers in the foreground. Co-ordinates were measured off the photographs after corresponding images had been identified. APPLICATION TO THE ANALYTICAL STEREOPLOTTER The analytical solution of monocularly read plate co-ordinates is n o t a satisfactory means of co-ordinating a large number of points on a single stereopair or for collecting data from a number of stereopairs. The approach used in the analytical solution was therefore adapted to an analytical stereoplotter. In this case, modifications were made to the software of a Qasco SD-4 analytical stereoplotter, a device which is described by Elfick and Fletcher (1982). If the apparent horizon is used to determine the camera orientations and if the ground co-ordinates of the cameras are known, the absolute orientation procedure, which requires the use of control points, cannot be followed with wave photographs. Two alternatives were considered: (a) the relative orientation could be followed by a special case of the absolute orientation; or (b) a solution, equivalent to combining the relative and absolute orientation stages, could be adopted. Maintaining separate relative and absolute orientations had two disadvan-
331
tages. Firstly, the relative orientation of the high oblique photographs is weak, presumably because it is not possible to select object points at the optimally located Von Gruber positions. (The photographic overlap varies with the distance of the waves from the cameras, and there is always a loss of usable area above the horizon on high oblique photographs.) Some attempts to obtain reltive orientations of high-oblique surf photographs led to results which were clearly unsatisfactory. Secondly, it is mathematically complex to deduce the absolute orientation parameters given relative orientation results (in which the right-hand camera is positioned with respect to the left-hand camera), given the ground co-ordinates of the two cameras and given two orientations of each camera. The weakness of the relative orientation is not apparent in the combined solution, in which the number of unknowns is reduced. Consequently, it was decided that the software of the Qasco SD-4 analytical plotter would be modified to enable a single-stage orientation to be carried out on the wave photographs. Description o f the revised software
The alteration of the stereoplotter software involved the modification of the relative orientation procedure to account for the change in the set of known and u n k n o w n parameters. The modified relative orientation involves the least-squares solution of the collinearity equations as used for the fully analytical solution. The standard absolute orientation routine normally provided with the plotter becomes redundant when the altered routines are being used. It should be made clear that the operator can still call up the original software, when the stereoplotter is needed for conventional work. In the standard SD-4 software, the relative orientation may be based on a minimum of six and a maximum of 10 points. The least-squares solution of the relative orientation is based on linearized collinearity equations, written in the form of the conventional observation equations, that is, each equation is in terms of only one observed plate co-ordinate. To minimize the a m o u n t of alteration needed, the revised software which undertakes the "combined relative and absolute orientations" is also based on observation equations. Accordingly, the values of w and K, which are determined for each camera from the horizon, are held fixed in the solution. According to a rigorous approach, they should be regarded as observations, but this requires a more complex solution, as the following explanation suggests. The set of linearized collinearity equations is written in matrix form as: BV+AX+C
= 0
(1)
where V is the vector of corrections to the observations; X is the vector of corrections to the a priori estimates of the required parameters; B and A are coefficient matrices; and C is the vector of constants. The solution in the Qasco SD-4 requires that:
332 B=I
The solution for X is then given by: (2)
NX + A Tc = 0
where N
(3)
= ATA
If, on the other hand, w and K are observations, then: B =/= I
and the solution for X is given by: X
= -N -1AT(BGBT)
-1 C
(4)
where N
= A T ( B G B T ) -1 A
(5)
and where G is the variance-covariance matrix for the observations. The stereoplotter software was modified to provide the simpler of the two solutions. The deficiency of this solution, in comparison to a solution in which co and K are included as observations, was tested; see below. Because the set of unknowns in the stereoplotter solution comprises the X, Y and Z co-ordinates of the object points and the ~ orientations of both cameras, the minimum number of object points needed for a solution without redundant observations is two. A solution with ten object points, for example, would provide eight redundancies. The results of orientations reported in this paper were checked using the computer program which was written for the fully analytical solutions reported earlier (Mitchell, 1983). The program was based on the same equations as those used in the SD-4 plotter, but the program used a different solution algorithm to that of the stereoplotter. This program also has the facility to incorporate ¢o and K as observations; its use in this manner is mentioned below. In addition, the program formed Z XX, the variancecovariance matrix for the parameters: ZXX = ao N-1
(6)
where Oo is a variance factor estimated by: ao = v T G - 1 V / r
(7)
and r is the number of redundancies. In the standard SD-4 software, the least-squares solution is obtained by an elimination m e t h o d applied to eqn. (2), and n o t by inversion of the matrix N, as used in eqn. (4). Consequently, the matrix ZXX is not presently formed by the stereoplotter software.
