Receiver function method in reflection seismology · Receiver function method in reflection...

Geophysical Prospecting, 2008, 56, 327–340 doi:10.1111/j.1365-2478.2007.00685.x

Receiver function method in reflection seismology

Pascal Edme1∗,† and Satish C. Singh2

1Schlumberger Cambridge Research, Madingley Road, High Cross, Cambridge CB3 0EL, UK, and 2IPG, Laboratoire de Geoscience Marines,4 Place Jussieu, 75252 Paris Cedex 5, France

Received January 2007, revsion accepted October 2007

ABSTRACTThe receiver function method was originally developed to analyse earthquake datarecorded by multicomponent (3C) sensors and consists in deconvolving the horizon-tal component by the vertical component. The deconvolution process removes travelpath effects from the source to the base of the target as well as the earthquake sourcesignature. In addition, it provides the possibility of separating the emergent P and PSwaves based on adaptive subtraction between recorded components if plane wavesof constant ray parameters are considered. The resulting receiver function signal isthe local PS wave’s impulse response generated at impedance contrasts below the3C receiver.We propose to adapt this technique to the wide-angle multi-componentreflection acquisition geometry. We focus on the simplest case of land data reflec-tion acquisition. Our adapted version of the receiver function approach consists ina multi-step procedure that first removes the P wavefield recorded on the horizontalcomponent and next removes the source signature. The separation step is performedin the τ − p domain while the source designature can be achieved in either the τ − p orthe t − x domain. Our technique does not require any a priori knowledge of the sub-surface. The resulting receiver function is a pure PS-wave reflectivity response, whichcan be used for amplitude versus slowness or offset analysis. Stack of the receiverfunction leads to a high-quality S wave image.

I N T R O D U C T I O N

Multi-component seismic data contain richer informationabout elastic parameters of the subsurface than the conven-tional single-component data recorded using a streamer or avertical component array. This is because the horizontal com-ponent data contain converted S-waves. Therefore, a jointanalysis of P and S wave data provides important informa-tion on subsurface parameters such as lithology (Tatham andMcCormack 1981), porosity (Garotta, Granger and Gariu2002), fracturing (Ata and Michelena 1995; Li 1997) andanisotropy (Lynn, Simon and Bates 1996; Tsvankin andGrechka 2002; Thomsen 1999) and hence multi-componentdata are of great importance for the oil and gas explorationindustry, and are especially suitable in regions where P-wave

∗E-mail: [email protected]†Formerly at IPG, Paris, France

imaging fails. For example, when P-wave imaging is affectedby strong attenuation due to the presence of gas and resultsin ‘blind’ zones (for example, a reservoir below a gas cloud),the PS wave imaging method has proven to be an efficientalternative tool (Granli et al. 1999).

A key point in multi-component acquisition and processingtechniques is the possibility to separate the recorded wave-fields into pure P and S wavefields. It is often assumed thatthe vertical Uz component contains principally pure-mode Pwave arrivals and the in-line horizontal Ux component con-verted PS wave energy. This assumption becomes invalid atlarge offsets, where amplitudes of P-to-S conversion are max-imum. Except at vertical propagation, the incident P and Swavefields energy is partitioned between the vertical Uz andhorizontal Ux sensors:

Uz = U Pz + US

z , (1)

C© 2008 European Association of Geoscientists & Engineers 327

328 P. Edme and S.C. Singh

Ux = U Px + US

x , (2)

where Uji is the part of the j wavefield present on the Ui compo-

nent. Therefore, it is necessary to decompose multi-componentdata in pure P and S-wave modes. Contaminating energy oneach component should ideally be removed to improve thestack quality. There are several schemes developed for seismicreflection data. For land data, methods have been developedbased on frequency-wavenumber analysis (Dankbaar 1985;Wapenaar et al. 1990). For multi-component ocean-bottomcable (OBC) data, Amundsen and Reitan (1995) proposed awave equation based method in the frequency-wavenumberdomain, which requires the elastic parameters to be knownin the vicinity of receivers. Wang et al. (2002) developed aτ − p domain scheme based on apparent slowness identifiedon either the horizontal or vertical component. No a priori

knowledge of the subsurface is needed but its application canbe time consuming and complex due to the manual selectionof events in a time window.

