Tipos de estrategia didáctica artificial cuántica

27cy

Ag(Xp,yp) 0 /N(xp,yp) .3.29

where /N(x,y) = [(x^+ .3.30

Making use of the spectra of the functions /n and Ag i.e. L n (u ,v) and AG(u,v), the Stokes’ integral becomes,

1

NAg(Xp,yp)= F^[AG(u,v) L n(u ,v) ] 27TY

.3.31

The spectra of the kernel functions are given analytically, so N becomes

1 NAg(Xp,yp)= F 2ny AG(u,v) . - 1 2ny AG(u,v)

q J

The FFT technique has some error sources associated with it.[Nagy,D.& Fury,R.J.,1990]. These are discussed in the sequel. The first is associated with the fact that FFT technique requires data to be gridded [Sideris,M.G.,1995], [Forsberg,R., 1990]. This necessitates some kind of gridding and averaging technique. So, instead of using the point data, in some cases one ends up using mean data. There is some information loss in moving from original point data to grid data or mean data. This case is investigated later on in this work. The second error is called the aliasing effects error. As mentioned earlier, measurements or signals can be considered to be randomly distributed or to be in some regular grid in finite spatial area. According to the sampling theorem, it is possible to recover the inter­ vening signal from certain sample data if the data is band limited. [Bracewell,R.N., 1986] In reality, it may not be band limited for gravity quantity although there may be a fre­ quency beyond which spectral contributions are negligible for practical purposes. On the other hand, there is usually the case of under sampling or lack of dense enough data to

Chapter 3: Geoid Determination Methods and Techniques

represent high frequency components present in the underlying quantity. Thus there is always an error in the gravity spectrum either due to insufficient sampling or the non- bandlimited nature of the gravity anomalies. This is called the aliasing effect. Aliasing can only be truly eliminated by sampling the signal at a rate at least twice as high as the highest frequency present in the data (if known). The practical solution to minimize or eliminate aliasing effects is to use as dense a data sample as practically possible. [Sideris, M.G.,1994]. Figure 3.5 shows the aliasing effect.

Figure 3.5 The Aliasing Ejfect.

The third error source is called the Spectral Leakage effect. This is due to capsize trun­ cation or data being in a finite domain (limited record length), which does not permit the long wavelength to be accurately represented. In other words, spectral leakage comes from the failure to examine the signal over its entire assumed period. This results in an erroneous Fourier transform, obtained as the convolution of the true Fourier Transform with a certain sine function, when using the Discrete Fourier Transform (DFT) technique. This effect is called leakage since some energy leaks from the main lobe to its side lobes. Spectral leakage can be minimized by tapering (bringing nonzero gravity values at bound­ aries smoothly to zero.) This is done effectively by using a window function other than the rectangular one, the spectrum of which will have insignificant side lobes. Usually, the

cosine taper window function is applied on the data. In practice, the given set of gridded

free-air anomalies is multiplied by a two-dimensional 80% or 50% cosine taper window. This effectively eliminates discontinuities along the edges of the grid and allows the data to take on some semblance of periodicity. Such a window must, however, be applied to the actual data and not to any artificial data such as that generated by filling out a record

Chapter 3: Geoid Determination Methods and Techniques

of zeros in the process of zero padding.{st& Bracewell, (ibid)) [Schwarz,K.P., Si­ deris,M.G. & Forsberg,R., 1990], [Vermeer,M., 1992], [Zhang,C., 1995].

In practice, in the Combination method, a global geopotential model is used as a reference field for the local gravity data.[Wang,Y.M., 1993]. This is effectively using a low-pass filter, where frequencies greater than F„ = 180/Nmax are cut off. Thus, those frequencies that may cause leakage are compensated by the global geopotential model effectively re­ ducing spectral leakage. Also, increasing capsize radius or record length to accommodate the longest wavelength in the data will also reduce the leakage problem. An intuitive ap­ proach to spectral leakage is the understanding that signals with frequencies other than those of the base frequencies of the DFT are not periodic in the observation window. This is linked to the fourth error source, that of Periodicity Effects. This due to periodic con­ tinuation of data assumed by FFT algorithm. The periodic extension of a signal not commensurate with the natural period, causes discontinuities at the boundaries of the observations. These periodicity effects cause leakage all over the entire observation data set at the discontinuities. This problem is minimized by Zero Padding [Zhao,S., 1989], [Harris,F.J., 1987], [Sideris,M.G. & Li,Y.C., 1993], [Gleason,D.M., 1990].

Another problem solved by zero padding is the Cyclic or Circular Convolution This is when the convolution of two discrete spectra is found to be erroneous.(e.g. convolution of Stokes’ and the terrain correction {tc) integrals.). In carrying out the convolution of the linear sequences one expects the resulting convolution to be a linear sequence as well. However, the use of discrete periodic FFT results in a periodic convolution or circular in 2D. Convolution of two m by « sequences produces a sequence of area 2m-1 by 2n-\.

Thus, each array should be dimensioned 2m by 2n (or larger array). This is where the zero values are assigned to each location in the extended areas of the space domain array. This is the process of Zero Padding.[Sideris,M.G., 1987], [Tsiavos,I.N., 1996]

The sixth error source is called Phase Shifting. This is caused by origin of the coordinate system for the data and the assumed origin of the subroutine of the FFT not being consistent. This is easily corrected for by a translation. [Gleason,D.M., 1990], [Si- deris,M.G., & Li,Y.C., 1993]. The problem of Singularity o f Kernel Function Origins is solved by using modified kernels or analytical spectra or using the multiband FFT analy­ sis as demonstrated in Sideris and Schwarz (1988). (see also [Forsberg,R.& Sideris,M.G.,

1993] and [Vermeer,M. & Forsberg,R., 1992]). The eighth and ninth error sources are the assumptions of the Planar Approximation in 2D FFT [Hees van,S., 1990] and the Spher­ ical Approximation in the 2D FFT, which are minimized by the spherical approximation

Chapter 3: Geoid Determination Methods and Techniques

in the first case and an ellipsoidal correction in the 2D FFT. [Sideris,M.G. & Forsberg,R., 1993], [Sideris,M.G. & She,B.B., 1995].

[Haagmans et.al,1993] introduced a new method for exact evaluation of convolution in­ tegrals on the sphere using ID FFT. The key to the ID FFT technique is that the kernel function values, like Stokes’, for a certain longitude difference between computation and integration point are the same for all computation points on one parallel, but different for computation points on different parallels. As a result, only an east-west convolution is carried out by means of FFT. The remaining north-south integration can be performed by a pointwise integration. The effect of this is that correct kernel function is used every­ where in the integration area. [Haagmans,R., de Min,E. & van Gelderen.,M, (ibid)]. The

ID spherical FFT approach gives the same results as those obtained by direct numerical integration. In addition, because it only needs to deal with a one-dimensional complex array each time, a considerable saving in computer memory results compared to the 2D FFT techniques. Moreover, the adoption of FFT makes it far more computationally effi­ cient than the classical direct numerical integration. [Sideris, M.G., & She,B.B., 1995], [Zhang,C., 1995].