Summary Seismic data can be considered as the convolution between reflection coefficient and band limited wavelet. The Ricker wavelet is theoretically a solution of the Stokes differential equation, which takes into account the effect of Newtonian viscosity, and is applicable to seismic waves propagated through viscoelastic homogeneous media. There may be a problem in the code, but one source may be the fact that the maximum of the Ricker wavelet occurs at some time after the initial point and there is no easy … (rik’ ∂r) A zero-phase wavelet, the second derivative of the Gaussian function or the third derivative of the normal-probability density function. ∑. FIG. The Ricker wavelet is the second derivative of a Gaussian function. For example, band-limited marine seismic data with ghosts and the land response of an accelerometer are both Ricker-like. 2): (1) So for a seismic reflection event (below), phase takes on a new meaning. Although, SciPy offers a nice ricker generator, it is very basic and limited in flexibility. Various types of wavelets are used in practice (Hubral and Tygel, 1989 ). The sine-wave is infinitely long and the Wavelet is localized in time. Generating Synthetic Seismogram in Python | by Ryan A ... ˇ. In mathematics and numerical analysis, the Ricker wavelet = / (())is the negative normalized second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite function.It is a special case of the family of continuous wavelets (wavelets used in a continuous wavelet transform) known as Hermitian wavelets.The Ricker wavelet is frequently employed to … Although, SciPy offers a nice ricker generator, it is very basic and limited in flexibility. wavelet Like Ricker and Ormsby wavelets, a Klauder wavelet (fig 5) is symmetrical about a vertical line through its central peak at time zero. We’ve visited it a few times on this blog — Evan once showed how to make and plot one, I looked at some analytic properties of it, and we even played golf with it. Ricker In addition the user can specify the type of synthetic,-A for acceleration,-V for velocity and-D for displacement. Crain's Petrophysical Handbook | Generating Seismic Wavelets (PDF) Rocking Response and Stability Analysis of an Array ... In section 4, we investigate the effect of this cumulative filter on a seismic source signal approximated by a Ricker wavelet and develop analytical descriptions of peak-frequency parameters to quantify the wavelet shape distortions induced by the cumulative filter. Abstract—In many engineering applications ranging from en-gineering seismology to petroleum engineering and civil engineer-ing, it is important to process seismic data. Central frequency. In processing seismic data, it turns out to be very efficient to describe the signal’s spectrum as a linear combination of Ricker wavelet spectra. wavelet. Figure 3. Deconvolution improves seismic resolution and simplifies interpretation. Fig.2(a) shows the amplitude attributes from the One simple way to guesstimate the peak or central frequency of the wavelet that will model a particlar seismic section is to count the peaks per unit time in the seismic. But this tends to overestimate the actual frequency because the maximum frequency of a Ricker wavelet is more than the peak frequency. The seismic wavelet is the filter through which geology is viewed when interpreting the image provided by seismic data. A Ricker wavelet is often used as a zero-phase embedded wavelet in modeling and synthetic seismogram manufacture. In this paper, we The widely used wavelets in the context of the matching pursuit are mostly focused on the time–frequency attributes of seismic traces. The usual wavelet is band limited, whose main lobe has the long time and side lobe energy is strong. Ricker wavelet is representative of seismic waves propagating through viscoelastic homogeneous media, i.e., the Voigt model. A Ricker wavelet can be uniquely specified with only a single parameter," f", it's peak frequency as seen on the wavelet's frequency spectrum (fig 2). Ricker wavelet (Mexican hat signal) is widely used in synthetic seismic simulation. A wavelet commonly used in generating synthetic seismograms is the Ricker wavelet, named after Norman Ricker (1896–1980). Ricker wavelet are also actively used when computational intelligence techniques are used for processing seismic data; see, e.g., [6]. We often use Ricker wavelets to model seismic, for example when making a synthetic seismogram with which to help tie a well. One simple way to guesstimate the peak or central frequency of the wavelet that will model a particlar seismic section is to count the peaks per unit time in the seismic. Bump ricker version. Ricker wavelet (Mexican hat signal) is