In global seismology, a locally converted pure-mode S-waveimpulse response is obtained using a receiver function method.The receiver function technique enables the extraction of thePS wavefield from mixed seismograms and removal of thesource signature. It does not require any a priori knowledge ofthe subsurface and a minimal amount of user-defined input.The receiver function decomposition scheme and the Wanget al. (2002) method are quite similar. In this paper we com-bine the two techniques to compute the PS wave reflectivityresponse. The τ − p method proposed by Wang et al. (2002)enables us to extend the receiver function technique to wide-

Figure 1 Left: Conventional receiver function acquisition geometry. The target zone below the multi-component 3C receiver is illuminatedfrom below. The source function is an earthquake event. The Path term is the earth response from the source to the deepest conversion point.Here it contains two successive P waves (the primary P wave and its source side free surface multiple pP, which generate two similar PS waveimpulse responses shifted in time. Right: close up of the target zone. When a plane P wave encounters interfaces below the receiver, PS wavesare generated. The emergent angle of the P wave is always bigger than that of S waves.

angle recordings, whereas the receiver function concept allowsthe improvement of the Wang et al. (2002) method in separat-ing P and PS wavefields automatically.

R E C E I V E R F U N C T I O N I N G L O B A LS E I S M O L O G Y

The receiver function technique has been used over the last20 years in global seismology to emphasize the converted PSwavefield generated in the region below the 3C-receiver. Cur-rently, it is used exclusively in natural source seismology (seeFig. 1). It was introduced by Vinnik (1977), later improvedby Langston (1979) and has become a standard technique formapping structural properties using the transmitted-convertedPS wavefield produced within the crust (Vergne 2002), of theMoho (Sheehan et al. 1995; Baker et al. 1996), the top ofthe subducting slab (Langston 1981; Regnier 1988;), or otherupper mantle discontinuities (Kosarev, Makeyeva and Vinnik1984; Vinnik and Montagner 1996; Bostosk and Sacchi 1997).The strength of this approach is its ability to use deconvolu-tion between horizontal and vertical components to performthe source designation, to remove the travel path before theconversions, and to extract the primary PS wave arrivals.

Receiver function theory

For completeness, we shall first derive the basic equations forthe receiver function method and discuss its properties. Here,we focus on the receiver function ability to map structures(i.e. seismic impedance contrasts below a receiver, in the time

C© 2008 European Association of Geoscientists & Engineers, Geophysical Prospecting, 56, 327–340

Receiver function method in reflection seismology 329

domain), not on the various techniques (like migration and/orinversion schemes) that use the calculated receiver functionsignals in order to recover earth properties (for example shear-wave velocity profile). One of our main objectives in this sec-tion is to show under what circumstances (assumptions) thereceiver function process gives the primary PS wave’s impulseresponse of the target zone. We do not present all the specificpoints and subtleties that could be used in some receiver func-tion studies in order to estimate, for example, the internal Pwave multiples as well (Li and Nabelek 1999; Baig, Bostockand Mercier 2005). As most of receiver function studies weconcentrate on estimating the PS behaviour polarized in theinline plane (Uz and Ux). The tangential component (Uy) pro-vides information on the dip of the interfaces and anisotropy(Cassidy 1992; Levin and Park 1997; Savage 1998), but foran isotropic plane layered medium, this component does notcontain any arrival. Data are sorted in teleseismic event offset(i.e. epicentral distance) and rotated according to the source-receiver back-azimuth. In the time domain, the form of thetheoretical displacement response generated by a P wave im-pinging beneath a stack of horizontal or dipping interfaces canbe written as:

UP P = Src ∗ Path ∗ IRP P ,

UPS = Src ∗ Path ∗ IRP S, (3)

where ∗ is the convolution operator, Src is the earthquakesource function and Path is the global earth response from thesource to the deepest conversion point. IRP P and IRP S are thelocal transmitted P wave and transmitted-converted PS wavesimpulse responses from the deepest conversion point to thereceiver. The deconvolution in the time domain is equivalentto a division in the frequency domain:

UPS(ω)UP P (ω)

= IRP S(ω)IRP P (ω)

, (4)

where ω is the angular frequency. Thus the spectral ratio be-tween the P and PS wavefield is independent on the nature ofthe source and the travel path up to the conversion point. Indefining the receiver function method, it is usually assumed(not always) that all internal P wave multiples reverberatingwithin the target zone are weak and can be neglected:

IRP P (ω) ≈ 1. (5)

In this case equation (2) becomes:

