Fourier Transforms of Functions on the Continuous Domain

Transcription

1 Chapter Fourier Tranform of Function on the Continuou Domain. Introduction The baic concept of pectral analyi through Fourier tranform typically are developed for function on a one-dimenional domain where the independent variable i time. On the other hand, the domain of many geodetic and geophyical ignal i the phere that approximate the Earth, which in ome application may alo be approximated locally by a plane. In order to extend the pectral concept to thee more general domain, thi chapter firt decribe the Fourier tranform in one dimenion, uing the independent variable, x, to emphaize and prepare for a multi-dimenional Carteian domain. Once firmly etablihed in one dimenion, the Fourier analyi generalize with no effort to higher dimenion, pecifically the plane. The adaptation to the pherical domain i le trivial and require a change in bai function from ine and coine to Legendre and inuoidal function, and the coordinate change to pherical polar coordinate. It i aumed that the domain of x i the entire real line. The domain in thi cae i continuou. If a function i ampled at dicrete and uniformly ditributed point on the real line, then the function i more aptly called a equence. In thi cae, the domain of the function i dicrete. Thee two type of function on the Carteian pace domain are further divided into periodic and non-periodic function and equence, giving four type in total. The pherical domain i then alo viewed a the domain for a ind of periodic function. Thi chapter et the foundation for pectral method uing function (periodic and nonperiodic on the continuou domain. The dicrete domain cae i taen up in Chapter 3. All function and equence are conidered to be real-valued, in accordance with application in geodey (and geophyic, although the entire theory of Fourier analyi i uually developed for complex-valued function, and mot of the reult hown here alo hold for complex function. Mot of the material in thi chapter i taen from variou textboo on Fourier and timeerie analyi. No pecific reference are made for every concept and reult. In everal cae detailed derivation are omitted becaue they are too lengthy, but uually they are at leat Fourier Geodey. Jeeli, January 7

2 plauible, if not evident. The tudent i encouraged to derive them. The lat ection i a bibliography of ome claical text that may be conulted for additional detail.. Fourier Serie Conider a real function, gɶ ( x, defined on the finite interval, x < P, on the real line and aume that it i extended periodically for ( ± = (, =,,, < x <, i.e., it period i P, gɶ x np gɶ x n. (. It i aumed that one may form the contant, P π π a = gɶ ( x co x dx, b = gɶ ( x in x dx,. (. P P P P P If the function i reaonably well behaved, the um of all thee coefficient, each multiplied by a gɶ x, correponding ine or coine function, reproduce the function, ( = = π π gɶ ( x = a + a co x + b in x, < x <. (.3 P P Thu, not only doe the infinite erie converge, it converge uniformly to gɶ ( x. It i alo poible to allow piecewie continuity of gɶ ( x (e.g., gɶ ( P gɶ ( may be allowed, but then the erie converge to the average of gɶ ( x on either ide of a tep dicontinuity, = = lim ( g ( ( co in ɶ π π x δ x + g ɶ x + δ x = a + a x + b x δ x + P P. (.4 where δ x + mean that the limit to zero i approached with δ x alway poitive. We will not conider thi ituation except in pecial cae, when neceary. Equation (.3 (or (.4 for gɶ ( x i called it Fourier erie repreentation and a, b are called Fourier coefficient. Later they are alo identified a the pectrum of the function, gɶ ( x. The inuoidal function are orthogonal over the interval, [, P ], which mean that for integer,, l, Fourier Geodey. Jeeli, January 7

3 P π πl in x co x dx = ; (.5 P P P π πl P in x in x dx = l + l P P ( δ δ P π πl P co x co x dx = l + + l P P ( δ δ ; (.6 ; (.7 where δ l i the Kronecer delta (equal to zero unle =l, in which cae it i equal to one. co πl x P or Multiplying both ide of the erie expreion, equation (.3, by either ( in ( x P πl, interchanging ummation and integration, and maing ue of their orthogonality, the coefficient, a, b, are again given by equation (.. An example of a periodic function repreented by a Fourier erie i illutrated in Figure.. In thi cae the erie i finite (all but a finite number of the coefficient are zero; and, it i noted gɶ = gɶ P, becaue a finite erie of continuou function i continuou. In that necearily, ( ( the cae of the aw-tooth function, gɶ ( x = ( x (Figure., where g ( lim g ( x.5mod P ɶ ɶ, the erie converge in accordance with equation (.4 to half it maximum amplitude at x = np. x P 7 3 Figure.: g ɶ π π ( x =.5 + a co x + b in x = = where the thic line i the function over one period and the coefficient are,,,,,,,,.5, 3.,, 3,.7 b, b, b =,.4, 3. { a a a a a a a } = { }, { } { } Fourier Geodey.3 Jeeli, January 7

4 ( g ɶ x Figure.: Saw-tooth function, g ( x = ( x x ɶ.5mod P ; P =. The argument of the ine and coine in the erie expreion (.3 for gɶ ( x are proportional to the quantitie, P. Thee are the harmonic frequencie of the function, where the fundamental frequency i P ( =. The term frequency, uually encountered with time-varying ignal, i ued here in connection with function depending on patial variable, and in geodetic or geophyical text it i alo called patial frequency. Another way to identify a contituent of the erie i by wave number (referring to the integer,. Frequency i the reciprocal of wavelength; that i, wave number i the number of wave of a particular length one can fit end-to-end within a unit in the patial (or, time domain. Thu, the unit of meaure of frequency i the invere of the unit of meaure for a wave contituent of the function. If wavelength i the length between pea of the wave, or a cycle, then frequency, f, may be thought to have unit of cycle per length unit. If the wave i viewed more in term of the phae of an angle, where a complete wave i π radian, then frequency, uually denoted, ω, ha unit of radian per length unit. One type of frequency can be converted to the other by noting that cycle i π radian; and ω = π f. We adopt the cyclical frequency convention here in order to intill more emphatically the notion of frequency a the reciprocal of wavelength, and, in addition, becaue it i directly related to wave number that i ued alo for periodic ignal. Formally, the term cycle i not a unit (SI convention, but it i ued to imply the context in which frequency i ued. Thu, in thi text a value of f i annotated by cy/ or cy/m to ditinguih it from radian frequency. It i clear from the preceding development that the ue of ine and coine add a level of inefficiency in notation that may be alleviated by combining both inuoid into a ingle function uing Euler formula, e iβ = co β + iin β. (.8 And, indeed, one may write the Fourier erie of gɶ ( x, equation (.3, a π i x P gɶ ( x = Ge P, (.9 = Fourier Geodey.4 Jeeli, January 7

5 where the Fourier coefficient, G, in general are complex number. The fact that the erie require both poitive and negative frequencie, or wave number, come from the combination of the ine and coine coefficient and hould not caue undue concern. Later it i hown that for real function, the coefficient with negative wave number i imply the complex conjugate of the coefficient with poitive wave number. From equation (.5 and (.6, it i traightforward to a, b, in equation (.3 and the find the correponding relationhip between the coefficient, { } coefficient, { G }, in equation (.9, a ib, > P G = a, = a + ib, < (. The invere relationhip i given by a = ( G + G, P i b = ( G G, P > (. The orthogonality of the inuoid, equation (.5 through (.7, alo condene to P π ( l i x P e dx = Pδ, (. l which i eaily proved by ubtituting equation (.8 and integrating. The antecedent factor, P, in equation (.9 mae the unit of G conitent with later tranform of non-periodic function. Thu, the unit of gɶ t per frequency unit, ince the unit of P G are the unit of ( are invere to thoe of frequency. The pectral domain in thi cae of periodic function contitute the et of frequencie, P, or wave number,. It i a dicrete, but countably infinite et, which i another way of aying that the et of bai function i dicrete and countably infinite, or that the function pace of gɶ i eparable. The et of coefficient, { a, b } or { G }, i nown more preciely a the Fourier pectrum of gɶ. It i alo the Fourier tranform of gɶ. The Fourier tranform terminology i ued generally for other type of function, a well (e.g., non-periodic function. Thu, to be more pecific in thi cae, or where it could lead to confuion, one hould ay that it i the Fourier erie tranform. Although the frequency domain i dicrete, thi Fourier tranform hould not be confued with the dicrete Fourier tranform that refer to dicrete periodic function for which the frequency domain i alo finite (Chapter 3. If gɶ i given, it pectrum formally i obtained uing the orthogonality, equation (.. Multiplying equation (.9 on both ide by e iπlx P and integrating yield Fourier Geodey.5 Jeeli, January 7