333
The use of the horizon Mitchell (1983) explained t ha t the horizon on high oblique photographs does n o t enable camera orientations to be deduced immediately if the (x,y) plate co-ordinates are defined in the conventional manner (x across the p h o to g r ap h to the right, y towards the top of the photograph). This was overcome in the analytical solution of the monocularly observed points by rotating the x and y plate co-ordinate systems on the photographs and making corresponding changes to ground co-ordinate system. For example, heights were given by the Y co-ordinates. To avoid difficulties caused by redefining the plate co-ordinate system in the SD-4 software, an alternative solution has been a dopt e d for the stereoplotter routines. The rotations ¢o, ~, K ab o u t the cameras' x, y, z axes are assumed to have occurred in the order (¢, co, K), instead of (co, ~, K). The r o t at i on matrix M ~ relating the ground and camera co-ordinate systems for the SD-4, has elements as follows: m,, = cos ~ cos K + sin co sin ~ sin m,2 = c o s w sin m13 = s i n ~ c o s ¢ s i n ~ - s i n ~ c o s K rn2, = sin co s i n O c o s ~ - c o s @ s i n K m22 = cos co cos K m23 = s i n ~ s i n ~ + s i n c o c o s ¢ cosK m3, = cos co sin
m32 = - sin m33
=
cos
¢o c o s
The relevant co-efficients of the corrections to the parameters in the linearized collinearity equations are as in Table 1, rather than as given in most t e x t and reference books for M~¢~ (e.g. Wolf, 1974, p. 533; Wong, 1980, p. 57). The orientations w and K for any camera can now be calculated from the apparent 'dip' and 'slope' of the horizon as seen on a photograph: = ~' + ~
(8)
where co' is the dip of the apparent (rather than true) horizon, and is given by: tan co' = - p / f
(9)
Aw is the correction for refraction and for the difference between the true and apparent horizons; p is the perpendicular distance from the horizon to the principal point, measured positive upward from the principal point, f is the camera focal length, and
334 TABLE 1 C o e f f i c i e n t s o f c o r r e c t i o n s t o t h e a priori e s t i m a t e s o f t h e p a r a m e t e r s in t h e linearized c o l l i n e a r i t y e q u a t i o n s f o r ith c a m e r a a n d for o b j e c t p o i n t A Parameter
C o e f f i c i e n t o f c o r r e c t i o n to e s t i m a t e s o f p a r a m e t e r in e q u a t i o n in x
in e q u a t i o n in y
f ( A X m~ : sin~ - A Y sinco sink
f ( A X m22 s i n ¢ - A Y s i n ¢ o cosK
r
r
+ A Z m ~ cos¢) X +--
( A X m 3 2 sinq~ r + AZ m ~ cos40
+ AZ m ~ cos~)
f ( ~ X m,3 - a Z m,~) r X
+-
+ Y ( A X m~2 sin~ - A y cos¢o
~Ycos~0
r
+ ~ Z m32 c o ~ ) f ( A X m23 r
( ~ X m3~ - ~ Z m ~ , )
+ y ( ~ X m ~ - ~ Z m~,)
r
K
X Aand
Z i
r
fq
fP
r
r
f --m~l r
x + -- m3x r
YA a n d - Y i
f x - m~2 + - r r
ZA and-Z i
--
f
m32
x m~3
AZ ms, )
+
r
--
f y --rn21 + - - m31 r r
f y m22 + -- m3~ r r
--
f m3a
r
--
r
y m:~
+
-
-
ma3
r
p = r n l l ~ X + m l ~ A Y + m 1 3 A Z ; q = m 2 1 A X + m ~ A Y + m2~,~Z; r = m 3 1 A X + m 3 2 A Y + m 3 3 ~ Z ; A X = X A - Xi; A Y = Y A - - Yi; a n d A Z = Z A - Zi.