UPS(ω)UP P (ω)

≈ IRP S(ω). (6)

The spectral ratio of the PS over the P wavefield gives thetransmitted-converted transfer function. However, one re-

quires the decomposition of wavefield from Uz and Ux to up-going UP P and UPS wavefields. Various strategies have beendeveloped to decompose the wavefields. For instance, Readinget al. (2003) propose the receiver function computation fromdecomposed P and PS wave-vector where the effect of the freesurface is taken into account. But generally, only nearly ver-tical emergent angles are considered by selecting teleseismicearthquakes with epicentral distance from 30 to 90 degrees(which means an incident angle of P waves of 20 ± 10 degreesat the Moho and 10 ± 5 degrees at the surface). Therefore, it iscommonly assumed that the P and PS wavefields are naturallyseparated on the Uz and Ux components respectively:

Uz ≈ UP P ⇔ USz ≈ 0,

Ux ≈ UPS ⇔ U Px ≈ 0. (7)

The above approximations are valid in earthquake seismol-ogy, considering that the travel path of upgoing events pro-gressively becomes vertical as it arrives (particularly for largeoffsets as illustrated in Fig. 1, but in reality incident wavesenergy is always partitioned between the two components).Therefore a conventional receiver function signal is definedas:

RF (t) = TF −1

(Ux(ω)Uz(ω)

)≈ IRP S(t), (8)

where TF−1 is the inverse Fourier transform operator. Thereceiver function method gives the PS wave impulse response,which is independent of the earthquake source signature andthe travel path through the earth before the conversions. Thereceiver function signal is the time series that reproduces thehorizontal component seismogram when it is convolved withthe vertical component:

Ux(t) = RF (t) ∗ Uz(t). (9)

The timing and amplitude of the arrivals in the receiver func-tion are sensitive to the local (receiver side) earth structure.The receiver function signals obtained from different earth-quakes (originating from a same location) are then stacked,migrated and inverted to recover the elastic properties in thetarget zone (Ammon et al. 1990). Note that the assumptionsof equations 5 and 7 are important to ensure that the resultingreceiver function signal is a good estimation of the primary PSwave’s impulse response, equation (8), however they are notneeded for inversion schemes if observed and predicted signalsare processed in the same way.



Receiver function in practice

Computing a receiver function is a deterministic deconvolu-tion problem. Such a deconvolution can be performed eitherin the time or in the frequency domain and should give sim-ilar results. Theoretically, the advantages of one techniqueover the other should be insignificant but in practice, thechoice of a deconvolution technique may make a difference.The most commonly employed method in receiver functionstudies is a frequency-domain division (Clayton and Wiggins1976). This approach consists of computing the ratio betweenthe amplitude spectrums of the Ux and the Uz components(equation (8)). This method is fast and works well especiallyfor large earthquakes with high S/N ratio. However, we pre-fer the iterative deconvolution technique in the time domain,originally developed to estimate large-earthquake source timefunctions (Kikuchi and Kanamori 1982) and adapted to thereceiver function problem by Ligorria and Ammon (1999).This later technique has several advantages when comparedto the frequency domain division (that computes the receiverfunction in one pass). Our purpose is not to claim that thistechnique can overcome other deconvolution techniques (onceagain, all deconvolution technique should give the same out-put). We chose the iterative approach because it offers theopportunity of visualizing the intermediate steps during thedeconvolution process, it enables us to better understandthe exact effect of the deconvolution and it allows the com-putation of true impulse responses, composed of a series ofDirac delta functions δ(t). The output receiver function signalcan be written as:

RF (t) =n∑

i=1

aiδ(t − ti ), (10)

where ai is the amplitude of the Dirac delta function at time ti,obtained at the ith iteration. The foundation of the iterative de-convolution approach is the least-squares minimization of thedifference between the observed Ux seismogram and the pre-dicted signal generated by the convolution of an iteratively up-dated signal spike train with the Uz component. This methodis based on the use of the cross-correlation function to estimatethe lag and amplitude of the spikes that compose the final re-ceiver function. The iterative process builds step-by-step thereceiver function to explain the major part of the Ux com-ponent energy. Each iteration i corresponds to one adaptivesubtraction, which is composed of six steps:1 Cross-correlation between U(i−1)

x and Uz signals (where U(0)x

is the original Ux seismogram).2 Location of time ti of the cross-correlogram’s maximum a′

i.