6 P ( π i x P G = g x e dx ɶ. (.3 iπ x P Since e i periodic with period, P, thee coefficient could be determined over any interval of length, P, which i eaily proved with a change in the integration variable, x x + a, for any real contant, a. The Fourier pectrum may alo be written in polar form in term of amplitude and phae, G i = A e φ, (.4 where the amplitude pectrum i A = G, (.5 and the phae pectrum i ( G ( G Im φ = tan. (.6 Re Determining the Fourier coefficient i alo termed an analyi of gɶ ( x ; and, if the pectrum i given then the recontruction of the function i called the ynthei of gɶ ( x. If the function i continuou over it period, with gɶ ( = gɶ ( P, then it and it Fourier pectrum are dual repreentation of the ame information one i equivalent to the other. The pectrum of a function diplay the ame information, but in a different domain the frequency domain, where it i often more ueful than in the pace- (or time- domain. The two relationhip, (.3 and (.9, contitute a Fourier (erie tranform pair for continuou periodic function, denoted a ( G gɶ x. (.7.. Propertie of the Fourier Serie Tranform The Fourier erie tranform pair (.7 may be manipulated uing everal linear operation whoe reult are ummarized with the following (non-exhautive lit of propertie. They can be proved with relative eae from the baic tranform pair, repeated here for convenience, π i x P gɶ ( x G e, ( = P = P π i x P G = g x e dx ɶ. (.8 Fourier Geodey.6 Jeeli, January 7

7 It i noted that while we aume only real-valued function, gɶ, all definition and propertie hold equally for complex-valued function.. Fourier tranform pair: g ( x G ɶ ; (.9 ɶ ; (.. Proportionality: ag ( x ag, a i any contant 3. Superpoition: g ( x + g ( x ( G + ( G ɶ ɶ, (. provided gɶ and 4. Symmetry: ( 5. Tranlation in x : ( 6. Tranlation in : ( gɶ have the ame period, P ; gɶ x G ; (. π i x ɶ P ; (.3 g x + x G e π i x 7. Differentiation: ( P g x e G + ɶ ; (.4 p p d π gɶ x i G p, (.5 dx P provided gɶ i differentiable at x up to order, p ; If gɶ i an even function, then o i it Fourier pectrum, and vice vera, ( = ( if and only if = gɶ x gɶ x G G, (.6 which follow from Fourier tranform pair (.9 and (.. If gɶ i a real-valued function (a in all our application, then it pectrum for negative wave number i the conjugate mirror image of it pectrum for poitive wave number (the pectrum i Hermitian; the convere i alo true: ( ( gɶ x = gɶ x if and only if G = G. (.7 * * Similarly, Hermitian function have real-valued pectra, ( ( * * ɶ ɶ. (.8 g x = g x if and only if G = G Propertie (.7 and (.8 are evident immediately by taing the complex conjugate of equation (.8. Combining propertie (.6 and (.7, if gɶ i both real and even, then o i it pectrum, and converely, Fourier Geodey.7 Jeeli, January 7

8 ( ( ( ( gɶ x = gɶ x and gɶ x = gɶ x if and only if G = G and G = G. (.9 * * Another important property (proved below i nown a Pareval theorem, P ( g x dx G ɶ ; (.3 = P = it importance lie in the interpretation of the left ide a the total energy of the function (over one period, which according to the right ide i alo ditributed over it pectrum. For example, the energy at frequency, P, i ( G + G P. It i another conequence of the fact that the (continuou function over the period, P, and it Fourier pectrum contain identical information, only expreed in different domain. If gɶ and gɶ have the ame period, P, Pareval theorem may be generalized, P * g ( ( ( ( * x g x dx G G ɶ ɶ, (.3 = P = which i obtained by ubtituting equation (.9 on the left ide, P π π P π P P i x * i ' x * i ( ' x P P P ( G e ( G e dx = ( G ( G e dx. (.3 ' ' = ' = = ' = Equation (.3 then follow immediately by the orthogonality, equation (.. Pareval theorem or it generalized form come in many flavor depending on the type of function. The univeral terminology of Pareval theorem i adopted in thi text for each of thee..3 The Fourier Integral Suppoe the period of a function become infinitely large and that the function i abolutely integrable, g ( x dx <. (.33 From equation (.8, one may write Fourier Geodey.8 Jeeli, January 7

9 P P i x x P π ( gɶ ( x = gɶ ( x e dx. (.34 P = Setting f = P with P = δ f and taing the limit, P, the frequency, f, change from being countably infinite to continuou over the entire real line, and the um become an integral, π ( = ( i f x i π f x g x g x e dx e df, (.35 which i alo nown a the Fourier integral equation or theorem. A ingle-integral form i obtained firtly by changing the integral on f to the limit a f of a definite integral over the interval, ( f, f ( lim ( f, and interchanging integration, f π f ( x x i g x g x e df dx =. (.36 f Then, with equation (.5 and noting that the integral of the ine function, being an odd function, i zero, one obtain, ( π f ( x x in g ( x = lim g ( x dx, (.37 f π x x where the integrand i alo well defined if x = x. The Fourier tranform of g ( x i defined by i ( ( ( π f x F g G f g x e dx ; (.38 = and, equation (.35 then how that the invere Fourier tranform converge to g ( x at point of continuity, ( ( ( iπ f x F G g x G f e df. (.39 = Analogou to the Fourier coefficient, equation (.3, ( G f i nown a the Fourier pectrum (or pectral denity of g. It i a function of continuou frequency, f, which, a before i defined here a cyclical frequency and, jut lie P, ha unit of cycle per unit of x. Fourier Geodey.9 Jeeli, January 7

10 The pectral domain in thi cae i continuou and infinite. Lie it dicrete couin, G ( f generally i complex even if g ( x i a real function and it can be decompoed into it amplitude pectrum, (( ( ( Im ( Re ( A f = G f + G f, (.4 and it phae pectrum, φ ( f ( ( Im G f = tan. (.4 Re G f Amplitude and phae together yield the pectrum in the form, ( ( G f i ( f = A f e φ. (.4 The amplitude often i diplayed a the proxy characterization of the complex pectrum, mainly to identify particular reonance or pectral trend of a ignal (ee the example, below. The phae pectrum play a role in characterizing the performance of filter a illutrated in Section g x per unit of frequency, whence it alternative The unit of G ( f are the unit of ( name a pectral denity. A band-limited function, g ( f x, i one whoe non-zero pectrum i limited to a finite pectral domain or a finite band of frequencie. For example, uually it i implied that the pectrum vanihe for all frequencie greater (in magnitude than ome f, ( G f =, f > f ; (.43 f The frequency, f, i alo called the cutoff frequency, although, thi may not alway define a definite boundary between non-zero and zero pectral content. There may be a tranition zone, ometime rather wide, that eparate the eential pectral domain of the function and it nearzero part. The bandwidth of the function i defined here by the domain of frequencie bounded by the cutoff frequency, ( f W g =. (.44 Subtituting equation (.38 into equation (.39, one obtain a in equation (.37, Fourier Geodey. Jeeli, January 7

11 f π ( g f ( x = g ( x e df dx f = π i f x x ( g x in ( π f ( x x x x dx (.45 which how that band-limiting a function according to equation (.43 create a coniderable ditortion of it original form (Gibb effect. g x, vanihe outide a ub-domain in pace, for Analogouly, a pace-limited function, ( example, T T gt ( x =, x >. (.46 Note that a pace- (or, time- limited function i not the ame a a periodic function ince it pecific definition of zero outide an interval preclude periodicity over the real line. An extremely important reult i that no function, other than the zero function, can be imultaneouly band-limited and pace-limited. There i a reciprocal tradeoff between limited frequency and limited pace domain, which i expreed by an uncertainty principle analogou to the one introduced by Heienberg in quantum mechanic. It i elaborated further in Chapter in connection with the bandwidth of filter and window function..3. Propertie of the Fourier Integral Tranform Many of the propertie of the Fourier erie tranform pair carry over directly to the Fourier integral tranform pair, equation (.38 and (.39, repeated here, ( = iπ fx i (, ( ( g x G f e df π fx G f g x e dx =. (.47 Additional propertie beyond thoe for the Fourier erie tranform enue if g ( x i alo differentiable, which i aumed here. While our application generally involve real-valued ignal, the following propertie hold alo for complex-valued function.. Fourier tranform pair: g ( x G ( f. Proportionality: ag ( x ag ( f ; (.48 ; ( Superpoition: g ( x g ( x G ( f G ( f + + ; (.5 Fourier Geodey. Jeeli, January 7

12 4. Symmetry: g ( x G ( f ; (.5 i f x 5. Tranlation in x : ( ( g x + x G f e π ; (.5 iπ fx 6. Tranlation in f : g ( x e G ( f f p 7. Differentiation in pace (time: g ( x ( iπ f G ( f p 8. Differentiation in frequency: ( iπ x g ( x G ( f d dx p p + ; (.53 ; (.54 p d ; (.55 p df 9. Duality: G ( x g ( f ; ( Similarity (caling: ( f g ax G, a ; (.57 a a The proof of tranform pair (.49 through (.55 are left to the reader. It i aumed in G f, i differentiable up to order, p. propertie (.54 and (.55 that g ( x, repectively ( Property (.56 aume that G a a function of x i abolutely integrable and ha bounded variation. It Fourier tranform i given by ( ( ( i π f x G x = G x e dx = g ( f F. (.58 The proof of equation (.57 follow from a change in integration variable, x = ax, x i π f i f x a π ( ( = ( = ( F g ax g ax e dx g x e dx a f = G a a (.59 The antecedent abolute value combine the eparate cae, a > and a <. A an example of the utility of thi property, conider the change of unit of a ignal from temporal to patial unit, a in the cae that involve the velocity of a enor ytem. The enor may yield quantitie depending on time (t, but it i deired to tranform thee to quantitie depending on ditance ( x. Auming a contant velocity, v, the change of variable i given by Fourier Geodey. Jeeli, January 7

