Ricker, Ormsby; Klander, Bntterwo -A Choice of avelets

Sengbush, R.L. 1983. Analog and digi­tal filters. In: Seismic Exploration Methods.IHRDC, Boston, p. 244-251.

Bibliography

Curtis, J.W. 1975. A multi-trace syn­thetic seismogram generator. Bulletin of theAustralian Society of ExplorationGeophysicists, v.6(4), p. 91-99.

Geyer, R.L. 1969. The vibroseis sys­tem of seismic mapping. Journal of theCanadian Society of ExplorationGeophysicists, v.6(l), p. 39-57.

Hosken, J.W. 1988. Ricker wavelets intheir various guises. First Break, v.6(l), p.24-33.

filter with maximal flatness in the passbandso that applying a Butterworth filter to aunit impulse function will generate awavelet such as in figure 7. Four parame­ters are needed to specify a Butterworth fil­ter, beginning with "fl" and "h", the lowerand upper frequencies of the bandpass.Then the lower and upper cutoff rates, thatis the slope of the amplitude response out­side the bandpass, are needed. They areexpressed as either "n", the order of theButterworth filter (a positive integer) or interms of "decibels/octave" which is "6n"(i.e. the cutoff rate can be described aseither "third order" or as "18decibels/octave"). As the cutoff rate of aButterworth filter increases, the waveletitself becomes more leggy. In an article byCurtis(l975) there is an appendix entitled"Generation of a Wavelet from ButterworthFilter Spectral Characteristics" which pro­vides a thorough mathematical treatment ofButterworth filters. MATLAB has theadvantage of containing a function forButterworth filter design in its signal pro­cessing toolbox so that Butterworthwavelets and their frequency spectrum (fig.8) can be generated quite easily.

Every time a synthetic seismogram isgenerated, the geophysicist must firstchoose the type of wavelet that will be usedand then the specific parameters needed todefine the wavelet. Making these choicesbecomes much easier with a thoroughunderstanding of the differences and simi­larities between Ricker, Ormsby, Klauderand Butterworth wavelets and how varyingtheir parameters changes the shape of thewavelet and its frequency spectrum.

Klauder(t) =real [sin(1tkt(T-t))/(1tkt) exp (21tifot)]

(1th)2 sinc2(1tf2t) - (1tfI? sinc2(1tf1t)(1tf2-1tfl) (1th-1tfl)

where k = (f2 - fl)/T (rate of change offrequency with time)

fo = (f2 + fl)/2 (midfrequency ofbandwidth)

i = ~1 (an imaginary number)

The frequency spectrum of a Klauderwavelet (fig 6) clearly shows the substantialsimilarities between a Klauder and anOrmsby wavelet.

The most distinctive characteristics ofButterworth wavelets is that they are mini­mum-phase and physically realisable. AButterworth wavelet will start at time zerowhile Ricker, Ormsby and Klauderwavelets all have their peaks at time zero.Butterworth defined a minimum-phase

Like Ricker and Ormsby wavelets, aKlauder wavelet (fig 5) is symmetricalabout a vertical line through its central peakat time zero. A Klauder wavelet representsthe autocorrelation of a linearly swept fre­quency-modulated sinusoidal signal used inVibroseis. It is defined by its terminal lowfrequency, "fl"; its terminal high frequency,"fl"; and the duration of the input signal"T", often 6 or 7 seconds. The real part ofthe following formula will generate aKlauder wavelet.

Ormsby(t) =

(1tf4? sinc2(1tf4t) - (1tf3? sinc2(1tf3t)(1tf4-1tf3) (1tf4-1tf3)

tion results in the creation of the Ormsbywavelet shown in figure 3. An Ormsbywavelet will have numerous side lobesunlike the simpler Ricker wavelet whichwill always only have two side lobes. TheOrmsby wavelet will also become moreleggy the steeper the slope of the sides ofthe trapezoidal filter. Four frequencies areneeded to specify the shape of an Ormsbyfilter and which are also used to identify anOrmsby wavelet (i.e. the 5-10-40-45 HzOrmsby wavelet of fig. 3). These frequen­cies are "fl", the low-cut frequency; "f2",the low-pass frequency; "f3", the high-passfrequency and" f4 ", the high-cut frequencywhich are all used in the following formulato generate an Ormsby wavelet.

Because of the simple inverse relation­ship between the peak frequency andbreadth of a Ricker wavelet, the sameRicker wavelet could be just as uniquelydescribed as a "31 ms Ricker wavelet" or asa "25 Hz Ricker wavelet".

In some texts you will see the Rickerwavelet's breadth, that is the time intervalbetween the centre of each of the two sidelobes, quoted as the reciprocal of the Rickerwavelet's peak frequency. This is only anapproximation and indeed will give awavelet breadth 28% wider than the truewavelet breadth. The correct formula forthe breadth of a Ricker wavelet is:

Ricker wavelet breadth = 'i{/1t/f= 0.7797/f

Ormsby wavelets are also zero-phasewavelets although Ormsby, an aerospaceengineer, actually defined a filter so suchwavelets should really be called Ormsby­filtered wavelets. The trapezoidal shape ofan Ormsby filter can be seen in the fre­quency spectrum shown in figure 4.Applying this filter to a unit impulse func-

Ricker wavelets (fig 1) are zero-phase ­wavelets with a central peak and twosmaller side lobes. A Ricker wavelet can beuniquely specified with only a single para­meter,"f", it's peak frequency as seen onthe wavelet's frequency spectrum (fig 2).The mathematical formula for a Rickerwavelet is given by:

Commercially available software usedto generate synthetic seismograms gives thegeophysicist a choice of up to four types ofstandardized wavelets as well as the optionof a user-defined wavelet. This review arti­cle will summarize the characteristics ofthese wavelets, illustrate the shape of thewavelets and their frequency spectrum andpresent the parameters needed to definethem as well as their mathematical formu­las.(In· all the following formulas, "t" istime in seconds, "f" is frequency in hertzand "1t" is the irrational number pi).Illustrations of the wavelets and their fre­quency spectrum were generated using thescientific numeric computation softwareMATLAB and the wavelets were chosen tohave as similar a bandwidth as possible forease of comparison.

by Harold Ryan, Hi-Res Geoconsulting

8 CSEG Recorder September, 1994

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September, 1994 CSEG Recorder 9