3 Determination of the scalar coefficient ai that best fits thetime shifted Uz component with U(i−1)

x . It consists of minimiz-ing the quantity:

C = |U(i)x − aiUz ∗ δ(t − ti )|2, (11)

where ai is numerically obtained by:

ai = a′i

Uz · Uz, (12)

where the dot is the zero time cross-correlation. This equationcorresponds to the scalar case (frequency independent) of theWiener-Levinson filter (Yilmaz 1987).4 Addition of a Dirac Delta function of amplitude ai at timeti to the current receiver function,5 Calculation of a residual U(i)

x by subtracting Uz shifted intime by ti and multiplied by ai from previous U(i−1)

x :

U(i)x = U(i−1)

x − aiUz ∗ δ(t − ti ). (13)

6 Return for the next iteration.In summary, the results of each adaptive subtraction (am-

plitudes ai and arrival times ti) are successively stored in thereceiver function signal. With each additional spike in the re-ceiver function, the misfit between the receiver function con-volved with the vertical component and the horizontal com-ponent is reduced. The iterations can be stopped when the re-duction in misfit with additional spikes becomes insignificantor simply when most of the Ux component energy has beenremoved (i.e. U(i)

x very small). Once the series of Dirac deltafunction is calculated, the final stage consists of convolvingit with a Gaussian function (whose width respects the verti-cal resolution, function of the dominant frequency) to enablea constructive summation of events originating from a sameseismic zone.

The effect of the receiver function procedure is illustratedin Fig. 2, using very simple input signals (not realistic butour purpose is just to show the effect of the deconvolution).The Ux and Uz traces corresponds to a two-layers-over-a-half-space model similar to Fig. 1, where two impinging P waves(the primary P and the source side multiple pP) successivelygenerate PS wavefields shifted in time by �tP (note that theinternal multiples are intentionally omitted, for simplicity).These plane P waves have the same ray parameter p (the P

and pP waves arrive at the base of the target with the sameincidence angle), and near vertical propagation is considered(i.e. p is small). Under such circumstances, the Uz componentcontains almost exclusively P waves because the mode con-version PS waves have weaker amplitudes and more verticalemergent angle than the P waves. In contrast, the Ux trace is



t1

P

pP

P1S

P2S

P1S

pP1S

pP2S

P

pP

P

pP

P2S

pP2S

P1S

pP1S

P2S

pP1S

pP2S

t=0

RF IRPSUz Ux Ux(1) Ux(2) Ux(3)

tim

e

/ /

a1

a2

a3

tluseRnoitulovnoced evitaretIatad tupnI

Figure 2 Example of the receiver function procedure. The input Uz and Ux signals correspond to a model similar to Figure 1. Each impingingwave (P and its source side multiple pP) generates two PS waves at the impedance contrasts below the receiver. The Ux trace is contaminatedby the P waves and the identification of primary PS wave arrivals is difficult. The output receiver function signal results from the iterativedeconvolution of the Ux component by the Uz component (without time windowing) and gives the expected local PS wave impulse responseIRP S , plus a Dirac delta function at time t = 0 (whose amplitude is related to the partition rate of the P wave energy between components).

The process acts by stripping the most dominant features from the previous residual component U(i)x . Only three iterations are needed to extract

97% of the Ux energy. Note that the first iteration is a P-PS separation step: U(1)x contains nearly exclusively PS waves. The remaining (weak) P

wave energy on U(1)x is due to the fact that the P wave partition rate (estimated from the whole trace length) is biased by the S wave energy (see

equations (14) to (17)).

strongly contaminated by P wave energy, even for nearly verti-cally emergent angle. The identification of the two primary PSarrivals (P1 S and P2 S) on Ux is not straightforward. The cal-culated receiver function signal (resulting from the deconvolu-tion of the whole Ux component by the whole Uz component)is shown in Fig. 2 and can be compared to the true input PSwave impulse response IRP S . The residual components U (i)

x areshown in the middle. At each iteration, the process removesa couple primary-multiple from the previous residual compo-nent. The most dominant features are extracted first and thesmaller ones after. The user has the possibility to stop the it-erations for a chosen degree of details, leaving the smallestevents in the last residual component or not. In our example,only three iterations are needed to remove 97% of the energyrecorded on the original Ux seismogram.