13 x = vt. (.6 Let the pectrum of the time ignal, gt ( t, be Gt ( f t, where t g ( x = g ( x v. Then with a = v the pectrum of g ( x i G ( f v G ( vf f i temporal frequency; and let x t x x x = t x, where f x i patial frequency. For example, uppoe that the enor i a gravity meter on board an aircraft. If G f = G vf ha unit [mgal/(cy/]. Now g t ha unit [mgal] and t ha unit [], then t ( t t ( x if v ha unit [m/], then x ha unit [m], f x ha unit [cy/m], gx ( it pectrum, G ( f, ha unit [mgal/(cy/m]. x x x till ha unit [mgal], and Aide from thi rather proletarian utility, the imilarity or caling property demontrate a deeper and fundamental relationhip between a function and it pectrum. Written a ( a g ax G a f a, (.6 a cale increae in the function implie a correponding cale decreae in the pectrum; and, an expanion of the patial domain implie a contraction of the frequency domain. The latter i analogou to the uncertainty principle noted in the previou ection by which a function and it pectrum cannot both be of limited extent. Jut lie periodic function, there are certain ymmetrie in the Fourier tranform pair if g i even, real, or Hermitian. Analogou to reult (.6 through (.9, one ha, ( ( if and only if ( ( g x = g x G f = G f ; (.6 * * ( ( if and only if ( ( g x = g x G f = G f ; (.63 * * ( ( if and only if ( ( g x = g x G f = G f ; (.64 * * ( ( and ( ( if and only if ( ( and ( ( g x = g x g x = g x G f = G f G f = G f.(.65 For quare-integrable function, Pareval theorem tate that * * g ( x g ( x dx = G ( f G ( f df, (.66 which i proved imply by ubtituting the invere Fourier tranform for g on the left ide and interchanging integral, * i π f x * i π f x * G ( f e df g ( x dx = G ( f g ( x e dx df = G ( f G ( f df.(.67 Fourier Geodey.3 Jeeli, January 7

14 If g ( x g ( x g ( x = = then Pareval theorem how that the total energy of a function i equal to the total energy of the pectrum, g ( x dx = G ( f df. (.68 The quared magnitude of the pectrum, G ( f, i alo called the energy pectral denity..3. Rectangle Function The rectangle function i one of the more ueful function in the tudy of pectral analyi. It will urface repeatedly in the dicuion of filter and pectral denity etimation, a well in any other application of practical data analyi ince obervation of a geophyical ignal are available only in a limited patial domain (or, for a limited duration in time. Thi practical reality i repreented by the product of the rectangle function and the data function, theoretically defined over the entire domain. The baic form of the rectangle function, hown in Figure.3, i defined by ( b x, x < =.5, x =, x > (.69 Alternative name for the rectangle function are box-car function and top-hat function, both b ± i not univeral, but evident from the figure with a little imagination. The definition of ( i illutrated here primarily o that the invere Fourier tranform converge to b( x for all x. However, for implicity and where the convergence quetion i not an iue, we may define b ± =. ( The Fourier tranform, according to equation (.38, i given by iπ f x iπ f x ( = F ( ( = ( = = co( π in = π f ( π f B f b x b x e dx e dx fx dx (.7 Fourier Geodey.4 Jeeli, January 7

15 or, ( inc( f B f =, (.7 where the o-called "inc" function of f (formally the cardinal ine function, inc ( f in ( π f =, (.7 π f ha thi pecial name and notation becaue it find frequent uage. An alternative definition alo inc x = in x x ; however, equation (.7 i the defining notation appear in the literature, ( ( ued here. In either cae, the apparent ingularity at the origin diappear, inc( =, (.73 which i proved mathematically by l Hôpital rule for the limit a x, and i evident from it graph (Figure.3. Figure.3: Rectangle function (left panel and it Fourier tranform, the inc function (right panel. It can be hown that the invere Fourier tranform, i.e., the right ide of i ( inc( π fx b x f e df =, (.74 Fourier Geodey.5 Jeeli, January 7

16 at x = ± i equal to ±, and thu the invere Fourier tranform i the ame a the rectangle function, a defined by equation (.69. Evaluated at x =, equation (.74 yield inc( f df = ; (.75 that i, the area encloed by the inc function in the frequency domain i alo unity, a it i for the rectangle function in the pace domain. The zero of the inc function occur at f = ± n, n. It can be hown that ( inc( x = b( f F, (.76 which follow from the duality property (.56 (here and wherever generic notation for the G g g =F G. direct and invere Fourier tranform are needed, we ue =F ( and ( Conider the more general rectangle function, b( x T T, for T > interval, ( T, T, that i non-zero over the, but caled to preerve unit area. By the propertie of imilarity, equation (.57, and proportionality, equation (.49, there i, ( πtf ( T x in b ( x = b B( Tf = = inc Tf T T πtf (. (.77 Thu, a the rectangle function bae hrin (T decreae, the main lobe of the inc B Tf are at f = ± n T ; the oppoite clearly hold: a T function expand (the zero of ( increae, the lobe of the inc function become narrower. In all cae, the Fourier tranform i unity at zero frequency. The rectangle function i the mathematical tool that truncate, or pace-limit, a function to a T, T, by finite domain. That i, a function, g ( x, eentially limited to a finite domain, ( the definition g T ( x g ( x, T x T, otherwie = (.78 i the ame a x gt x = b g x T ( ( ; (.79 here the pecial definition at ± T i not conidered important. Thu, for example, the effect of truncation can be tudied by analyzing the pectral propertie of the rectangle function. Fourier Geodey.6 Jeeli, January 7

17 Equation (.39 how that generally the entire pectrum i needed to capture the complete function. If the function i mooth then it i reaonable to aume that it i well repreented with a partial pectrum that exclude the high-frequency component that would otherwie reflect the roughne in the function. However, at point of dicontinuity the convergence of the Fourier integral to the function i low; and, ince the intermediate integral of the converging tranform, equation (.37, are continuou they mut ocillate trongly near the dicontinuity in order accommodate the tep in the function. Thi i called Gibb effect that i demontrated for the b x, in Figure.4 for cut-off frequencie, f = and f =, in equation rectangle function, ( (.45. Figure.4: Demontration of Gibb effect at the dicontinuitie of the rectangle function. b x, approach b( x a f. Approximation, ( f Gibb effect i preent in any rendition of a general function by it band-limited approximation if the function varie more rapidly than can be accommodated by the limited pectrum. Thi i illutrated alo in connection with filtered function in Chapter, ince a filter i nothing more than a band-limiting operation..3.3 Gauian Function Unlie the incongruity in hape of the rectangle function and it Fourier tranform, the Gauian function and it tranform are virtually identical. Defined by ( γ x e π x =, (.8 it Fourier tranform i, again, a Gauian function (Figure.6, Fourier Geodey.7 Jeeli, January 7

18 Γ ( f f = e π. (.8 The area under both γ ( x and ( f domain, one define γ ( ( x β π ( x β Γ i unity. For caled width, but till unit area in the pace = e, β >, (.8 β with Fourier tranform, uing the imilarly property (.57, given by Γ ( β π ( β f The area under ( f = e. (.83 ( β Γ i β. The width of ( β γ, may be characterized by the pan between it inflection point, x = ± β π ; and, with the ame characterization, the width of it Fourier tranform i ( π β. A the width of the Gauian function increae, the width of it Fourier tranform decreae; and, vice vera; thu, again illutrating the reciprocal tradeoff between pace-limited and band- (or, frequency- limited function, even in an approximate ene, ince mathematically the Gauian function i neither pace- nor frequency-limited. Figure.5: Gauian function, γ ( x, and it Fourier tranform, ( f Γ. Fourier Geodey.8 Jeeli, January 7

19 .3.4 Dirac Delta Function Conider again the rectangle function, ( b x T T, with < T (Figure.6, recalling that the area under any of thee caled function i unity. A T approache zero, the magnitude of ( T ( ( b x = b x T T at x = approache infinity, while it approache zero everywhere ele. Mathematically, one define where ( T ( T ( δ x = limb x, (.84 ( x, for all x δ =, (.85 and it i infinite at x =. ( x δ i called the Dirac delta function, alo imply the delta function, or the impule function. Becaue it doe not have a finite value at the origin, it i δ x i not a legitimate function. However, the infinite limit i not ometime aid that ( arbitrary, but defined in the ene of equation (.84 a the limit of a function that ha unit area; δ x may be called a function. It could alo be and, for thi reaon, or with thi undertanding, ( ( β defined a the limit of the Gauian function, γ ( x, equation (.8, a β. Even though δ ( x i not a well-behaved function it Fourier tranform exit. Indeed, from equation (.77 and (.73, lim T B( Tf = lim ( inc( Tf T =, for all f (.86 and, one may formally write ( δ ( x ( f = F. (.87 Fourier Geodey.9 Jeeli, January 7