tan
K = -(Y2
- yl)/(x2
- xl)
(10)
where (Xl, Yl) and (x~, Y2) are points on the apparent horizon. When placed in the plate-carriers of the SD-4 plotter, the photographs may have any orientation relative to the stereoplotter's ( x , y ) plate coordinate system. Therefore, the value calculated for K is not necessarily that of the camera in the field, unless fiducial marks are utilized. The values of w and ~ obtained in the orientation procedure are not affected by the value of K. To improve the accuracy estimates of K, three points spread along the horizon are used for each photograph. The values of K are then calculated by:
335
Y2-Yl)
1(y3-Y2 + 2 x3 -- X 2
tan K =
X2
(11)
Xl
where points ( x l , y ~ ) and ( x 3 , y 3 ) a r e near the edges of the frame and (x2, Y2) is close to the centre. The value of co' is calculated from not one but five points, spread evenly along the horizon: tan
co'
1 =
-p/f
-
~
(YiCOS~-xisin~)
(12)
~
f
i=1
5
PRACTICAL TESTS WITH THE ANALYTICAL PLOTTER
A series of tests was undertaken on the stereoplotter, using the modified software, to estimate the accuracies and precisions which could be achieved in realistic situations. Details of four stereopairs which were used in the tests are given in Table 2. Each pair of photographs was taken in the vicinity of Newcastle in New South Wales (150 km north of Sydney). Stereopair number 1 did n o t involve ocean waves, and the photographs were not taken synchronously. The surf photographs of pair number 2 provided a clear object with plenty of detail, and were taken from camera positions optimized for accurate results; on the other hand, pairs 3 and 4 were taken in poor weather and in less-than-optimum situations. TABLE
2
S t e r e o p a i r s u s e d i n t e s t s o f m o d i f i e d SD-4 s o f t w a r e Pair n u m b e r :
1
2
3
4
Date:
January 1982
March 1982
August 1983
August 1983
Cameras:
Wild P-32, 64 r a m f o c a l length
N i k o n 3 5 m m n o n - m e t r i c c a m e r a s w i t h N i k k o n u s lens, n o m i n a l 50 m m f o c a l l e n g t h , c a l i b r a t e d t o be 51.5 m m
Object:
Test fence with co-ordinated object points. Ocean and h o r i z o n visible
Breaking surf waves, taken from beach (see Fig. 1)
Film:
120 ram, black and white, nford Pan F
35 m m , b l a c k a n d w h i t e , K o d a k A S A 1 0 0
Inshore waves, some breaking, taken from point close to s h o r e
Inshore waves, mostly unbroken, taken f r o m headland
Weather:
Clear a n d s u n n y
Clear a n d s u n n y
Dull, cloudy, rain in vicinity
Distance between cameras:
4.4 m
9.2 m
7.1 m
22.4 m
Height of cameras a b o v e sea-level:
Not applicable
2 m
12 m
45 m approx.
Synchronization method:
Not applicable
C a m e r a s e x p o s e d b y air h o s e s f r o m a single s o u r c e
336 C a l i b r a t i o n o f t h e t w o n o n - m e t r i c c a m e r a s led t o t h e use o f a focal length o f 51.5 m m in all c a l c u l a t i o n s involving these c a m e r a s . This f o c a l length was c o n s i d e r e d t o be c o r r e c t t o b e t t e r t h a n 0.5 m m . S y n c h r o n i s e d e x p o s u r e o f t h e n o n - m e t r i c c a m e r a s was a c h i e v e d b y t h e o p e r a t o r , w h o s q u e e z e d a r u b b e r b u l b w h i c h was c o n n e c t e d b y t h i n airhoses to each c a m e r a . T h e s i m u l t a n e i t y o f this p r o c e d u r e h a d b e e n e s t i m a t e d p r e v i o u s l y t o b e q u i t e s a t i s f a c t o r y (Mitchell, 1983). Figure 1 s h o w s b o t h c a m e r a s o n t h e b e a c h at Newcastle, in t h e configurat i o n u s e d f o r p h o t o g r a p h i n g pair n u m b e r 2.
Fig. 1. Non-metric cameras positioned for photography of surf waves referred to as stereopair number 2.
