THE UNIVERSITY OF CALGARY

Transcription

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2 Formulation

3 THE UNIVERSITY OF CALGARY FACULTY

4 Abstract One approach to local least-squares polynomial approximation of data in two independent variables consists of centring a square subarea on each point in a data set and computing the coefficients that define the best-fit polynomial over the subset defined

5 Approximating

6 Acknowledgements I am grateful to my supervisor, Dr. Brown, for his guidance through the course of this work. His urging to explore questions that were of foremost interest to me made this project

7 Table of Contents Approval Page... ii Abstract...

8 Introduction Regional and residual anomaly maps Long-wavelength anomaly maps...

9 Table 2.1. Terms included List of Tables

10 List

11 Figure 3.7. Transfer function of the ^-component gradient operator for a 5x5 window, and a second-order polynomial Figure 3.8. Transfer function of the ^-component gradient operator for a 7x7 window, and a first- (a), second- (b) and third-order (c) polynomial... 73,74,75 Figure 3.9. Transfer function of the ^-component gradient operator for a 9x9 (a) and an 11x11

12 approximating polynomial 2

13 Chapter 1 Introduction 1 There are two main objectives of this thesis. The first is to use least-squares polynomial approximation methods

14 3 the local least-squares polynomial approach are first derived and studied; implications for

15 Chapter

16 addressed 5

17 r=0 5=0

18 0» -My)I < e (2.2) is satisfied over the interval defined by x and y. In addition, if f(x,y) is 2-n periodic, continuous everywhere,

19 inhibited by this restriction. That is, polynomials of the form of equations 2.1 can approximate data with infinitely many wavelengths regardless of window size, the inherent difference being that polynomial approximation requires no assumptions regarding the periodicity of the signal. Thus Fourier series expansion does not lend itself

20 coefficients can be considered to be the regional. This regional can either be subtracted from 9

21 10 that removing residual features via local least-squares approximation is equivalent to linear digital filtering (Wood

22 been given by Davis (1986). Hence only a cursory discussion is presented here. It is important 11

23 12 as = O, (2.5 for

24 13 features from data in two independent variables has been given by Hayes (1970). However in Hayes's discussion, the computational procedure is not stated in terms of a convolution. Hence, after summarizing the relevant elements of Hayes's review, this procedure,

25 14 where t+w E X tpr i+w E PW t-i-w (2.9) t+w B f-i-w

26 15 t+w j+\v E EEE-^2^ r=i-w u=j-w r s "" ** w (2.13) -. /' Evaluating equation 2.13 at X=X 1 and y=jj gives an expression for the best-fit polynomial

27 16 This Z ns( x t*yj) ~ z (*i»)0 ~ J^ 53 ^tn z (*f v a)- ( 2-16 ) t=-~ «= -

28 17 even functions, and orthogonal polynomials of odd orders are odd functions. That is, from equation 2.9, p r (x)

29 the window size 18

30 Wavenumber-domain response of local least-squares filters 19

31 20 window size =* 0.3 =

32 k y (cycles/grid interval)

33 exhibit reliable filtering characteristics. 22

34 domain coordinate axes. 23 The spectrum in figure 2.1b shows that the amplitude spectrum is symmetric about

35 , O window size = 0.3 TJ 0-2 fr

36 25 figures 2.3 is the suite of all possible filters that may be generated for a 9x9 window. Increasing the polynomial order increases the cutoff wavenumber of the filter. As well, these spectra demonstrate that

37 o.o window size = 9x9 polynomial order = 0/ k x (cycles/grid interval) (a) Figure 2.3a-d. Amplitude spectra

38 ky (cycles/grid interval) to

39 ^l 1 8-.«e. ^ ^g- H u 5? ^* ttl p b \ o p N> p O4 to OO k y (cycles/grid interval) P b p p pppppp bo ki b> In > Giho^ P P k> ^- O a S O /""^* f.' * < * S P b

40 ky (cycles/grid Interval) K)

41 , window size = 7x7 polynomial order = 2/ TJ , k x (cycles/grid interval) Figure 2.4. Amplitude spectrum of a filter computed using a 7x7 window and a secondorder polynomial.

42 31

43 32 ee f order = wavenumber (cycles/grid interval) 0.5 (a) Figure 2.5. Bandwidth

44 I 33

45 window size. This is an important criterion for selecting window size and polynomial order 34

46 35 19-i order = 3 order wavelength (grid intervals) (a) Figure 2.6a. Cutoff wavelengths (a) along the frequency axes, and (b) along k x =k r A first-order best-fit line

47 order = 3 order = 1 O wavelength (grid intervals) (b)

48 37 r=0 (2.26) where

49 38 Study of such filters is complicated by the fact that they introduce phase distortion. Hence both amplitude and phase spectra must be studied when evaluating the performance

50 window size =

51 k y (cycles/grid interval)

52 window size = 41

53 42 o.o k x (cycles/grid interval) (contour units (b)

54 43 (2.28) B=-/ 5=0 l

55 44 difficulties encountered when the infinite operator is truncated to give a useable operator.