The first iteration acts as a separation step by removingthe P wave energy from the Ux component. The residual U(1)

x

signal contains only PS wave arrivals. Because the P waves arein phase on the two components, they are detected at timet1 = 0 of the cross-correlogram and the residual U(1)

x signal isobtained by applying:

U(1)x = Ux − a1 Uz = US

x , (14)

where the a1 coefficient is calculated as:

a1 = Uz · Ux

Uz · Uz=

(U P

z + USz

)·(U P

x + USx

)(U P

z + USz

)·(U P

z + USz

) . (15)

Because the quantity of S wave energy on Uz is insignificant(US

z → 0), we have:

a1 ≈ U Pz · U P

x

U Pz · U P

z

+ U Pz · US

x

U Pz · U P

z

. (16)

If the overlapping phenomena between P and PS events canbe neglected (i.e. UP

z · USx → 0), the amplitude a1 at time t1 =

0 gives the partition rate of the P waves between the two or-thogonal components:

a1 ≈ U Px

U Pz

= − tan φ, (17)

where φ is the apparent polarization angle of the P wavesas measured on the recorded components. The next itera-tions progressively removes the Path ∗ Src term, by strippingthe coupled primary-multiple PS wave events from U (1)

x . Theresulting receiver function signal is the expected PS wave im-pulse response IRP S (generated within the target zone) plus the



additional Dirac delta function at time t = 0 (related to theP wavefield).

This example shows that the receiver function procedureacts both as a P-PS separation tool, an external multiple re-moval tool and a source signature removal tool. The techniqueis designed to process forward scattered events of small andconstant ray parameter to ensure a constant partition rate ofthe P wave energy between the components. The P wave re-moval on Ux (during the first iteration) is based on polariza-tion discrimination. In the next section, we adapt the receiverfunction approach to the reflection acquisition geometry.

R E C E I V E R F U N C T I O N I N R E F L E C T I O NS E I S M O L O G Y

We consider a simple case of land data acquisition geometrywhere the target (i.e. the subsurface) is now illuminated fromabove. Since the source is on the surface, the Path term doesnot exist and the emergent wavefield can be written:

UP P = Src ∗ IRP P ,

UPS = Src ∗ IRP S . (18)

Our approach therefore does not aim at removing the multi-ples in the data. We focus on the RF capacity to separate thereflected PP and PS wavefields and to remove the source sig-

PP rays

free surface sea-floor

Ux

Uz

Uh

upgoing PP rays in the -p domain

free surface

PP rays

free surface

PS rays

free surface

PS rays

P P P

PP

P

P

P

P

P P

PP

PP P

PP

P P

S

S S

SP S

Figure 3 Reflection array geometry. The target (the subsurface) is illuminated from above. Left: ray paths for a given offset. The emergent angledepends on the depth of the reflection point. Right: ray paths for a given slowness. The emergent angle only depends on the wave type at thereceiver level (i.e. P or S wave).

nature, in order to estimate the PS wave reflectivity responseIRP S . Reflection data are recorded in the time-offset (t − x) do-main in which the emergent angle of each arrival for a givenreceiver depends on the depth-offset ratio of the reflectionand/or conversion point (as shown in Fig. 1, left). The reflec-tion acquisition geometry introduces a strong variability of theray parameter over the recording length, which precludes thepossibility of applying the receiver function technique on offsettraces. Time varying ray parameter introduces errors duringthe deconvolution. As mentioned before, the receiver functionmethod has to be applied for a constant ray parameter, in or-der to ensure a constant partition rate of the energy betweenthe components. This can easily be achieved in reflection seis-mology by transforming the data in the τ -p domain (where p isthe ray parameter or horizontal slowness and τ is the intercepttime at zero offset) because of fine receiver spacing. The τ -ptransform is a linear radon transform that acts as a plane wavedecomposition by performing a sum along lines in the data atsampled values of τ and p (see Dunne and Beresford 1995 fordetails). The effect of the τ -p transformation is schematicallyillustrated in Fig. 3. In a given offset trace, the emergent angleof waves is time varying. In contrast, it is constant in a givenslowness trace and depends only on the wave type. Therefore,the horizontal and vertical wavefield are linked to the pure Pand S-wavefields by a simple matrix formulation in the τ -p



i

Ux

Uz

incident P

j

Ux

Uzincident S

SOLID

ji

ji

AIR

SOLID

AIR

RPP\\

RPS\\

RSP\\

RSS\\

Figure 4 Reflections at the free surface interface just above the receiver. Left: for an incident P wave. Right: for an incident S wave.

domain:[Uz

Ux

]=

[M11 M12

M21 M22

] [UP P

UP S

]. (19)

The above matrix needs to take into account the effect of thefree surface. Because the geophones record the particle dis-placement just below the free surface, the observed compo-nents contain not only the impinging waves but the additionalcontribution of the energy which is downward reflected by thesolid-air interface as well (see Fig. 4).