(a) Testing the software In t h e first test, w h i c h was a i m e d at v e r i f y i n g t h e s o f t w a r e m o d i f i c a t i o n s , a pair o f p h o t o g r a p h s t a k e n w i t h a Wild P-32 m e t r i c c a m e r a was used. A fence, o n w h i c h t h e r e w e r e 12 c o - o r d i n a t e d p o i n t s , a n d b e y o n d w h i c h the o c e a n and h o r i z o n w e r e visible, was p h o t o g r a p h e d f r o m t w o c a m e r a s t a t i o n s 15 m f r o m t h e fence. T h e c a m e r a s t a t i o n s w e r e 4.4 m a p a r t . A n o r i e n t a t i o n was u n d e r t a k e n in t h e s t e r e o p l o t t e r using t e n p o i n t s o n t h e fence. T h e f l o a t i n g m a r k was t h e n driven t o t h e t e s t p o i n t s . T h e m e a n o f t h e a b s o l u t e values o f t h e errors in height d i f f e r e n c e s b e t w e e n eight pairs o f t e s t p o i n t s was 7.5 m m . T h e test, t h e r e f o r e , did n o t reveal a n y flaws in t h e software.
337
(b) Precision of camera orientation deduced from the horizon The precisions of the orientations w and K were estimated by orienting the stereopair n u m b e r 3 ten times in the stereoplotter. The precision {one standard deviation) of ¢z was 0.0001 radians (+ 30 seconds of arc) and for K, +0.001 radians (+4 minutes of arc). It is w ort h noting t hat the focal length of a camera must be measured to 0.1% accuracy if a 0.1% accuracy is to be ensured for w. See eqn. (9).
(c) Precision of the orientation of the stereoscopic model The reliability of results obtained from the ocean photographs could only be estimated from the internal consistency of the results, since no independent means o f measuring the water surface at the instant of p h o t o g r a p h y , was available. The precision of the combined relative and absolute orientation procedure was tested in the modified SD-4 pl ot t er using stereopairs numbers 2 and 3. The precision was assessed by the error which the orientation procedure was likely to introduce into height differences measured on the stereoscopic model. The precision of the orientation of the model was also indicated by the variances of the cameras' ¢ orientations, which are derived in the model orientations. On stereopair n u m b e r 2, ten object points, comprising five pairs of nearby points, were selected as suitable for the test. The points were, for example, droplets of spray and discernible particles of foam; t h e y were between 30 m and 90 m f r o m the cameras. An orientation of the stereopair was then effected in the plotter, and the co-ordinates of the ten points were read, three times each. A mean height difference was then calculated for each of the five pairs o f points. This procedure was t hen repeated with t w o further orientations, the same ten points being observed. The three height differences for each of the five pairs were used t o estimate t hat the standard deviation o f a height difference was appr ox i m at el y + 20 mm. However, the mean height differences suffer from an error due to locating an object point and setting the floating mark on it, so the error from this source had to be isolated. All 45 height differences were used to estimate that, for any height difference, the standard deviation representing the latter error source was + 10 mm. It was concluded therefore t hat the standard deviation of a height difference arising f r om the orientation procedure was only 15 mm. Using the same logic for height differences measured between all points (not only between close points), the standard deviation for a height difference was calculated to be + 40 mm. The three orientations also provided six estimates o f the ¢ orientation of the left and right cameras, f r om which a standard deviation of -+0.03 radians {+ 2 degrees) for ¢ was calculated. One o f the three orientations was t hen checked using the observed plate co-ordinates as data in the independent c o m p u t e r program. The variance-
338 covariance m a t r i x c o m p i l e d in this m a n n e r gave s t a n d a r d deviations for ¢ o f + 0 . 0 0 6 radians (left) and + 0 . 0 0 8 radians (right). These values are smaller t h a n the practically e s t i m a t e d values b y a f a c t o r o f a b o u t f o u r t o five. The variance-covariance m a t r i x also enabled standard deviations t o be calculated for t h e height differences b e t w e e n the ten points used for the s t e r e o - m o d e l o r i e n t a t i o n . T h e m e a n o f these 45 standard deviations was + 14 m m , which is a b o u t one-third the standard deviation d e d u c e d f r o m the t h r e e o r i e n t a t i o n s themselves. S t e r e o p a i r n u m b e r 3 (see Fig. 2) was used in a separate test. T h e o p e r a t o r used, as closely as c o u l d be judged, the same ten points on t h e w a t e r surface to carry o u t five separate o r i e n t a t i o n s o f the model. The p o i n t f u r t h e s t to sea was 60 m f r o m the cameras. T h e deviations o f all 45 height differences f r o m their means given b y the five orientations, were used t o estimate t h a t the standard deviation for a height d i f f e r e n c e was + 0.2 m. P r e s u m a b l y , this figure includes an error d u e to re-identifying the points.