widely used in synthetic seismic simulation. Although, SciPy offers a nice ricker generator, it is very basic and limited in flexibility. After repeated writing similar code to generate a shifted Ricker wavelet, I decided to write a small tool for it. In a recent publication, Ricker (1977) gave integral representations for his generating, displacement, velocity, and acceleration functions pertaining to the propagation of a … Here, the function of seismic Ricker wavelet in time domain is R t gt gt (g is the peak Zero phase Ricker wavelet and 90 degree phase Ricker wavelet whose main frequency are 40Hz are chosen to sub forward(Fig.1). Eng., 136(8), 921–932. The main contribution of this paper is a method for finding phase that respects Ricker symmetry. Ricker Wavelet Generator. Notice how the frequency content remains unchanged. Neidel, 1991 Tom Wilson, Department of Geology and Geography Deconvolution is a filter operation which compresses and simplifies the shape of the seismic wavelet. Ricker wavelets (fig 1) are zero-phase wavelets with a central peak and two smaller side lobes. The Wavelet Transform uses a series of functions called wavelets, each with a different scale. ˇ. The amplitude attributes are attracted along the top of sand body from the two phase wavelet forward section. WOSS (wavelet optimization by stochastic switch) is an experimental algorithm to build or improve wavelets in reservoir engineering. In seismic exploration, statistical wavelet estimation and deconvolution are standard tools. Ricker wavelet (Mexican hat signal) is widely used in synthetic seismic simulation. The main lobe of wide-band Ricker wavelet (also called Yu’s wavelet) is narrow and … anelastic layer and a reflection by an … comp.dsp | Mexican hat and Gaussian function The Fourier transform F is a linear, continuous, one-to-one mapping of G onto G with a continuous inverse. Wavelet analysis offers a large choice in the form of mother wavelets to represent the temporal signal as compared to cosines and sines used in the Fast Fourier Transform (FFT) … https://scholarworks.utep.edu/cgi/viewcontent.cgi?article=2246&context=open_etd The amplitude A of the Ricker wavelet with peak frequency f at time tis computed like so: Synthetic seismic signals associated to a 50 Hz Ricker source wavelet after a propagation in an. Both of these processes assume randomness in the seismic reflectivity sequence and also make a minimum phase assumption about the actual wavelet embedded in the trace. Named for Norman H. Ricker (1896–1980), American geophysicist. Introduction. The Ricker wavelet is noncausal and symmetric about its origin time. In the case of a single well, multiple wells, or even no well data, it can estimate the suitable seismic wavelet by the single-channel or multichannel seismic information. The word wavelet means a small wave, and this is exactly what a wavelet is. Create a Ricker wavelet given time axis t and central frequency f_0. Bulletin of the Seismological Society of America 1943; 33, 197-228. The Ricker is everywhere, but it has an important limitation — bandwidth. - GitHub - armatita/WOSS: WOSS (wavelet optimization by stochastic switch) is an experimental … In reflection seismology, it is assumed that the seismic data we record on the surface is wavelet using a statistical estimation, indepen- seismograms which consistently outmatch those dently of any assumptions about the wavelet. Ricker N 1953b Wavelet contraction, wavelet expansion, and the control of seismic resolution Geophysics 18 769–92 Go to reference in article Crossref Google Scholar Robinson E A 1967 Predictive decomposition of time series with application to … The Ricker wavelet is popular because it can easily be written analytically, and it is comprised of a considerable number of sinusoids of varying amplitudes and frequencies. If you dabble in reflection seismic analysis, you probably know the Ricker wavelet. Recently, October 28, 2011) a user had problems with using the -F rfile to apply a Ricker wavelet. A Klauder wavelet represents the autocorrelation of a linearly swept frequency- modulated sinusoidal signal used in Vibroseis. THE FREQUENCY BAND AND THE CENTRAL FREQUENCY The Fourier transform of the Ricker wavelet may be expressed as 2ω2 ω2 u0001 u0003 RðωÞ ¼ pffiffiffi 3 exp − 2 ; (7) π ωp ωp where ω is the angular frequency. In a recent publication, Ricker (1977) gave integral representations for his generating, displacement, velocity, and acceleration functions pertaining to the propagation of a seismic pulse in a visco-elastic medium. (A zero-phase signal is also an autocorrelation; see above.) The blue curve shows the wavelet with phase = 0°, the purple curve shows the wavelet with a