20 Figure.6: The function ( b x T T with T. The vertical line connecting the function value at x = ± T only emphaize the area and do not define the function at thee point. From the definition that δ ( x i the limit of rectangle function all having unit area, one ha, δ ( x dx =. (.88 Furthermore, it can be hown that δ ( x x g ( x dx = g ( x, (.89 which i nown, for evident reaon, a the reproducing property of the Dirac delta function. Since the rectangle function i an even function, o i the delta function, ( x δ ( x δ =. (.9 According to equation (.88, the unit of δ ( x are /(unit of x. The invere Fourier tranform of i the delta function, and thu one alo ha the reult, δ π ( = = i f x i π f x x e df e df, (.9 Fourier Geodey. Jeeli, January 7

21 where the econd equality i jutified by equation (.9. Thi alo how that, formally, iπ f x ( = e dx = δ ( f F. ( Fourier Tranform Uing the Delta Function Fourier integral tranform of periodic function were pecifically excluded ince they are not abolutely integrable over the entire real line; i.e., inequality (.33 would not hold. Formally, however, it i poible to define uch a tranform for periodic, a well a dicrete function, or equence, with appropriate ue of the Dirac delta function. With a change in variable, firt replacing x by T ; ubequently replacing f by x, the econd equation (.9 become π i x T e dx = δ. (.93 T Subtituting the Fourier erie, equation (.9, into the Fourier tranform, equation (.38, one find, = π i x iπ f x P iπ f x P F ( gɶ ( x = G e e dx = G e dx P P = = = Gδ f P P (.94 That i, the Fourier integral tranform of a periodic function with Fourier coefficient infinite equence of impule caled by G G i an P and paced along the frequency axi at the dicrete frequencie P (Figure.7. The unit of the Dirac delta function are the invere of frequency unit in thi cae; they cancel the unit of P. The Fourier tranform of a periodic function, formulated a equation (.94, mae it conitent with the Fourier integral tranform of non-periodic, abolutely integrable function of bounded variation. It i inightful perhap only in that ene, and erve toward a unified mathematical development of the Fourier tranform for a larger cla of function. However, equation (.9 and (.3 are the preferred Fourier (erie tranform pair for periodic function, a they avoid the ue of the Dirac delta function. Fourier Geodey. Jeeli, January 7

22 Figure.7: Fourier tranform (amplitude pectrum of gɶ ( x, hown in Figure., where { G, G, G, G, G, G, G, G } {.7, 3,, 3.,.5 i3, i.4, i,.5} ± 7 ± 6 ± 5 ± 4 ± 3 ± ± = ± Secondly, conider the periodic function that i an infinite equence of identical rectangle function, each having T a it bae, = x x dɶ ( x = b, (.95 T T where x > T i the pacing between their center (Figure.8, left and, hence, the period of dɶ x a T while each rectangle retain unit area, it dɶ ( x. Defining ( x to be the limit of ( become a train of impule, a given by equation (.84 (ee alo Figure.8, right, ( = lim ( = δ ( x dɶ x x x. (.96 T = Thi function, ( x, (again in a formal etting i called the ampling function, or Dirac comb, becaue if multiplied with ome arbitrary function, it ample the latter uing impule, which ha ome utility when developing the Fourier tranform of dicrete function (Chapter 3. Fourier Geodey. Jeeli, January 7

23 Figure.8: Left: Infinite equence of rectangle function, dɶ ( x, where vertical line are included only to emphaize the area under each rectangle. Right: Sampling function, ( x. Being periodic, with period, x, dɶ ( x, given by equation (.95, can be repreented a a Fourier erie, where the Fourier coefficient are given by equation (.3, which i an integral x, x, which cover only the = over any period of d ( x term in equation (.95, yield ɶ. Chooing the interval, [ ] x π x i x x D = b e dx. (.97 T T x Now, ince x T >, the rectangle function, ( b x T, i zero outide the integration interval; and, the integration can be extended to ± without changing it value. With a uitable change of integration variable ( x = x T, and uing the reult (.7 and the definition (.7, one obtain, π π i x i Tx x x x D = b e dx = b( x e dx T T T = inc x (.98 Formally, the Fourier integral tranform of dɶ ( x, according to equation (.94, i an infinite x. Uing equation (.94 for dɶ ( x and equence of impule having magnitude, D ubtituting equation (.98 into the right ide yield Fourier Geodey.3 Jeeli, January 7

24 ( dɶ ( x = T F = inc δ f. (.99 x x x In the limit a T, and with equation (.96 and (.73, one ha the reult, F δ ( x x = δ f x x = =. (. The Fourier tranform of the ampling function i, again, a ampling function, but now in the frequency domain..4 Two-Dimenional Tranform in Carteian Space Geodetic and geophyical data may be collected in one patial dimenion, uch a topography or potential field quantitie meaured on profile uing airborne enor ytem, and can certainly be analyzed in that ingle dimenion. However, it i more typical that uch data ultimately are amaed to repreent a ignal whoe domain i more naturally the plane, or the phere, or even (though rarely three-dimenional pace. All the concept of Fourier erie and integral tranform carry over into thee higher dimenion with no effort if the underlying coordinate ytem i Carteian. The reaon for thi facility i the eparability of the bai function of thee pace, the complex exponential, into product of univariate function correponding to the individual coordinate, x, x,. Therefore, the following can be tated without much further theoretical dicuion, where it i motly a traightforward matter of including an additional coordinate, ummation, and integral..4. Two-Dimenional Fourier Serie Tranform Periodic function in two Carteian dimenion are periodic with repect to each coordinate with poibly different period, P and P, ( ±, ± = (, gɶ x n P x n P gɶ x x, for any integer, n and n. (. The orthogonality relationhip for the exponential (cf. equation (. i P P l l iπ x + x P P e dx dx = PP δ l δ l, P, P, (. Fourier Geodey.4 Jeeli, January 7

25 for all integer,,, l, l. Then, the Fourier coefficient are defined by P P π i x + x P P (,,, ɶ, (.3 G = g x x e dx dx < <, where the frequencie in the two coordinate direction are P and P, repectively. The unit of gɶ x, x divided by the unit of both frequencie. For continuou G are the unit of (, function, the correponding uniformly convergent Fourier erie i i π x + x P P,,,, PP = = ( gɶ x x = G e < x x <. (.4 Periodic function defined on higher-dimenioned Carteian pace have analogou erie expanion and Fourier erie tranform with obviou generalization. The propertie of proportionality, uperpoition, tranlation, and differentiation (propertie (., (., (.3, and (.5 for two- (or higher- dimenioned function are completely analogou and need not be repeated. The ymmetry property in two dimenion i gɶ ( x, x G,, gɶ ( x, x G,, g ( x, x G, ɶ, (.5 where the proof are identical to thoe for the tranform pair (.. Reult imilar to propertie (.6 through (.8 are the following. Directly from the ymmetry property, ( ±, ± = (, if and only if ±, ± =, gɶ x x gɶ x x G G, (.6 where the ign of the pace variable on the left match thoe of the wave number on the right. For real-valued function, ( * ( *, =, if and only if, =, gɶ x x gɶ x x G G ; (.7 and, for real-valued pectra, ( * ( *, =, if and only if, =, gɶ x x gɶ x x G G. (.8 Finally, combining tatement (.6 and (.7, one obtain * ( = ( ( ± ± = ( gɶ x, x gɶ x, x and gɶ x, x gɶ x, x if and only if. (.9 G = G and G = G *,, ±, ±, Fourier Geodey.5 Jeeli, January 7

26 again, where the option on ± mut match on both ide of the mutual implication. The * pectrum i real only if the ign are negative for both dimenion; that i, G = G and G, = G, implie that G = G = G. * *,,,,, The Pareval theorem for two-dimenional function follow in the ame manner a for equation (.3, P P * * g ( x, x g ( x, x dxdx ( G ( G ɶ ɶ, (.,, = P P = = provided that both gɶ ( x, x and g ( x, x repective dimenion. ɶ have the ame period, P and P, in their.4. Two-Dimenional Fourier Integral Tranform The tranition to non-periodic function in two Carteian dimenion now follow the line of reaoning developed for one dimenion. We require abolute integrability over the x, x -plane, g ( x, x dxdx <, (. a well a bounded variation in the two dimenion. The Fourier integral equation, analogou to equation (.35, for continuou function then hold, g ( x, x = g ( x, x e dxdx e df iπ ( fx + fx iπ ( fx + fx. (. Thu, the Fourier tranform pair for continuou function i (, (, π ( i f x + f x G f f g x x e dx dx =, (.3 (, (, π ( + i f x f x g x x G f f e df df =. (.4 The unit of (, G f f are the unit of g divided by both frequency unit. Other propertie for the two-dimenional Fourier tranform pair naturally follow from the propertie of the one- Fourier Geodey.6 Jeeli, January 7