Fig. 2. Stereopair number 3. See Table 2 for details. Again, the observations f r o m one o f the o r i e n t a t i o n s were recalculated using the c o m p u t e r program. T h e EXX m a t r i x f o r m e d b y this p r o g r a m enabled standard deviations for all height differences t o be calculated; their m e a n value was + 0.04 m or a b o u t one-fifth the s t a n d a r d deviation d e d u c e d f r o m the r e p e a t e d orientations. T h e standard deviation for ¢ (which is free f r o m the p o i n t r e l o c a t i o n error) was calculated f r o m the five o r i e n t a t i o n s to be + 0 . 0 0 8 radians (+ 30 minutes); the E x x m a t r i x gave a value o f + 0 . 0 0 1 6 radians for b o t h l e f t - a n d right-hand cameras. Thus, the actual standard deviations are five times the p r e d i c t e d size, in this case. T h e possibility t h a t the size o f the practically d e t e r m i n e d standard deviations were larger t h a n the p r e d i c t e d value because ¢o and ~ were being held fixed in the solutions, has been e x a m i n e d . It was suspected t h a t holding these o r i e n t a t i o n s fixed (rather t h a n treating t h e m as observed values) c o u l d result in an artificially w e l l - c o n d i t i o n e d N matrix. This, in t u r n , c o u l d produce unrealistically small e l e m e n t s in the inverse o f N, and h e n c e in the E x x m a t r i x also. T o test this hypothesis, the w and K values were t r e a t e d as
339
observations, with standard deviations of +0.0001 and +0.001 radians, respectively. The final co-ordinates and estimated variances were n o t significantly different f r om those obtained when the orientations were fixed.
(d) Precision of abstraction of wave detail The precision with which co-ordinates of wave points could be observed on the fully-oriented model was examined using stereopairs numbers 2 and 4. On pair n u m b e r 2, the top edge of a steeply form ed wave, a b o u t 55 m from the cameras, was delineated twice. The floating mark was set on the wave crest at intervals, usually where the shape of the crest was judged by the o p er ato r to change. The two delineations are depicted in Fig. 3. T h e y suggest a precision certainly bet t er than + 0.05 m in this case. -v
kl
o 5
1'0 Co-ordinate
in d i r e c t i o n
2'0
1'5 of camera
base
(metres)
Fig. 3. D e l i n e a t i o n o f c r e s t o f s u r f w a v e o n s t e r e o p a i r n u m b e r 2.
After an orientation of stereopair n u m b e r 4 (which could n o t be oriented rigorously because the camera heights were not measured exactly, because o f the difficulty of levelling the 45 m to the sea surface) a profile across the water surface was abstracted four times. The X co-ordinate of the floating mark was held fixed, so that all profiles followed the same path in a direction away from the camera base. The floating mark was moved an arbitrary distance (averaging a bout 2 m) by the operator, and then set on the water surface. Figure 4 shows the stereopair and Fig. 5 shows the four profiles. A precision of a b o u t + 0.2 m for the delineation seems to have been achievable for this particular pair of photographs, for points up to 150 m from the camera. An apparent slope of a bout six percent on the water surface has been removed f r om the profiles. This grade can be attributed to an error in the camera heights (see Discussion), but its existence should n o t have influenced the precision of the profiling. DISCUSSION
A height difference between two points on the water surface, when measured during wave data abstraction with the stereoplotter, will have
340
Fig. 4. Stereopair number 4. See Table 2 for details.
-r' ¢0
I~-
Points from four profiles shown: ~r
o o
~
Points
in firs! p r o f i l e a r e j o i n e d
× ~lo
o
~_A
v O
110
120
o
~
i
T
°
o
i
150
~o
Distance from cameras (metres) J
Fig. 5. Water surface profiles from stereopair number 4.