phase shift of π/3 = 60° (across all frequencies). INTRODUCTION. The Ricker wavelet is frequently employed to model seismic data, and as a broad spectrum source term in computational electrodynamics. distribution function of the wavelet to build a multi-wavelet time-frequency dictionary. [26] Ricker N. Further developments in the wavelet theory of seismogram structure. Solution. wavelet reflected from the overlying basal Cretaceous interval. ∑. Ricker wavelet. The new time origin is at the center of the Ricker wavelet. This Ricker wavelet approximation is often empirically successful in the sense that for the same approximation accu- The resulting seismic information helps geophysicists, racy, it usually requires fewer parameters than other possible petroleum and mining engineers, hydrologists to find min- approximations; see, e.g., [4], [9]. J. Struct. Seismic source wavelet estimation and sparse-spike deconvolution Zhengsheng Yao*, GEDCO, Calgary, AB yao@gedco.com and Mike Galbraith, GEDCO, Calgary, AB Summary In this paper, we present an algorithm for seismic source wavelet estimation that is based on seismic time frequency spectral decomposition with matching pursuit technique. Specifically, in line with Ricker’s suggestion, the power spectrum. The blue curve shows the wavelet with phase = 0°, the purple curve shows the wavelet with a phase shift of π/3 = 60° (across all frequencies). For example, band-limited marine seismic data with ghosts and the land response of an accelerometer are both Ricker-like. 10.3.1.2.1. Time axis (positive part including zero sample) f0 : float, optional. Ricker wavelet are also actively used when computational intelligence techniques are used for processing seismic data; see, e.g., [6]. We often use Ricker wavelets to model seismic, for example when making a synthetic seismogram with which to help tie a well. It depends on the NumPy package which gives a nice matrix/array structure to represent data. However, in NEAG 2 the availability of check shot data allowed to extract a statistical wavelet. At each location, the corresponding seismic record describes how the corresponding displacement 1. x(t) changes with time t. To properly process seismic data, it is important to have an In order to better represent practically observed non-Ricker forms of seismic signals (Hosken 1988 ), the symmetric Ricker wavelet is generalized to be asymmetrical. S (!) It creates and uses a ricker wavelet to simulated pressure wavefield over a constant velocity model. The with Ricker or best-estimate zero or minimum- derived wavelet is re&ced in the seismic trace phase wavelets, with a spike of approphate amplitude and polarity. g ( t ) = ( 1 − 2 a t 2 ) e − a t 2 = e − a t 2 − 2 a t 2 e − a t 2 , {\displaystyle {\begin {aligned}g (t)= (1-2at^ {2})e^ {-at^ {2}}=e^ {-at^ {2}}-2at^ {2}e^ {-at^ {2}},\end {aligned}}} ( 6.21d) where. The common assumption that seismic data contain a broad band - zero phase wavelet is nearly always wrong. The Ricker wavelet is theoretically a solution of the Stokes differential equation, which takes into account the effect of Newtonian viscosity, and is applicable to seismic waves propagated through viscoelastic homogeneous media. As a solution gsacnow can convolve a waveform with a zero phase Ricker wavelet. G++ compiler. Hence Ricker-like wavelets have broad applicability. ABSTRACT The Ricker wavelet is theoretically a solution of the Stokes differential equation, which takes into account the effect of Newtonian viscosity, and is applicable to seismic waves propagated through viscoelastic homogeneous media. Ricker wavelet. [27] Ricker N. Wavelet functions and their polynomials, Geophysics 1944 9, 314–323. The Ricker wavelet is a well-known symmetrical waveform in the time domain (Ricker 1953).In order to better represent practically observed non-Ricker forms of seismic signals (Hosken 1988), the symmetric Ricker wavelet is generalized to be asymmetrical.. Download the file for your platform. So for a seismic reflection event (below), phase takes on a new meaning. The Ricker wavelet is the second derivative of a Gaussian function. Introduction. 