27 dimenional tranform pair, equation (.49 through (.57. For example, the tranform pair for partial derivative of non-negative integer order, p, p, are p p g x x i f i f G f f p p x p p (, ( π ( π (, x p p ( iπ x ( i π x g ( x, x G ( f, f p p f f, (.5 p p, (.6 auming that correponding derivative exit. A in the cae for the -D periodic function, it i neceary only to elaborate on the ymmetry propertie. Analogou to tranform pair (.5, one ha (, (,, g ( x, x G ( f, f, g ( x, x G ( f, f g x x G f f. (.7 Similarly, for real and/or even function, the following reult are eaily proved. (, (, if and only if (, (, g ± x ± x = g x x G ± f ± f = G f f ; (.8 (, * (, if and only if (, * (, g x x = g x x G f f = G f f ; (.9 (, * (, if and only if (, * (, g x x = g x x G f f = G f f ; (. * ( = ( ( ± ± = ( * G ( f, f = G ( f, f and G ( ± f, ± f = G ( f, f g x, x g x, x and g x, x g x, x if and only if (. * where, in the latter cae, the pectrum i real only if g ( x, x g ( x, x g ( x, x = g ( x, x. Finally, for quare-integrable function, = and g ( x, x dxdx <, (. Pareval theorem tate that g ( * ( ( * x, x g x, x dxdx = G f, f G ( f, f dfdf. (.3 Fourier Geodey.7 Jeeli, January 7

28 .4.3 Special Function in Two Dimenion The rectangle function in two dimenion i defined a (, b x x, x < and x <, x and x, or x and x = = =, x = and x = 4, x > or x > (.4 where the pecial value at the edge and corner of the cube (Figure.9 are pecified o that the invere Fourier tranform converge everywhere to the -D rectangle function. The notational ditinction between the one-dimenional and the two-dimenional rectangle function i achieved here by the number of their argument; and, no confuion i anticipated. The Fourier tranform i imply the product of inc function, ( b( x, x = B( f, f = inc( f inc( f F. (.5 For arbitrary extent, T, T, one ha the tranform pair a in equation (.77, T T x x b x x b T f T f B T f T f (, (, =, inc( inc ( = (, T T T T. (.6 Figure.9: The two-dimenional rectangle function (left, equation (.4, and it Fourier tranform (right, equation (.5. Fourier Geodey.8 Jeeli, January 7

29 Figure. how the two-dimenional Gauian function and it identically haped Fourier tranform, γ ( x, x (, ( x + x =, (.7 e π ( f f Γ f f e π + =. (.8 With different caling in the two coordinate direction, they are given by ( β, β ( (( x ( x x, x e π β + γ = β, β >, β >, (.9 Γ ββ ( β, β (, (( ( f f f f e π β + = β, (.3 imilar to the one-dimenional Gauian function and it tranform, equation (.8 and (.83. ( β, β ( The volume encloed by γ x, x i unity for any poitive value, β, β. ( β Figure.: Gauian function, γ ( x, x (top, =, β = γ ( x, x ( β Fourier tranform, Γ ( f, f (top,, β ( f, f Γ = = (bottom, right. (bottom, left, and their Finally, the generalization of the Dirac delta function to two dimenion i Fourier Geodey.9 Jeeli, January 7

30 ( δ x, x =, if x or x, (.3 uch that it Fourier tranform i ( ( F δ x, x =, for all f, f. (.3 One alo ha and δ ( x, x dxdx =, (.33 δ ( x ' x, x ' x g ( x ', x ' dx ' dx ' = g ( x, x. (.34 Following the ame argument a for equation (.9, the invere Fourier tranform i given by δ (, π ( + i f x f x x x e df df =. (.35.5 The Hanel Tranform Some problem in geodey and geophyic are implified ignificantly if a two-dimenional function on the plane or it pectrum can be approximated or modeled by an iotropic function, one that doe not depend on direction. Certain function, uch a ernel of convolution (Chapter, are defined pecifically by phyical law to be iotropic. For example, the reciprocal ditance function that i fundamental in potential theory i independent of direction. In thee cae, the correponding Fourier tranform on the plane implify to a ingle integral. g x, x, depend only on the ditance,, from the origin, Suppoe that a function, ( (, g ( g x x =. (.36 The appropriate two-dimenional coordinate on the plane are the polar coordinate,, ω, related to the Carteian coordinate by x = co ω, x = inω, (.37 Fourier Geodey.3 Jeeli, January 7

31 where ω i an angle reconed counter-clocwie from the x -axi (Figure., and = x + x. (.38 The domain for thee coordinate i defined by < and ω π ; and, a differential area element i given by dxdx = ddω. x ω. x Figure.: Polar coordinate,, ω, of the point with Carteian coordinate, x, x. g x x, With a correponding change in integration variable the Fourier tranform of (, equation (.3, become π ( (, = ( = ( π f ( ν ω i co F g x x g e dωd π iπ fco( ν ω g e dω d (.39 where f = f co ν, f = f inν, (.4 and thu, f = f + f. (.4 The integral in the lat line of equation (.39 with repect to ω i (Gradhteyn and Ryzhi, 98, p.95; noting that the integrand i periodic Fourier Geodey.3 Jeeli, January 7

32 π iπ f co( ν ω e dω = π J ( π f, (.4 for any angle, ν, where J i the zero-order Beel function of the firt ind. Therefore, (, ( π ( ( π G f f = G f = g J f d, (.43 which i called the Hanel tranform (of order zero of g ; alo the Fourier-Beel tranform. If g i a real ignal then ince the Beel function i alo real, the Hanel tranform i real. Note that although it i a function of only one frequency (the radial frequency, f, the unit of G ( f till contain the invere quare of frequency unit. The Hanel tranform defined above i jut a pecial cae of the two-dimenional Fourier tranform, where the function being tranformed ha circular ymmetry, given by equation (.36. One then ha alo the invere Hanel tranform, derived from the invere Fourier tranform in exactly the ame way. The reult for function defined on the infinite plane i ( π ( ( π g = G f J f f df. (.44 Being a pecial cae of the Fourier tranform, the Hanel tranform ha imilar propertie. Proportionality (.49 and uperpoition (.5 are exactly the ame and need not be repeated. Slight variation in the other propertie are hown in the following.. Hanel tranform pair: g ( G ( f. (.45. Duality: G ( g ( f ; ( Similarity (caling: ( f g a G, a > ; (.47 a a Pareval Theorem in thi cae derive directly from the correponding theorem for function of two independent variable, equation (.3, * * g ( g ( d = G ( f G ( f f df (.48 It i eay to how that if Fourier Geodey.3 Jeeli, January 7

33 π g ( = g ( x, x dω, (.49 π ω = then the Hanel tranform of thi angular average i the correponding average of the pectrum, G f, f, in the frequency domain, ( G ( f = π g ( J ( π f d = G ( f, f dν π π. (.5 = ν = The iotropic rectangle function i alo called the cylinder function, defined to encloe unit volume,, < π bc ( =, = π, > π (.5 The Hanel tranform of b ( c ( = π ( π B f J f d where π ( π f π c i given by π π f J ( π f d ( π f π f J ( π f f ( π f (.5 π = = = J f π = ( π f d xj x = xj x. It i dx lim J x x =. For a different cylinder radiu, µ = µ π, J i the firt-order Beel function of the firt ind, and ( ( ( noted that B ( =, ince ( c x ( µ >, the imilarity property (.47 how that ( µ ( ( b = b µ µ (.53 c ha the Hanel tranform, c Fourier Geodey.33 Jeeli, January 7

34 ( µ Bc ( f = Bc ( µ f = J ( π µ f = J ( π µ f. (.54 µ f π µ f π Both the two-dimenional Gauian function and Dirac delta function are iotropic and their correponding Fourier tranform, equation (.8 and (.3, repectively, are alo their Hanel tranform. Note that the generalized Gauian function, equation (.9 i iotropic only if β = β = β p, γ ( β p ( ( π β β p p πβ p f = e Γ ( f = e. (.55 β p.6 Legendre Tranform While Carteian coordinate are adequate for many local analye in geodey and geophyic, global data et are increaingly prevalent, epecially ince meaurement ytem on board Earth-orbiting atellite are generating voluminou amount of data related to the olid Earth, the ocean, and the atmophere. Becaue of the near-pherical hape of the Earth, it i thu advantageou to develop a pectral analyi of function on a phere. The (continuou patial domain for function on the phere i denoted, Ω, and i defined a the et of point having pherical polar coordinate, θ, λ, repreenting, for the Earth, the geocentric co-latitude and longitude, {(,,, } Ω = θ λ θ π λ π. (.56 One could alo ue geocentric latitude, φ, intead, where φ = π θ, and there i no conventional preference. Since the pherical patial domain i finite, the bai function for the tranform to the pectral domain form a countable et; in other word, the correponding function pace i eparable. Depite the obviou periodicity (period, π in longitude, we abandon the notation for periodic function ince for mot application there i no need to ditinguih between the finite pherical domain and a correponding infinite domain. However, it i poible to develop twodimenional Fourier erie for g defined on the patial domain, Ω, in the form of equation (.4 uing x θ and x λ, with period, P = π and P = π. Thi i done to derive ome reult on aliaing in Chapter 3, and we revert to the notation, gɶ, for thi particular intance. The polar region can be awward in thi cae, and one mut impoe the contraint, gɶ (, λ = cn and gɶ ( π, λ = cs, for all λ, where c N, c S are contant. No uch multiplicity occur with the ue of aociated Legendre function, Pn, m ( co in ( mλ and co ( mλ (or θ (ee below and together with im e λ they form a complete bai for function on the phere, which i the preferred bai for pherical pectral analyi. The nomenclature for the pectral Fourier Geodey.34 Jeeli, January 7