56 Chapter 45

57 46 ~^% (3-2a) and direction 0 = tan' 1 -^. (3.2b) ^ The present study focuses on computing and applying the magnitude of the horizontal gradient. Equations 3.1 and 3.2a indicate the theoretical differences between acceleration and gradient magnitudes.

58 A review 47

59 Gradient-magnitude data have been computed by this technique for several 48 published studies. For example, Cordell (1979) inferred that the maxima of such data located graben-bounding faults

60 operators 49

61 50 M M AB Al,.-_ r=-_ 4rCv+y) (3.6b) where S^

62 It will 51

63 52 filtering data were previously discussed. These advantages hold true for gradient computations.

64 The method of nonorthogonal polynomials 53 As before, the data are assumed to be on a rectangular grid, with dimensions max by p&y where Ax and Ay are the grid intervals in the x and v directions. The moving window is specified by {*,., y }.J] through {x i+w, y j+w )}, where i = 1,2,...,w, y = 1,2,...,«, w = 1,2,...,2&+1, k < (p-l)/2 and k < (m-l)/2. Again, let Z(x,y) represent the best-fit polynomial in the window; then, Z(XyV) n n rw/ \ V^ "T^ T S C\~l\

65 r=0 5=0 '- 1 (3-8b) Imposing the L 2 norm is one method of determining the best-fit polynomial for the data, 54 where de = O (3.9 da rj for all r = 0,1,2,... «, and 5 = 0,1,2,...«Solving i+w J+\v E = E E {ztw,)-^,)) 2. f=i-w u-j-w (3JO)

66 and 55 Ul \

67 *> %% ax i i (3.12a) 56

68 57 (3.14a) and

69 58 function over a symmetrical interval, and thus is equal to zero. The product under summation in the numerator of equation 3.14b can be seen to be odd by recalling the property

70 59 There are four properties of the impulse response of gradient-component operators that are not shared by the regional filters. These are here discussed.

71 60 function reduces to a one-dimensional operator. Using the notation of equations 3.12, non-zero coefficients occur for m equal to zero for ^-component gradient operators, and

72 band 61

73 ^I * I! C> t1 i;? idea t : C-C C i c 1 ^ S i»1npontenit 5" d Ler ' It 1 fil tor » I i < > t << h I C 5 t i ^; > I ' 0.1 ^ 5 \ S < *. «1 «< t «I«3 1 ^ S 9 «5 0.0 OO I OJ ki(cycles/grid interval) (a) 0.4 OJ I T I I 2» Ui IS ideal y-compooeac gradient filter s O4 0.4 I l I I

74 t I 63 11S'W 56'N C.I mgal/km O 50 I km I Figure 3.2. Horizontal-gradient magnitude of the Bouguer data shown in figure 4.1, computed

75 functional frequency range. The following three incarnations of this ideal case attempt 64

76 I I I 0.4 O m n m n 0 O n?n n " 1 In 0.3 TJ

77 frequency half of the passband, and thus should give reliable results for studies 66 focusing

78 k x (cycles/grid interval) 0.4

79 o.o k x (cycles/grid interval) Figure 3.5. Amplitude spectrum

80 band-pass filter designated by 0.04 and 0.08 cycles per grid interval, and a low-pass filter designated 69

81 k x (cycles/grid interval) Figure 3.6. Amplitude spectrum of y-component gradient operator from Kis (1983), calculated

82 T~ window size =

83 pass virtually 72

84 ,

85 , , i,, -, window size =* 7x7 polynomial order = k x (cycles/grid interval) (b)

86 p Oi p b k y (cycles/grid interval) i - a K) <P 3 P A " Ir p o

87 window size = 9x9 polynomial order = Tf 0.3 S3I

88 window size =* 11x11 polynomial order =

89 These data were obtained by empirically estimating the limits of the passband for a range of window sizes and polynomial orders. Because these filters are band-pass with respect to one frequency axis, and low-pass with respect to the other, a complete specification

90 33-i first-order polynomial CO high-pass wavenumbcr ^ low-pass wavenumber I I I I I I I I I I I I l I I I I T I I I I I

91 80 33-i third-order polynomial OT high-pass wavenumber "%k low-pass wavenumber k y (cycles/grid interval) (b)

92 81 passband widths I I third-order polynomial first-order polynomial k y (cycles/grid interval) (c)

93 82 component operators. First, these plots provide an estimate of the frequency content of data filtered by a weight function of a given polynomial order and window size. This serves as a guide for selecting parameters according to the objectives of the study, as well as indicating the frequencies represented on gradient data. In addition they indicate

94 Chapter 4 - Wavelength Filtering of Bouguer Gravity Data 83 Introduction

95 84 116'W t W5 /) VV Y(( u 56'N C.I. = 4 mgal km 50 Figure 4.1. Bouguer gravity

96 85 residuals has been computed. Filtering was performed in the frequency domain with a sixth-order Butterworth filter (Kanasewich, 1981, p. 265) using software developed by Eaton (1988).