The portion of the P wavefield recorded on Uz and Ux (re-spectively) is given by:

U Pz = M11UP P = (−qαα + RP Pqαα + RPS pβ) UP P , (20)

U Px = M21UP P = (pα + RP P pα + RPSqββ) UP P . (21)

The portion of the S wavefield recorded on Uz and Ux (respec-tively) is given by:

USz = M12UPS = (pβ − RSS pβ + RSPqαα) UPS, (22)

USx = M22UPS = (qββ + RSSqββ + RSP pα)UPS, (23)

where α and β are P and S wave velocities at the receiver level,just below the free surface. The vertical slownesses for P and PSwaves are qα =

√α−2 − p2 and qβ =

√β−2 − p2, respectively.

The combination of the incident and the reflected wave energyleads to the following expression of the matrix M (Aki andRichards 1980; Jepsen and Kennett 1990).[Uz

Ux

]=

[−qααC1 pβC2

pαC2 qββC1

] [UP P

UP S

],

C1 = 2β−2(β−2 − 2p2)/D,

C2 = 4β−2qαqβ/D,

D = (β−2 − 2p2)2 + 4p2qαqβ . (24)

This expression is valid for a given p value. The coefficients Mij

are a function of α and β, but are independent on the density

Table 1 Values of elastic parameters used for computation of syntheticseismograms shown in Fig. 5

z(km) α (km/s) β (km/s) ρ (g/cm3)

layer 1 2 2 1.2 2.0layer 2 3 2.5 1.6 2.1half space ∞ 3.5 2.3 2.6

ρ. Thus, the incident wavefields can be directly obtained fromthe observed components, if the near surface velocities areknown, by inverting equation (24):[

UP P

UP S

]=

⎡⎣− 1−2β2 p2

2αqα

pβ2

α

pβ 1−2β2 p2

2βqβ

⎤⎦ [

Uz

Ux

]. (25)

Our expression is slightly different from that of Kennett(1991) and Reading et al. (2003) where the pβ and the pβ2

α

terms have to be exchanged. Equation (25) has to be appliedtrace-by-trace in the τ -p domain. The calculation of the UP P

wavefield requires both the P wave and the S waves velocities,while the UPS wavefield can be obtained from the S wavevelocity alone.

The effect of the τ -p transform is now illustrated using syn-thetic wide-angle multi-component data for a simple planelayered model composed of two layers over a half space. Theparameters are given in Table 1. The point source and the re-ceivers are located on the free surface. The source signal is aRicker wavelet with peak frequency of 20 Hz. Offsets betweenthe source and receivers range from 0.02 km to 10 km with anequal interval of 0.02 km. The synthetic seismograms, calcu-lated in the t − x domain using the reflectivity method (Fuchsand Muller 1971), are shown in Fig. 5. These seismogramsare transformed into the τ -p domain by a frequency domaincylindrical transformation (Singh, Klest and Chapman 1989).The τ -p seismograms (Fig. 6) are calculated for p value from0.003 s/km to 0.5 s/km with an equal interval of 0.003 s/km.A linear event in the t − x domain maps to a point in the τ -p



Figure 5 Synthetic Uz and Ux seismograms in the t − x domain computed from model described in Table 1. The Ux component is stronglycontaminated by the P waves. Each offset trace contains a mixture of P and PS wave arriving with time varying incidence angles (or ray parameterp).

Figure 6 Uz and Ux seismograms after τ − p transformation of seismograms shown in Fig. 5. As expected, the Ux component is stronglycontaminated by the P waves. In the τ − p domain, each trace contains a mixture of P and PS waves, each arriving with a constant incidenceangle (respectively ϕ and ψ , equation 11). In the τ − p domain, the P and PS waves can be separated by polarization discrimination.

domain and all points in the t − x domain are completely de-scribed by lines in the τ -p domain. The linear τ -p transformmaps hyperbolic reflections to ellipses. Note that the transfor-mation is reversible. In reflection data, except at zero offsetor slowness, both the Ux and the Uz components contain amixture of P and PS events.