35 tranform varie in the literature and a particular claification i adopted here with alternative noted, and no confuion i anticipated. Although the geodetic and geophyical application of pectral analyi on the phere clearly refer to the Earth, one may aume without lo in generality a unit phere, that i, a phere whoe radiu i unity (with no particular unit of meaure. A the need arie, it i eay to cale the phere to any radiu with defined unit of meaure; however, the pherical pectrum depend in the firt place on the angular variation of the function. The pherical pectral analyi thu alway refer to the domain, Ω, and never to a domain that include a radial dimenion; there i no frequency or wave number correponding to ditance from the center of the phere. The pherical function in geodey that depend on jut one angular coordinate invariably are ernel function in a convolution (Chapter ; that i, they depend on the angular ditance, ψ, from ome given point on the phere. Angular ditance refer to the length in radian of a great circle arc on the unit phere ubtended at it center by the angle, ψ ; it range of value i ψ π. Thu, any egment on a meridian (great circle containing the pole, being alo an interval in co-latitude at contant longitude, i uch an angular ditance. An interval of longitude at contant co-latitude, however, doe not define an angular ditance, or great circle arc, unle the co-latitude happen to be θ = π (the equator. Thu, pherical function of one angular coordinate may alo be denoted, g ( θ, with the origin for θ at the north pole. Imagine rotating the origin of the θ, λ -coordinate ytem (the north pole to a given point, ( θ, λ ; then, the angular ditance to any other point on the phere i it co-latitude with repect to that new pole (Figure.. More generally, the coordinate of a point, ( θ, λ, relative to a pole (origin at ( θ, λ are defined by the pherical polar coordinate, ψ, ζ, given by coψ = coθ coθ + inθ inθ co( λ λ, (.57 ( inθ in λ λ tanζ = inθ coθ coθ inθ co λ λ ( (.58 Equation (.57 i the law of coine and equation (.58 come from the formula, ( ( inψ in π ζ = inθ in λ λ, (.59 ( ( inψ co π ζ = inθ coθ coθ inθ co λ λ, (.6 where the firt i the law of ine and econd i eaily derived. Note that the angle, ζ, i lie a longitude (poitive counter-clocwie with repect to the reference meridian through the pole at θ, λ. ( Fourier Geodey.35 Jeeli, January 7

36 , with repect to the pole and meridian at Figure.: Coordinate, ψ, ζ, of the point, ( θ, λ ( θ, λ, obtained by firt rotating the x -axi to the meridian plane of (, the z -axi to the point, ( θ, λ, reulting in the x, y, z -ytem. θ λ, followed by rotating.6. One-Dimenional Legendre Tranform A univariate function on the phere i defined to depend olely on co-latitude, denoted in thi g θ, with domain equal to a emi-circle, ψ π. On thi cae by g ( ψ (later alo by ( i interval the orthogonal polynomial, e ψ, < <, form a complete et of bai function. However, in anticipation of the two-dimenional bai function on the phere, we ue another et of orthogonal polynomial that form a complete functional bai on the emi-circle. Thee P y, n, with argument, y = coψ ( y. When are the Legendre polynomial, ( n i generalized to the phere they avoid the aforementioned problem with e ψ at the pole. The argument, y, hould not be confued with the global Carteian y -coordinate. The Legendre polynomial are olution to Legendre differential equation, d dy d + + =, (.6 dy ( y Pn ( y n( n Pn ( y and are generated, for example, by Rodrigue formula, Fourier Geodey.36 Jeeli, January 7

37 n d n Pn ( y = ( y, n. (.6 n n n! dy A more practical and numerically very table recurion formula for P ( n y i ( ( ( ( (, ; (, ( n + P y = n + yp y np y n P y = P y = y. (.63 n+ n n Value of the Legendre polynomial at the interval endpoint, y = ±, are ( ( ( n P =, P =, for all n ; (.64 n n and, the orthogonality on thi interval i P ( y P ' ( y dy = δ n n n n '. (.65 n + Graph of the Legendre polynomial for mall n are hown in Figure.3. g ψ, by We define the Legendre tranform of a univariate function, ( π Gn = g ( ψ Pn ( coψ inψ dψ. (.66 The et, { G n }, i alo nown a the Legendre pectrum of g ( ψ. If ( the invere Legendre tranform converge to g ( ψ for ψ π, ( ψ = ( + ( coψ g ψ i continuou then n n. (.67 n= g n G P Together, equation (.66 and (.67 contitute the one-dimenional Legendre tranform pair. The integer, n, i called the degree of the Legendre polynomial and alo identifie a wave G, in thi cae are the ame a number of the Legendre pectrum. The unit of the pectrum, { } n the unit of the function, g. Fourier Geodey.37 Jeeli, January 7

38 .6. Two-Dimenional Fourier-Legendre Tranform It may be aumed without lo of generality in application that function, g, on the phere are defined on the finite domain, Ω (equation (.56, with the contraint that g ( θ, = g ( θ, π for any θ. Since they may be extended periodically in λ for < λ < with period π, the bai function depending on λ have the form of the coine and ine function, or alternatively the complex exponential function. For the co-latitude, however, the pecial geometry of the phere call for the aociated Legendre function, depending on y = coθ and generated by n+ m ( ( m d ( n Pn, m y = y y n n m, (.68 + n! dy for integer, n, m n, called degree and order, repectively. Specifically, thee are the aociated Legendre function of the firt ind and olution to the aociated Legendre differential equation, d d m ( y Pn, m ( y + n( n + P n, m ( y =. (.69 dy dy y Comparing equation (.6 and (.68, the zero-order ( m = aociated Legendre function P y = P y. are Legendre polynomial, ( ( n, n The aociated Legendre function are orthogonal on θ π, or y, ( n + m! Pn, m ( y Pn ', m ( y dy = n n' n ( n m! δ, (.7 + for any m n. A uitable normalization i uually included to eae calculation and mathematical manipulation, ( ( n m ( n + m n +! P y = P y n, m n, m ε m! ( where (allowing for m < in later formula, (.7 ε m, < m n =, m = (.7 Figure.3 illutrate the ocillatory behavior of thee function in co-latitude. Recurrence relation are given by Fourier Geodey.38 Jeeli, January 7

39 where ( α ( β ( Pn, m y = n, m ypn, m y n, mpn, m y, m n, n, (.73 α n, m = ( n ( n + ( n m( n + m, β n, m = ( n + ( n + m ( n m ( n 3( n + m( n m, (.74 and the tarting function are ( ( P y = n + yp y, n ; (.75 n, n n, n n + Pn, n ( y = y Pn, n ( y, n ; (.76 n P, ( y =, P, ( y 3 y =. (.77 The recurion (.73 become numerically degenerate for very high degree, and pecial numerical technique are required to obtain ufficient preciion for n > 5. Fourier Geodey.39 Jeeli, January 7

40 ( θ P co 4 ( θ P co 8 θ ( θ P co 3 ( θ P co 7 P ( coθ P ( co θ 8, 8,8 P ( θ co 8,6 θ P ( θ co 8,4 Figure.3: Legendre polynomial, Pn ( co normalized aociated Legendre function, ( order. θ (top, for elected even and odd degree; and Pn, m coθ (bottom, for degree, n = 8, and elected n, m coθ, in combination with the inuoidal function of longitude, called the pherical harmonic function, It can be hown that the aociated Legendre function, P ( Y ( θ, λ P ( coθ n, m n, m co mλ, m n = in m λ, n m < (.78 form a complete orthogonal bai for function on the phere. Thi i the form ued in geodey and geophyic. The particular notation that include negative integer, m, i only a matter of Y θ λ, to repreent both the ine and coine convenience that allow a ingle function, ( n, m, component and that avoid the ue of complex function. The normalization of the Legendre Y θ λ are function alo yield a particularly imple orthogonality relationhip. In fact, the ( orthonormal on the unit phere, n, m, Fourier Geodey.4 Jeeli, January 7

41 4π Yn, m ( θ, λ Yn ', m' ( θ, λ dω = δ n n' δm m' (.79 Ω where dω = inθ dθdλ. The two-dimenional Fourier-Legendre tranform of a ignal, g, on the phere i defined by ( θ, λ n m ( θ, λ Ω,, ; (.8 Gn, m = g Y, d n m n n 4π Ω and under the uual condition of bounded variation and abolute integrability, the correponding two-dimenional Fourier-Legendre erie converge to point of continuity of g, n ( θ, λ ( θ, λ g = G Y. (.8 The et, { n, m} n= m= n n, m n, m G, i called the (two-dimenional Fourier-Legendre pectrum of g ( θ, λ ; it i real ince both g ( θ, λ and the bai function are real. The unit of the pectrum are the unit of the function. The integer, n, m, correpond roughly to wave number (frequencie in latitude and longitude. For any particular degree, n, an increae in the order, m, decreae the number of zero, or wave, of the aociated Legendre function, Pn, m ( coθ. Pn, ( co about n wave; while Pn, n ( co (Figure.3. The pherical harmonic function, Y ( θ λ θ ha n zero or θ ha only one wave, a it vanihe only at both pole n, ± n,, thu have zero only in longitude with correponding wave effectively dividing the phere into ector; whence their name, Y θ λ, are independent of longitude ectorial harmonic. The pherical harmonic function, ( n,, and the wave in co-latitude divide the phere into zone; they are called zonal harmonic. All Y θ λ, < m < n, teellate the phere with wave and are called other pherical harmonic, ( n, m, teeral harmonic (Figure.4. Fourier Geodey.4 Jeeli, January 7

