557 



facilities, however, were not available for the pres- 

 ent study and the conventional technique was used. 

 In the three-dimensional space which is reproduced 

 in the stereoscopic viewing devices a right-handed 

 system of coordinates was defined with the y-axis 

 in the direction of flight and the z-axis upward. 

 During the analysis the sea surface was read at a 

 square grid with spacing Ax = Ay , which was chosen 

 such that aliasing in the spectrum would be limited 

 to only a fraction of the total wave variance. For 

 each stereo pair the analysis was carried out in a 

 square field as large as possible and the elevations 

 were determined relative to an arbitrary plane of 

 reference. In the subsequent numerical analysis the 

 linear trend was removed through a least-squares 

 analysis. The fields obtained from a series of 

 stereo pairs were initially arbitrary in shape but 

 fairly close to a rectangle. Later they were clipped 

 or extended to a square of one common size of Lj,'Ly 

 as required in the spectral analysis. Sections where 

 no stereo information was available (mainly in the 

 areas of extension) were filled with zeros. 



h(x) were available from the stereo analysis in a 

 number of square fields and these fields were con- 

 sidered to be realizations of the ensemble. They 

 were Fourier transformed with a multi-dimensional, 

 multi-radix FFT procedure [Singleton (1969) ] and 

 the final estimates were obtained by averaging the 

 results over the available realizations. The sea 

 surface data were not tapered and the spectral 

 estimates were not convolved; consequently the 

 spectral estimates are "raw" estimates. In analogy 

 with time series analysis [e.g., Bendat and Piersol 

 (1971)] the reliability is represented by a x^~ 

 distribution with 2n degrees of freedom, where n 

 is the number of fields. The resolution denoted 



by Akj. • Ak„ is on the order of (L^ 



Ly) 



-1 



The k,6-Spectrum 



->- 

 The transformation of the k-spectrxom to the k, 



e-spectrum is formally given by Eq. 4. 



E(k,e) = E(k) J 



(4) 



3. TRANSFORMATION AND STATISTICAL SIGNIFICANCE 



The sea surface data from the stereophotogrammetric 

 analysis were Fourier transformed to estimate the 

 two-dimensional wave number spectrum (k-spectrum) . 

 To inspect the directional characteristics as a 

 function of wave number, the k-spectrum was trans- 

 formed to the wave-number, direction space to pro- 

 duce the k, 6-spectrum. The k-spectrum was also 

 transformed to the frequency domain. 



The li-Spectrum 



The definition adopted here for the two-dimensional 

 wavenumber spectrum E(k) is given by Eqs. 1, 2, and 

 3. 



where 



-J- H(k) 



E(k) = lim < -^ > 



A-*" 



H(lt) = I// h(J) e-^2"^-^d3|2 



A = // dx 

 R 



(1) 



(2) 



(3) 



where k = magnitude of k, 9 = orientation of k and 

 where the Jacobian Jj = k. Computing the values of 

 E(k,e) at a regular grid in the k,9-plane requires 

 the estimation of E(k) at corresponding values of 

 k. This was done by bi-linear interpolation of 

 E(k) at the proper values of k (see Figure 1) . 



The directional resolution can be estimated by 

 considering the angular distance between two 

 neighbouring, independent estimates of E(k) on a 

 circle in the k-plane centred in k = 0. On this 

 circle with arbitrary radius, k, approximately 

 2irk/Ak independent estimates of E (it) are available 

 and the directional increment between these estimates 

 in radians is Ak/k. This would be a fair approxima- 

 tion of the directional resolution if all pictures 

 were oriented in the same direction. But actually 

 the orientation is a random variable due to the heli- 

 copter motion during the sortie. The directional 

 bandwidth to be added will be on the order of twice 

 the standard deviation (Og) of the helicopter yaw. 

 The final expression for the directional resolution 

 (A9) is given in Eq. 5. 



AS - Ak/k + 2o, 



(5) 



and <> denotes ensemble averaging. Observations of 



The resolution in k will be on the order of the 

 increment between estimates of E(k) in the k-plane 

 which is L"-' = ly''- 



The reliability of the estimates of E(k,e) can 

 again be expressed in terms of a x2_^-Lstj-j_]-,Qtion but 

 the number of degrees of freedom is not uniformly dis- 

 tributed over the k,6-plane. It constitutes an un- 



E, Es Et 



1 



I 



I 



4.E0 

 I 

 I 

 I 



' I 

 I 

 I 



I 



E, Es E2 



E is estimate of E(k) 



E_ linearly interpolated between E and E. 



E linearly interpolated between E_ and E, 

 D 3 4 



E„ linearly interpolated between E and E 



U JO 



+ gridpoint in k,9 plane transformed to 



k-plane 



grid in k-plane 



FIGURE 1. Bi-linear interpola- 

 tion in the k-plane. 