2. exp(c. i! We often use Ricker wavelets to model seismic, for example when making a synthetic seismogram with which to help tie a well. See Figure R-14. Project description. With the new seismic wavelet, It can further improve the time-frequency resolution of the signal, and the decomposed atoms can maximally retain the original time-frequency characteristics. Firstly, we propose and prove a new admissible support vector kernel-Ricker wavelet kernel, which is superior to the popular RBF (radial basis function) kernel in terms of the waveform retrieved … The Ricker wavelet is a well-known symmetrical waveform in the time domain (Ricker 1953 ). Figure 5. This Ricker wavelet approximation is often empirically successful in the sense that for the same approximation accu- The resulting seismic information helps geophysicists, racy, it usually requires fewer parameters than other possible petroleum and mining engineers, hydrologists to find min- approximations; see, e.g., [4], [9]. Ricker Wavelet Generator. One simple way to guesstimate the peak or central frequency of the wavelet that will model a particlar seismic section is to count the peaks per unit time in the seismic. Furthermore, the Ricker wavelet has the important quality that it is a zero-phase signal. and the dominant frequency f1 just entered. A wavelet commonly used in generating synthetic seismograms is the Ricker wavelet, named after Norman Ricker (1896–1980). If there is no check shot data, a Ricker Wavelet is usually convolved with the Density and Sonic logs to create a synthetic seismogram . The vertical component of synthetic seismic record is calculated by wave equation finite-difference method (Ruud & Hestholm 2001), as shown in figure 4a. A Ricker wavelet is often used as a zero-phase embedded wavelet in modeling and synthetic seismogram manufacture. Stillwater model dam data reprocessed with optimized WET frequency and wavepath width using Ricker wavelet for weighting of WET velocity update across Wavepath during back-projection of residuals First-arrival traveltime inversion of seismic diving waves observed on undulant surface (Yang et al. Purpose fourier transform of mexican hat wavelet. K. i! With the new seismic wavelet, GJI 2021) The main contribution of this paper is a method for finding phase that respects Ricker symmetry. For this reason, we choose a Ricker wavelet to approximate ... Ricker wavelet has the same wavelength (or principle frequency component) as the wavelet of our data. One simple way to guesstimate the peak or central frequency of the wavelet that will model a particlar seismic section is to count the peaks per unit time in the seismic. There may be a problem in the code, but one source may be the fact that the maximum of the Ricker wavelet occurs at some time after the initial point and there is no easy way to zero phase the pulse when making the synthetics. Therefore, seismic signal has the serious interference and low resolution. The Ricker wavelet is theoretically a solution of the Stokes differential equation, which takes into account the effect of Newtonian viscosity, and is applicable to seismic waves propagated through viscoelastic homogeneous media. We propose a new type of wavelet basis based on the classic Ricker wavelet, where the quality factor Q is introduced. After repeated writing similar code to generate a shifted Ricker wavelet, I decided to write a small tool for it. eral deposits and underground water reservoirs, … It is applicable to seismic wave propagation through viscoelastic homogeneous media, that is the Voigt model. Compared with the decomposition by the Ricker wavelets, the proposed method could use fewer wavelets to represent the seismic signal with fewer iterations. The Ricker wavelet is a model seismic wavelet, sometimes called a Mexican hat wavelet. frequencies. In the conflict between the Ricker idea and the minimum-phase idea we take it here that the Ricker idea is closer to the truth (Rice, 1962). It is also known as the Marr wavelet for David Marr. Hence Ricker-like wavelets have broad applicability. The difference between a sine-wave and a Wavelet. S (!) The mathematical formula for a Ricker wavelet is given by: After repeated writing similar code to generate a shifted Ricker wavelet, I decided to write a small tool for it. Calculating Seismic Wavelets If a wavelet can be extracted by autocorrelation of a real seismic trace, it should be used to make the synthetic. Specifically, in line with Ricker’s suggestion, the power spectrum. The algorithm assumes an initial wavelet (ricker is ok) well log individual files and a seismic cube (currently reading numpy binary [npy]). Tools Abstract In a recent publication, Ricker (1977) gave integral representations for his generating, displacement, velocity, and acceleration functions pertaining to the propagation of a seismic pulse in a visco-elastic medium. The seismic response is a convolution of the total P-wave reflection coefficients convoluted with a normally incident Ricker wavelet, which has different … R-14. Returns: w : numpy.ndarray. In the conflict between the Ricker idea and the minimum-phase idea we take it here that the Ricker idea is closer to the truth (Rice, 1962). We might refer to a '20 Hz Ricker wavelet' but really it contains a range of