42 Figure.4: Zonal, ectorial, and teeral pherical harmonic function, Y8, (, Y8,8 ( θ, λ (middle, and Y8,7 ( θ, λ (right, repectively. θ λ (left, An alternative form of the two-dimenional Fourier-Legendre tranform pair may alo be defined, although it i not a popular in geodey and geophyic. If the bai function combine in term of the complex exponential in longitude, then the Fourier-Legendre tranform i c c Gn, m = g ( θ, λ Yn, m ( θ, λ dω, (.8 4π Ω where we define! λ Y θ, λ = n + P ( coθ e = ε P ( coθ e! ( ( ( n m ( n + m c im imλ n, m n, m m n, m. (.83 The ign of the exponential, differing from definition in other text, uch a Arfen (97, p.57, here i conitent with the Fourier tranform function defined by equation (.38. The complex-valued pherical harmonic function are alo orthonormal, 4π c * c ( Yn, m ( θ, λ Yn ', m' ( θ, λ dω = δm m' δ ; (.84 n n' Ω and, the correponding Fourier-Legendre erie i n ( * n m c c ( θ, λ ( θ, λ g = G Y. (.85 n= m= n n, m, The relationhip between the real and complex pherical harmonic function i readily een to be Fourier Geodey.4 Jeeli, January 7

43 ( θ, λ = ε ( co λ in λ ( coθ Y m i m P c n, m m n, m ( θ λ ( θ λ Yn, ( θ λ ( θ λ ( θ λ Yn, m, iyn, m,, m n = ε m,, m = Y, + iy,, n m n, m n, m (.86 Subtituting equation (.86 into equation (.8 and comparing to equation (.8 c G, immediately yield the relationhip between the coefficient, { G n, m} and { n, m} G ig, m n n, m n, m c n, m ε m n, G = G, m = G + ig, n m n, m n, m (.87 c c,, n m n m c n, m n, ε m G + G, m n G = G, m = c c i ( G G,,, n m n m n m (.88 It i ometime more convenient to analyze a function or manipulate it Fourier-Legendre G, with repect to pectrum in term of the complex bai function, but the real pectrum, { n, m} the real pherical harmonic, Y n, m, can alway be found eaily from { c n, m} G uing equation (.88. Note that the degree and abolute order, or the wave number, of the complex pectrum correpond exactly to thoe of the real pectrum. g θ, λ = g θ, then by the orthogonality of e imλ If the function i independent of longitude, ( (, equation (., the Legendre pectrum include only zonal harmonic coefficient, G n,, G = for m ; and, the erie revert to the one-dimenional Legendre erie, equation i.e., n, m (.67. Since the Legendre polynomial generally are not normalized and, by equation (.7, ( ( Pn, y = n + Pn y, a comparion of equation (.67 and (.8 yield Gn, = n + Gn. (.89 One may alo view g ( θ, λ a a function defined on the rectangle, (,, { } W = θ λ θ < π λ < π, and extended periodically over the infinite plane, < θ <, λ < <. Then, ditinguihing thi pecial function a g ( θ, λ Fourier erie tranform pair i, a in equation (.4 and (.3, ɶ, the uual Fourier Geodey.43 Jeeli, January 7

44 m i F g (, G, me π π θ π θ λ λ + π = m= ɶ, (.9 F, m π = π λ= θ = ( θ, λ m iπ θ + λ π π G g e dθdλ = ɶ. (.9 F A relationhip between the pectra, G, m and G n, m, i derived in Section 4.5. gɶ θ, λ i continuou, which include the meridian, Equation (.9 hold only where ( λ =, π, a aumed here. However, for a function that i continuou on the phere, it planar analogue ha a tep dicontinuity at the pole, θ, π ( erie converge to g ( λ + g ( π λ = ( gɶ (, λ g ( π, λ ɶ, where the Fourier ɶ, ɶ,. Generally, any repreentation of a pherical ignal ( θ + λ i m uing the bai function, e, a in equation (.9, mut enure that the following contraint are atified, gɶ (, λ = cn, gɶ ( π, λ = cs for all λ, where c N and c S are contant. Thee contraint are automatically incorporated in the pherical harmonic. That i, from equation (.68, ( Pn, m ± =, m, (.9 and, therefore, the pherical harmonic do not permit multiple value of g ( θ, λ different longitude only the Legendre polynomial, P ( coθ, contribute. n ɶ at the pole for.6.3 Propertie of Fourier-Legendre Tranform Proportionality and uperpoition propertie for the one- and two-dimenional Fourier-Legendre tranform are analogou to correponding propertie (. and (. of the Fourier erie tranform. Many of the additional propertie for Fourier erie tranform alo hold with repect to the longitude coordinate, λ. Thu, uing the complex bai function, the following hold, where. through 4. are obtained directly from propertie (., (.3, and (.5. g θ, λ G n m : (Complex Fourier-Legendre tranform pair (.93 c. (, c. Symmetry: (, c im 3. Tranlation in λ : ( θ λ λ g θ, λ G n m ; (.94 g, + G e λ ; (.95 n, m Fourier Geodey.44 Jeeli, January 7

45 c 4. Differentiation: g ( θ, λ ( im G, 5. Surface Laplacian: ( ( θ λ ( where p d p p n m, (.96 dλ provided g i differentiable with repect to λ up to order, p ; g n n + G, (.97, θ, λ n, m (, = + cotθ + θ λ θ θ in θ λ (.98 i the (calar Laplace-Beltrami operator of econd derivative on the phere, alo nown a the urface Laplacian. From equation (.69, (.7, and (.78 with y = coθ it i readily verified (Exercie.5 that = +. (.99 ( Y, n, m ( θ, λ n( n Yn, m ( θ, λ θ λ Applying thi to equation (.8 thu prove the tranform pair (.97. Per degree, n, there are n + real and independent coefficient, G, c n + complex coefficient,, G c ( G * c n, m n, m G, are not independent (if (, n m n m g θ λ i real, ince, of order, m. The =. (. However, having both real and imaginary part, they carry a many independent component. The Pareval Theorem for the pherical function i eaily derived (Exercie.6 uing the orthogonality of the pherical harmonic function, equation (.79, 4π n = n= m= n g ( θ, λ h( θ, λ dω Gn, mh n, m, (. Ω where all quantitie are real-valued; for the complex Fourier-Legendre pectrum, 4π n = n= m= n * c ( ( ( * c g θ, λ h θ, λ dω Gn, m H n, m. (. Ω The total energy of a (generally complex function on the phere i then given by 4π n n n= m= n n= m= n c g ( θ, λ dω = Gn, m = Gn, m. (.3 Ω Fourier Geodey.45 Jeeli, January 7

46 An important property for pherical harmonic function i the addition theorem, n Pn ( co ψ = Yn, m ( θ, λ Yn, m ( θ, λ n +, (.4 m= n which alo hold for the complex pherical harmonic, equation (.83, n c * c Pn ( co ψ = ( Yn, m ( θ, λ Yn, m ( θ, λ n +. (.5 m= n If θ = θ and λ = λ, then from equation (.57, ψ = and, with equation (.64, n n + = P coθ. (.6 m= ( ( n, m The Fourier-Legendre pectrum of the derivative of a function on the phere with repect to co-latitude cannot be derived from the pectrum of the function. Mathematically the derivative of the aociated Legendre function either i not a linear combination of aociated Legendre function of the ame order, ( n + ( n + m( n m in θ d P n, m ( co n co P n, m ( co P n, m ( co d θ = θ θ θ n θ ; (.7 or, it i a linear combination of aociated Legendre function of different order, d P n m n m P dθ + + ( coθ = ε ( + ( + ( coθ n, m m n, m ( n m ( n m P ( coθ n, m+ (.8 In either cae, the partial derivative, Yn, m ( θ, λ θ, i not a linear combination of the pherical harmonic, Y ( θ λ ; conequently, the Fourier-Legendre pectra of g ( θ, λ θ and g ( θ, λ p, q, are not related analytically. Even the derivative with repect to longitude i uually caled by inθ, which mae the differentiation property (.96 of the tranform unavailing in many application. Fourier Geodey.46 Jeeli, January 7