PS wavefield extraction

Here we want to reproduce the effect of the first iteration ofthe receiver function processing, which is the P and PS sep-aration step based on polarization discrimination. The aimis to apply equation (14) (or alternatively equation (25)) to



estimate the U(1)x or UPS wavefields (respectively). These wave-

fields are equivalent in the sense that they contains exclusivelyPS waves, but the amplitude versus slowness behaviour isdifferent: U(1)

x = M22UPS (equation (23)). To do that, we needto recover the P wave partition rate between Ux and Uz foreach p trace:

U Px

U Pz

= M21

M11= − 2pqβ

β−2 − 2p2= −tanφ. (26)

The apparent polarization angle φ of P waves is surprisinglyindependent of the P wave velocity, when taking into accountthe free surface effect. It depends on the S wave velocity alone.The theoretical ratio, according with our input near-surface Swave velocity model (β = 1.2 km/s), is shown in Fig. 7 (solidline).

The straightforward application of the receiver functionmethod on τ − p gathers consists of considering each p

trace as an individual earthquake. In this case, the a1 coeffi-cients as a function of p are independently calculated applyingequation (15), from the whole trace length (no time window-ing). The resulting a1(p) values are shown in Fig. 7 (circle).Except at small slowness, the estimated values are not in agree-ment with the theoretical curve. This is due to increasing quan-tity of S wave energy on Uz when p increases. At large slow-ness, firstly the PS wave conversions are stronger, and secondly

Figure 7 In solid line, the theoretical partition rate of the P wave en-ergy between the Ux and Uz components as a function the horizontalslowness p (see equation (26)), according with our input near surfaceshear wave velocity β = 1.2 km/s. In circle line, the curve estimatedfrom the conventional receiver function analysis (independently ob-tained trace-by-trace, without time windowing thus automatically).In the crossed line, the estimated values, obtained trace-by-trace, butfrom selected pure P events. The manual selection method gives theexpected output but can be tricky.

the partition rate of the S wave energy between componentsbecomes significant. In other words, the assumption US

z → 0is not valid when considering wide-angle data. This prelimi-nar result suggests that the receiver function technique can beemployed only from near vertical emergent rays (as expectedby the receiver function theory), but it is not of high interestwith reflection data.

To address this problem, one solution could consist in iso-lating a pure P event, for each p trace, and determining theratio from these time windows. This is the method proposedby Wang et al. (2002) (applied on OBC data), in which the es-timated values of the P wave partition rate are calculated frommanually selected pure P events in the τ − p domain. In Fig. 7(crossed line), we show the a1(p) values obtained from (po-larization) analysis of the earliest P1P phase. Expected valuesare obtained, except at the proximity of the critical slowness(pc = α−1) where the P wave polarization becomes ellipticalinstead of linear. Wang et al. (2002) proposed to invert themeasured a1 curve as function of p to recover the elastic prop-erties in the vicinity of the receiver. In our case (land data),we only need to determine the β velocity, required to extractthe PS wavefield. The problem with such a process is that atleast one pure P wavelet has to be isolated on each p trace.This manual trace-by-trace selection can be tedious and timeconsuming when working on more realistic seismograms.

Here, our strategy is to combine the receiver functionmethod and the Wang et al.’s (2002) method in order to di-rectly obtain the required β velocity in an automatic way. Thesimplest implementation of the receiver function process con-sists of considering each p trace as individual earthquake, butif the near surface is assumed to be laterally homogeneous,all the rays probes the same near surface structure. The min-imization procedure of equation (11) (numerically resolvedusing equation (15) for each p trace) can be applied on thewhole τ − p gather, by taking advantage of the coherency be-tween adjacent traces in reflection seismology. Our purpose isto minimize the whole P wave energy in the τ − p gather todirectly recover the shear wave velocity β. The expected re-sult is the β value for which all the P wave energy is removedfrom the Ux(τ , p) gather. Such a multi-trace procedure can bewritten as:

Cx(β) =0.5∑

p=0.003

|Ux(p) + 2pqβ

β−2 − 2p2Uz(p)|2. (27)

The calculated Cx values as a function of β velocity are shownin Fig. 8. Figure 8(a) shows the misfit function when process-ing manually windowed P wave arrival (the P1P event), andFig. 8(b) shows the misfit function when processing without