errors due to two sources: one, the orientation of the stereoscopic model, and the other, the setting of the floating mark on the water surface. Results of the tests suggest that the magnitude of the first error will vary with the separation between the points. In the models tested, the standard deviation of a measured height difference due to this error was from + 20 m m (for adjacent points) to + 200 mm (for widely separated points). The error due to the setting of the floating mark on the surface was given a standard deviation of +2 0 0 mm. Therefore, the overall precision of a height difference was a b o u t + 300 mm. These results should apply to situations similar to those of the test cases. Error simulations can be used to compare the precisions to be expected in various ot he r camera configurations (Mitchell, 1983). During the orientation procedure, a strong solution should be sought by using points which are located to provide good ray intersection geometry. Normally, such points will be close to shore, provided that subsequent measurements on the model are to be made in this vicinity. Reading coordinates a n u m b e r of times will normally improve p h o t o g r a m m e t r i c results, but when obtaining wave heights, there may be restrictions on the time which the analysis can realistically take. A slope of the sea-surface which has been noticed on some surface profiles has been found to be a useful
341 indicator of a poor orientation solution. The use of the apparent slope of sea-surface to strengthen the solution could be investigated further. It is worth noting that: (a) standard deviations estimated by variancecovariance matrices derived from least squares solutions, appear to be optimistic (by a factor of five in the worst case encountered during the tests); and (b) horizontal co-ordinates were measured to precisions which were inferior to those of the heights (as the cameras were configured for heighting accuracy), but which were adequate for fixing points on the water surface. Although the results of a number of experimental orientations have not been reported here, the tests which have been described are easily reproducible. Indeed, better results could have been achieved with longer camera bases, with cameras closer to sea-level, in better weather conditions, with larger format cameras, and probably with colour film. Useful experience with observations in high-oblique photographs has been obtained during this study. Three-dimensional vision was found to be easily attainable. Nevertheless, further investigations should be undertaken into the tolerance of a photogrammetric operator for sustained viewing of such stereomodels. It is envisaged that a practical wave-measuring procedure would involve: (a) permanent mountings for a pair of cameras so that the cameras' locations and orientations would remain fixed for a series of photographic pairs; (b) with a suitable analytical plotter, fiducial marks on the photographs would enable a number of stereopairs to be oriented by reference to the previous orientation and without the need for repeating the complete orientation procedure; and (c) the plotter would be programmed to help the operator extract the required results (such as profiles across the surface). Further work on other aspects of the m e t h o d is desirable in order to assess its practicality: photographic limitations, the relative merits of metric and non-metric cameras and the scope of data that can be extracted from stereomodels of the ocean surface. CONCLUSION Because a precision of +0.3 m is adequate for many wave-measuring purposes, the use of terrestrial photogrammetric methods to measure the heights of waves in the near-shore region appears to be worth pursuing. The convenience of model orientation and data abstraction which can be achieved using an analytical plotter for which the software can be modified, renders any other plotter too unwieldy. A full assessment of the analytical plotter's role in collecting wave data would require an examination of its convenience, versatility and cost, but it would appear that, with limited further development, it should be possible to use p h o t o g r a m m e t r y to collect wave data over short periods, in special events, or to supplement or check alternative data collecting methods.
342 ACKNOWLEDGMENTS T h e w o r k d e s c r i b e d in this p a p e r was p a r t o f a p r o j e c t w h i c h has received financial s u p p o r t u n d e r t h e Australian R e s e a r c h G r a n t s S c h e m e . T h e Q a s c o SD-4 S t e r e o p l o t t e r in t h e D e p a r t m e n t o f Civil E n g i n e e r i n g and S u r v e y i n g at t h e U n i v e r s i t y o f N e w c a s t l e , Australia, was d o n a t e d b y Q a s c o Pty. L t d . o f S y d n e y . M u c h o f t h e o b s e r v i n g on t h e SD-4 s t e r e o p l o t t e r f o r the p u r p o s e s o f this s t u d y , was u n d e r t a k e n b y Mr. H.T. Kniest.
REFERENCES Elfick, M.H. and Fletcher, M.J., 1982. The Qasco SD-4. Photogramm. Eng. Remote Sensing, 48 (6): 925--930. Maresca, J.W. and Seibel, E., 1976. Terrestrial photogrammetric measurements of breaking wave and longshore currents in the nearshore zone. Proc. Conf. Coastal Eng., 15th, Honolulu, pp. 681--700. Mitchell, H.L., 1983. Wave heights in the surf zone. Photogramm. Rec., 11(62): 183-193. Silvester, R., 1974. Coastal Engineering, 1. Elsevier, Amsterdam, 457 pp. Wolf, P.R., 1974. Elements of Photogrammetry. McGraw Hill Kogakusha, Tokyo, 562 pp. Wong, K.W., 1980. Chapter II. In: C.C. Slama (Editor), Manual of Photogrammetry (4th Edition). ASP, Falls Church, Va. 1056 pp.