47 .6.4 Cap Function The rectangle function on the phere i defined by practical neceity in term of jut one variable, the angular ditance ( ditance in radian along a great circle arc on the unit phere Ω = ψ, ζ ψ ψ, ζ π, a from the center of a pherical cap, ( { } b ( ψ ( ψ 4 π, ψ < ψ A = π, ψ = ψ A, ψ > ψ (.9 where ψ > i the radiu of the cap that ha area (ee Figure., π ψ ψ σ π inψ ψ π ( coψ. (. A = d = d = Becaue ( ( b ψ ψ depend only on an angular ditance, it Legendre tranform i of the onedimenional ind, equation (.66. Subtituting a formula that relate the Legendre polynomial to it derivative, d n + Pn y = ( Pn + y Pn y, n >, (. dy ( ( ( ( the Legendre pectrum of ( b ψ i, ( ψ 4π Bn = Pn ( coψ inψ dψ A ψ = ( Pn ( coψ c Pn + ( co ψ, n coψ n + (. with ( B ψ =, ( = ( + co. (.3 B ψ ψ ( ( The Legendre erie of b ψ ψ i, from equation (.67, Fourier Geodey.47 Jeeli, January 7

48 ( ψ ( coψ, (.4 ( ψ ( ψ = ( + b n B P n n n= which converge to the defined value of particular definition of ( ( ( ( b ψ ψ at the edge of the cap. It i noted that the b ψ ψ enure that the zero-frequency value of it pectrum i unity, in agreement with the preceding definition of the rectangle and cylinder function. It alo mean that the average value of 4π ( b ψ over the phere i unity, ( ψ b ( ψ dω =. (.5 Ω Recurion formula for ( B ψ n may be derived from correponding recurion formula for the Legendre polynomial, P n. For example, ubtituting the formula (.63 into the integral of equation (., and uing equation (., one obtain ( ψ n ( ψ n ( ψ Bn = Bn co ψ Bn, n, (.6 n + n + with tarting value given by (.3. The pectrum of the cap function for ψ = i hown in Figure.5. The firt zero-croing occur at n = 9, which agree with the firt zero of the pectrum for the cylinder function, equation (.54, π f µ = 3.837, auming that the Carteian frequency i related to harmonic degree by f n ( π R cylinder radiu i = ( R. For the Earth, 637 m about of arc. In fact, the pectra, ψ. µ B ( π Rψ c ( f and = (Section.7 and the R =, thi i = µ.5 m, or ( B ψ n, are aymptotically cloe for mall Fourier Geodey.48 Jeeli, January 7

49 Figure.5: Fourier-Legendre pectrum of ( ( b ψ ψ for ψ =..6.5 Gauian Function on the Sphere The pherical analogue of the Gauian function i ( β β ( co ( e β γ ψ = ψ, ψ π, (.7 β e where β >. Again, it i defined o a to enure unity for it zero-frequency pectral value. Alo, being a function that depend only on the central angle, ψ, it Legendre tranform i given by equation (.66 that become with y = coψ, Γ ( β n β βe β y = e P n ( y dy. (.8 β e Subtituting the recurion formula, equation (., and integrating by part, one obtain ( β n ( β ( β Γ = Γ + Γ, n, (.9 n n n β with tarting value Fourier Geodey.49 Jeeli, January 7

50 ( β Γ =, Γ + β ( β e = β e β. (. Since the Gauian function i continuou for all ψ, it Legendre erie converge to γ everywhere. Alo, ince Γ unity, 4π ( β ( β ( ψ =, the average value of the Gauian function over the phere i ( β γ ( ψ dω =. (. Ω A in the Carteian cae, the parameter, β, define the width or ignificant extent of the = e, then Gauian curve. If ignificant extent i defined by the angle, ψ, for which γ ( ψ γ ( ψ = ( β. Thu, for the ame extent a the cap function, one ha β ( coψ co =. Other way of defining ignificant extent are elaborated in Section Figure.6 how the β = coψ and ψ =. The pherical Gauian function and it Legendre tranform for ( parameter, β, defined for the pherical Gauian i not the ame a iotropic Gauian function (ee the tranform pair (.55, γ ( p ( p β defined for the planar ( p = e. (. β β π β p With a planar approximation of a phere of radiu, R, that i, for mall ψ R and large β, ( β ( one ha γ ψ γ ( β p (, if π R β =. (.3 β p where the ubcript on the pherical Gauian function and the ditance argument,, for the planar Gauian function have entirely different meaning. Fourier Geodey.5 Jeeli, January 7

51 Figure.6: The pherical Gauian function (left and it Fourier-Legendre pectrum (right for β = coψ and ψ =. (.6.6 Spherical Dirac Delta Function The Dirac delta function for the phere, again, by neceity i defined only a a function of ψ and on the bai of the exitence of it Legendre tranform. Conider the Gauian function on the phere and it limit a the parameter, β, approache infinity, while the ignificant width, ( β ψ = co, approache zero. The volume of the Gauian function, equation (., remain contant at 4π. From equation (.7, l Hôpital rule for limit of indeterminate expreion give β lim γ ψ = lim =, < ψ π. (.4 β ( β ( β β β ( coψ ( e e At ψ =, the limit i infinite; hence, we may define the pherical Dirac delta function on the phere by ( ( β ( δ ψ = lim γ ψ, ψ π. (.5 β The exitence of the Legendre pectrum of δ ( ψ then follow from equation (.9. That i, ( β ( β lim Γ = lim,, (.6 β n Γ n n β Fourier Geodey.5 Jeeli, January 7

52 where, from equation (., i ( β lim Γ = and β ( β lim Γ β =. Thu, the pectrum of δ ( ψ π δ ( ψ ( coψ inψ ψ, for all, (.7 Dn = Pn d = n and formally one may write n. (.8 n= δ ψ ψ ( = ( n + P ( co Equation (.5 and (. confirm that ( β ( ( β δ ψ dω = lim γ ( ψ dω = lim Γ =. (.9 β β 4π 4π Ω Furthermore, the correponding reproducing property i 4π Ω δ ( ψ g ( θ, λ dω = g ( θ, λ, (.3 Ω where the relationhip between ( θ, λ and ( θ, λ i given by equation (.57 and ( From Sphere to Plane When analyzing a geophyical ignal pectrally on the pherically approximated Earth, it i often numerically legitimate to further approximate the urface locally by a plane. If both global and local pectra of a ignal have been determined with ignificant overlap in their correponding pectral band, one may wih to combine or compare them. In any of thee ituation, it i important to tranlate properly between frequency in the planar pectral domain and wave number in the pherical pectral domain. There are a number of approache to find uch a relationhip, baed eentially on the aymptotic relationhip between Legendre polynomial and Beel function, x lim Pn co = J ( x, for x >, (.3 n n which i the ame a Fourier Geodey.5 Jeeli, January 7

53 x lim P co n = J ( x, for x >. (.3 n n( n + Equation (.6 with y = coψ and y = inψ ψ become inψ Pn ( coψ = n( n + Pn ( coψ inψ ψ ψ. (.33 On the phere of radiu, R, define a ditance, approximately, ( + R, by the angle, ψ = R. Then, n n Pn co P co n = R. (.34 R R Now, uing the propertie of the derivative of Beel function, dj ( x dx J ( x d xj ( x dx = xj x, it i eaily verified that ( ( = and J ( π f = ( π f J ( π f, (.35 where the introduced frequency, f, correpond to the ditance,, a in equation (.37 and (.4. Finally, let x = π f and approximate, from equation (.3, π f P n co J ( π f. (.36 n( n + Then, a comparion of equation (.34 and (.35 how that f ( + n n. (.37 π R Thi approximation i valid in principle only for the high degree or high frequencie. However, it i reaonable even for the lower degree where it i alo lightly better than n f. (.38 π R Fourier Geodey.53 Jeeli, January 7

54 A note of caution i required here. The high-degree Fourier-Legendre pectrum of the function on the phere repreent it global variation at hort wavelength, wherea the determination of the high-frequency Fourier pectrum of that function over jut a local area reflect i local hort-wavelength character. Thu, while the pectral domain may be approximated by equation (.37 at high frequencie, the pectra of a global function and of it retriction to a local area may be quite different. For example, the high-degree component of the Earth entire topography, repreented in pherical harmonic, are not the ame a the highfrequency component of the topography retricted to the Colorado Rocy Mountain, nor to the topography of the northern Midwet of the U.S. (Figure.7. However, in nominal cae, the global and local pectra eem to match reaonably well. Figure.7: Comparion of global and local topographic pectra a azimuthally averaged amplitude value. The global pectrum i derived from a pherical harmonic erie model, DTM6 (Pavli et al.. The alpine pectrum repreent terrain data of the Colorado Rocy Mountain. The lowland topography pectrum i typical of the gently rolling terrain of the northern Midwet U.S..8 Example of Fourier Tranform Pair In thi ection, a few commonly ued one- and two-dimenional Fourier tranform pair are lited. Mot can be obtained by conulting a good Table of Integral. Some of thee were encountered already in previou ection, and other are eaily derived from thee uing the propertie of tranform. The two-dimenional Fourier tranform pair are alo Hanel tranform pair in thee example, where the following abbreviation are noted, = x + x, f = f + f. (.39 Fourier Geodey.54 Jeeli, January 7