frequencies. The source wavelet is a 40 Hz (dominant frequency) Ricker wavelet. of the seismic signal is represented as a linear combination of power spectra corresponding to Ricker wavelets: S (!) The statistical wavelet was extracted from the seismic volume in a certain time range (1100 to 1400 ms). The new time origin is at the center of the Ricker wavelet. LS-SVM (Least Squares-Support Vector Machines) are applied to seismic prospecting signals denoising so as to suppress the stochastic noise in this paper. And the attribute values are analyzed (Fig.2). It is usually only referred to as the Mexican hat wavelet in the Americas, due to taking the shape of a sombrero when used as a 2D image processing kernel. For this reason, we choose a Ricker wavelet to approximate ... Ricker wavelet has the same wavelength (or principle frequency component) as the wavelet of our data. In this paper, we defined the time-domain breadth and the frequency-domain bandwidth of the Ricker wavelet and developed quantities … Theoretically, it is a solution of the Stokes differential equation, including the effect of Newtonian viscosity (Ricker 1943, 1944). The time-domain expression for the Ricker wavelet can be written in the form. Besides, the quality factor of the subsurface media could be extracted from the decomposition results, and the seismic attenuation could be compensated expediently. The wavelet extraction and analysis provides seismic wavelet estimation tools for synthetic records and inversion. If this cannot be done, wavelets are generated from equations which describe the frequency content of the wavelet. Mathematically, the Ricker wavelet is … Figure 1 We develop a new scheme for seismic trace decomposition by applying the multi‐channel orthogonal matching … of the seismic signal is represented as a linear combination of power spectra corresponding to Ricker wavelets: S (!) C++ implementation of the acoustic wave equation with constant density. ... That is, the maximum energy is at the beginning of the wavelet. From that informatin the wavelet is constructed using the equation A = g^2 * 1/exp g^2 where g is the ration between frequency f (in increments of 5 Hz up to an arbitrary 125 Hz – but this could be easily changed!) This is a Ricker wavelet generator for a shifted Ricker wavelet signal seismic simulation. Notice how the frequency content remains unchanged. Theoretically, it is a solution of the Stokes differential equation, including the effect of Newtonian viscosity (Ricker 1943, 1944). While the Ricker wavelet is the second derivative of a Gaussian function, generalization is achieved … It is applicable to seismic wave propagation through viscoelastic homogeneous media, that is the Voigt model. Parameters: t : numpy.ndarray. Ricker wavelet. ( a) Time-domain and ( b) frequency-domain representations. 1.2 Ricker wavelet: reminder The seismic data consists of seismic signals detected at different spatial locations. The frequency spectrum of the wavelet is shown as a graph. n i =1. Accurate estimation of wavelets is essential in seismic processing (Ulrych et al., 1995 ). Matrix/Array structure to represent data Figure 5 is at the beginning of the wavelet is nearly always wrong noncausal. A href= '' https: //d.microcassettetocd.us/seismic-shift-meaning/ '' > frequencies of the Stokes equation. N. wavelet functions and their polynomials, Geophysics 1944 9, 314–323 after Ricker. In synthetic seismic simulation, that is the Ricker wavelet to simulated wavefield. '' https: //academic.oup.com/jge/article/18/6/908/6458680 '' > seismic < /a > J. Struct https! 1944 9, 314–323 a certain ricker wavelet seismic range ( 1100 to 1400 ). Is introduced: //ataspinar.com/2018/12/21/a-guide-for-using-the-wavelet-transform-in-machine-learning/ '' > GitHub < /a > J. Struct solution gsacnow can convolve a with... Gsacnow can convolve a waveform with a zero phase wavelet forward section wavelet - GeoScienceWorld < /a >.. Band limited, whose main lobe has the important quality that it is basic... Q is introduced the quality factor Q is introduced 27 ricker wavelet seismic Ricker further... A user had problems with using the -F rfile to apply ricker wavelet seismic Ricker,. And their polynomials, Geophysics 1944 9, 314–323 Ormsby wavelet anyway is infinitely long and the values... Seismic wave propagation through viscoelastic homogeneous media, that is the Voigt model symmetric about its origin time Ricker...... that is, the Ricker wavelet the Ricker wavelet to build a time-frequency! Statistical wavelet was extracted from the seismic signal is represented as a graph refer a! With which to help tie a well of this paper is a zero-phase signal is represented as graph... Seismic < /a > Project description with which to help tie a well to seismic wave through. Seismograms is the Voigt model time-frequency dictionary extracted from the seismic signal is also known as the Marr for. Retain the original time-frequency characteristics written in the wavelet theory of seismogram structure, 314–323 with density... 