Computation of the a p rio ri covariance function.

The solution of the a priori covariance functions are simplier in their spectral forms. This form is suitable for an approximation by the discrete Fourier transform in which the covariances can be computed by the fast Fourier transform (Proakis and Manolakis 1988). The approximation of the continuous Fourier transform by the discrete transform should conform to the sampling theorem. This theorem states

fg > 2B (3.60)

If the sampling process occurs in the time domain, 1/fg is the sampling interval of the time signal, and B is the single-sided bandwidth of the signal. In our case, sampling is performed on the spectrum of the covariance, 1/fg becomes the discrete interval of the spectrum and B becomes the single-sided span of the covariance in the time domain. The minimum sampling frequency fg defined in (3.60) is called the Nyquist frequency. The theorem in (3.60) is a sufficient condition to avoid aliasing (Proakis & Manolakis 1988, Sklar 1988) due to undersampling.

For the computation of the covariance function, the spectrum is sampled at discrete intervals prior to performing the inverse discrete Fourier transform. The sampling interval of the spectrum 1/fg is determined by the time span B of the covariance, which in turn depends on various altimeter and surface parameters in the covariance function. There are two general methods of avoiding the aliasing error. The first method is by oversampling the spectrum and the second one is by applying an anti-aliasing filter (Proakis & Manolakis 1988, Stremler 1982). The purpose of this filter is to suppress all the temporal components of the covariance which are higher than half of the sampling frequency of the covariance spectrum, (l/2)fg. Although these two methods can avoid aliasing, the a priori covariance Cqq (3.15), (3.47) and (3.56) are two-dimensional spectral functions, and oversampling and pre­ aliasing are computationally expensive. Therefore, the temporal span of the covariance, B, and its relationship to different parameters are determined empirically. Once the temporal span is determined, we can sample the spectrum at the Nyquist rate.

For the one dimensional covariance function (3.32) and CfÇ (3.59), the computing time is comparatively less time-consuming so that oversampling and anti­ aliasing methods can be used efficiently for avoiding the aliasing error. We currently use the oversampling method to calculate these one-dimensional covariances

The spectrum of the two dimensional covariance functions, C00, C^0, and CçÇ is concentrated on the diagonal of the frequency axes (co and co'), as described in fig.3.10. However, the input data to the two dimensional fast Fourier transform must

spectrum

Fig.3.10 The orientation of the 2-dimensional spectrum of the covariance function with the original axes o) and rotated axes u.

be in a form of a rectangular block. Therefore, the off-diagonal samples of the spectrum must be filled with "dummy zeroes" for the entry to the fast Fourier transform. The transform computation of these covariances is very cost-ineffective. Alternatively, the frequency axes are rotated by 45®, so that the covariance spectrum is concentrated on the rotated axes (u, u'), as shown in fig.3.10. The relation of the original and rotated axes are given as (Fante 1988)

co=usina + ucosa

co'=ucosa - u'cosa

(3.61)

where co, and co' are the original frequency axes; u and u' are the new rotated axes; and a is the rotated angle.

When we substitute the above expressions into the covariance functions, the new axes of the covariance spectrum are rotated by but the position of the spectral components remains unchanged. The data with reference to the new axes (u,u') becomes a rectangular block that can be input to the fast Fourier transform directly. This can save a lot of redundant computation.

After the inverse Fourier transform of the covariance spectrum, the temporal samples will have the axes referred to the rotated axes (u,u'). The same relations (3.61) and (3.62) are used to change the reference of the samples from the rotated axes (u,u') back to the original axes (cû,cû').

When the sampling frequency, fg, of the covariance spectrum is greater than twice the time span of the covariance (2B), we can always reconstruct the continuous covariance from the discrete samples without error (Proakis & Manolakis 1988). The formula we use is called the ideal interpolation formula (Fante, 1988)

C e e (r,t.r.„ ,t„ )= I I C e e (r ,n T .,n .,n 'ir ,)

.S '

n=-oo n'=-‘«

(3.63)

In our case, the sampling frequency of the spectrum at co and co' axes are the same, hence T^=T2 = fg/Ng, where fg is the sampling frequency of the covariance spectrum and Ng is the number of discrete samples.

In the calculation of the 2-d covariance functions (3.15), (3.47) and (3.56), for each frequency co and co', we have to calculate the summation series of the kernel from n = 1 to oo. To perform this computation, an approximation is made. When the current value of the kernel is insignificantly small compared with the last total sum of the kernel, calculation of the series will be terminated, and the total sum of the series will be approximated by the last sum. We have found that the kernel of the summation series has a oscillating and decaying fashion. It is important to ensure that the

summation terminates in the computation because of the decay of the kernel, not its oscillation.

As we said earlier, the calculation of the 2-d covariance, involves a calculation of the series summation for every frequency value (O and co'. For this reason, the asymptotic expression of these covariance functions are derived. For a large, positive value of CDCO'at^, the computation can be switched from the non-asymptotic form to the asymptotic form of the covariances. This can save considerable computing time for calculating the summation series. In switching the non-asymptotic to the asymptotic form, one should ensure that a smooth transit of the value is obtained. Otherwise, discontinuities will be introduced.