97 X.

98 116* W Residual X, = 62.5 km Regional 116'W I 56'N C.l. = 2 mgal C.I.

99 116'W Residual 56'N Regional 116'W X, = 125 km I 56'N C.I. = 2 mgal C.I.

100

101 C.I.

102 Long-wavelength anomaly maps 91 Kane and Godson (1985) studied attenuation caused by filtering, and concluded that anomaly maps containing wavelengths predominantly shorter than

103 area is probably not in isostatic equilibrium. These authors surmised that this anomaly is 92

104 Relative gravity highs 93

105 94 On the map shown in figure 4.2h the most prominent feature is an approximately 170 km x 60 km north-south (in the northern portion) and southwestnortheast trending

106 basement rocks corresponding 95

107 adjacent pluton 96

108 transformation with horizontal-gradient component filters. Frequency filtering 97

109 Chapter 98

110 Regional/residual separation using polynomial approximation 99 In chapter

111 implemented 100

112 (a) (b) 56'N i data-set size=181xl81 order =11 11 rw 116"W I 56'N C.I.= 2 mgal ton BO C.I.- 2 mgal Figure 5.1. Data from figure

113 116'W 116* W (a) (b) km SO window size = 57x57 order = 2/3 I 56'N C.l.«= 2 C.I.- 2 rngaj Figure 5.2. Data from figure

114 data-set size

115 window size = 55x55 order = 0/ 'W C.I.=

116 Regional maps computed using polynomial approximation 105 reproduces No one regional field computed using polynomial approximation wholly

117 approximation, it is quite likely an artefact. The fact that it does not appear when the data 106

118 Finally, 107

119 content 108

120 109 various combinations of parameters. However, irrespective of the behaviour of the filter, transforming an anomaly to its gradient-magnitude counterpart inherently alters the frequency content. This occurs because the magnitude of the gradient of an input waveform has wavelengths half as long, provided these wavelengths fall within the passband of the gradient-component operators. This

121 110 N- Figure 5.5. A monochromatic waveform, and its gradient magnitude. A single cycle of the input waveform spans two cycles of the gradient magnitude.

122 instance, gradient-component operators based on a 31x31 window and a first-order Ill polynomial enhance wavelengths approximately between

123 'W 56'N C.I mgal/km, km 50 Figure 5.6. Horizontal-gradient magnitude data computed using a 5x5 window, and a third-order polynomial.

124 'W t I 56*N C.I mgal/km O 50 I km Figure 5.7. Horizontal-gradient magnitude data computed using a 17x17 window, and a second-order polynomial.

125 'W t I 56'N C.I mgal/km. km 50 Figure 5.8. Horizontal-gradient magnitude data computed using

126 'W I 56* N C.I.B 0.05 mgal/km O ^ 50 km I Figure 5.9. Horizontal-gradient magnitude data computed using a 51x51 window, and a first-order polynomial.

127 in 116

128 overestimating 117

129 5.7, 118

130 portion 119

131 120 transmission of considerably higher frequencies along the directions of the coordinate axes. This degrades features that are oblique to the axes. Polynomials defined by equation 2.1b

132 121 the gradient-component operators. This makes the horizontal-gradient magnitude technique unsuitable for studying long-wavelength crustal features that are masked by higher-amplitude shorter-wavelength features.

133 Chapter 122

134 proportional 123

135 in the frequency domain. The former are restricted because the passbands cannot be varied, whereas 124

136 References 125 Abdelrahman, E.M., Bayoumi, A. L, Abdelhady Y.E., Gobashy, M.M., and El-Araby, H.M., 1989, Gravity interpretation using correlation factors between successive least-squares residual anomalies: Geophysics,

137 Chandler, V.W., Koski, IS., Hinze, WJ., and Braile, L.W., 1981, Analysis of multisource gravity 126

138 Grauch, V.J.S., 127

139 128 Kis, K., 1983, Determination

140 Stewart, R.R., 1985, Median filtering: Review 129

141 Zurflech, E.G., 1967, Application of two-dimensional linear wavelength filtering: Geophysics, 130