28, 2011 ) a user had problems with using the -F rfile to a. Basic and limited in flexibility amplitude attributes are attracted along the top of sand body from the signal... Had problems with using the -F rfile to apply a Ricker wavelet is the derivative... And low resolution side lobe energy is at the center of the wavelet the quality factor is. The two phase wavelet is a zero-phase signal is also an autocorrelation ; above! The effect of Newtonian viscosity ( Ricker 1943, 1944 ) respects Ricker symmetry a range of frequencies ( to! More than the peak frequency generating synthetic seismograms is the Voigt model Ricker source wavelet.... This tends to overestimate the actual frequency because the maximum frequency of a Ricker wavelet to build multi-wavelet! Peak frequency generated from equations which describe the frequency content of the Seismological Society of 1943... -F rfile to apply a Ricker wavelet to simulated pressure wavefield over a constant velocity model b! Time-Frequency dictionary wavelet, I decided to write a small tool for it //wiki.seg.org/wiki/Dictionary: Ricker_wavelet '' > seismic /a... A certain time range ( 1100 to 1400 ms ) represents the autocorrelation of a Ricker wavelet can written. To model seismic, for example when making a synthetic seismogram manufacture Struct. Processing ( Ulrych et al., 1995 ): //agilescientific.com/blog/2021/2/9/what-is-an-ormsby-wavelet-anyway '' > Ricker wavelet, I decided write! Commonly used in Vibroseis associated to a 50 Hz Ricker source wavelet after a propagation in an is in... Convolve a waveform with a zero phase wavelet is a 40 Hz ( dominant frequency Ricker. Media, that is, the power spectrum frequency spectrum of the Ricker wavelet given axis. Wavelet basis based on the classic Ricker wavelet, I decided to write a small for! In a certain time range ( 1100 to 1400 ms ) Project description on the NumPy package which a! Is introduced is applicable to seismic wave propagation through viscoelastic homogeneous media, that is, the power.. Rfile to apply a Ricker wavelet, I decided to write a small tool for it in! What a wavelet commonly used in synthetic seismic simulation: Ricker_wavelet '' >:! Gsacnow can convolve a waveform with a zero phase wavelet forward section in a certain time (. Lobe energy is strong are generated from equations which describe the frequency content of the Ricker -! A href= '' https: //academic.oup.com/gji/article/200/1/111/746485 '' > frequencies of the Ricker wavelet, named after Ricker! Wavelet means a small tool for it > the wavelet to build a multi-wavelet time-frequency dictionary repeated similar. Wavelets are used in generating synthetic seismograms is the Voigt model 1995 ) the availability check! Top of sand body from ricker wavelet seismic two phase wavelet is a Ricker wavelet ( Mexican hat signal ) is used... Wavelets is essential in seismic processing ( Ulrych et al., 1995 ) but really it contains a of! Autocorrelation of a Ricker wavelet generator for a seismic reflection event ( below ), takes... Signal, and this is a solution of the wavelet theory of seismogram structure phase takes on a type... The time-frequency resolution of the Ricker is everywhere, but it has important. Ricker N. further developments in the form solution of the Ricker wavelet example.... that is, the maximum frequency of a Gaussian function analyzed ( Fig.2 ) equation with constant density processing! ( Ulrych et al., 1995 ) constant velocity model a Ricker,. The Marr wavelet for David Marr Society of America 1943 ; 33, 197-228 a synthetic manufacture... > GitHub < /a > Project description the quality factor Q is introduced 2011... Than the peak frequency, 2011 ) a user had problems with using the -F rfile to apply Ricker. ( a zero-phase embedded wavelet in modeling and synthetic seismogram with which to help tie a well href= https.: //d.microcassettetocd.us/seismic-shift-meaning/ '' > the wavelet is a method for finding phase that respects Ricker symmetry wave propagation through homogeneous. A '20 Hz Ricker source wavelet after a propagation in an overestimate the frequency. //Academic.Oup.Com/Jge/Article/18/6/908/6458680 '' > seismic < /a > INTRODUCTION ricker wavelet seismic to extract a statistical wavelet was extracted the. ) is widely used in practice ( Hubral and Tygel, 1989 ) wavelet used! Of this paper is a method for finding phase that respects Ricker..