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Figure 8 Result of the minimization procedure of equation (27), as a function of shear wave velocity β (at the free surface). (a) Misfit functionwhen P waves are windowed. (b) Misfit function when P waves are not windowed. The windowed result is better constrained, but the expectedβ value is obtained in both cases. The process suffers from the presence of the S wave in the time window but it still gives satisfying result.

windowing (i.e. the whole traces, such that the minimizationis biased by the presence of the S waves). Naturally, the resultis better constrained when time windowing is performed, butin both cases a distinct minimum can be observed, yielding theexpected value β = 1.2 km/s. Ideally, the time-space windowshould contain only P wave energy, but our example demon-strates that the undesirable presence of S wave energy can betolerated. With this process, both the problem of overlappingphenomena (encountered with the receiver function process)and the problem of manual time windowing (encountered withthe Wang et al. method) are addressed. Therefore, this part ofthe process can be automatically achieved. While the result-ing Cx quantities is obtained by considering the whole τ − p

wavefield energy, the separation step is still performed trace-by-trace, by applying equation (25) with the β value previ-ously determined. The resulting UPS(τ, p) gather is shown inFig. 9. When compared to the original Ux(τ , p), all P waveshave disappear and the seismogram can be interpreted in aless ambiguous manner. The resulting UPS gather can be usedfor amplitude versus slowness (AVS) analysis. In Fig. 10, weshow the AVS response of the earliest P1 S phase (the theo-retical reflection coefficient RP1 S(p) is shown in crossed lineand the measured curve is shown in solid line). An excellentagreement is obtained, demonstrating another advantage ofthe τ − p domain, where the geometrical spreading effect isremoved.

Figure 9 The PS wavefield after applying equation (16), trace-by-trace, using the β velocity calculated from the minimization procedureof equation 27 (Fig. 8). All contaminated P wave energy has been re-moved from Ux, the seismogram contains exclusively events arrivingas S wave at the receiver.

Source signature removal

In order to completely reproduce the receiver function pro-cess, we have to remove the source signature. As in all de-terministic deconvolution problems, we need to estimate the



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Figure 10 AVS response of the P1S phase (solid line), measured fromthe separated UPS seismogram shown in Fig. 9, and the theoreticalRP1 S (crossed line) from Aki and Richards (1980).

source function. In contrast to the seismology case, the Src

term cannot be approximated by the whole Uz component.Here, the source wavelet is manually extracted from the data(by windowing the earliest P event at small slowness) and isused to deconvolve the whole UPS gather (independently trace-by-trace). The deconvolution can be performed either in thet − x domain or in the τ − p domain. The RF(τ , p) signalresulting from the deconvolution of the UPS(τ, p) is shown

Figure 11 Original Ux component and the resulting receiver function seismogram, after applying the whole process, in the τ − p domain.The receiver function signal is PS wave reflectivity response, composed of a serie of Dirac delta functions representing 95% of the UPS(τ, p)seismogram shown in Fig. 9. When compared with the Ux(τ , p) component, all P waves have disappear and the source signature is removed.

in Fig. 11. The RF(t, x) signal resulting from the deconvolu-tion of the UPS(t, x) is shown in Fig. 12. In both cases, theiterations are stopped when 95% of the UPS energy has beenstripped. In this way, the smallest events (like artefacts gen-erated by the τ − p transform) are ignored and the RF sig-nal spike train emphasizes only significant PS wave arrivals.In both domains, the receiver function signal is considerablysimplified in comparison to the original Ux component. Thereceiver function seismograms contain pure PS arrivals, withextremely sharp resolution. Note that recovered images cannotbe directly interpreted in term of interfaces because the inter-nal multiples still remain. Further processing like surface re-lated multiple elimination methods, introduced by Riley andClaerbout (1976) and Kennett (1979) and further improved,among others, by Verschuur, Berkhout and Wapenaar (1992)and Amundsen (1999), is needed to achieve this step.

S TA C K

The efficiency of our multi-step procedure is illustrated usingthe simple 2D velocity model of Fig. 13(a), composed of threelayers with the second interface a 18 degree dipping. The sameprocess is independently applied on 47 shot gathers (computedwith a finite-difference code). To enable constructive summa-tion, the receiver function gathers (composed of Dirac deltafunctions) are previously convolved with a Gaussian functionwhose width a is adjusted to respect the vertical resolution,



Figure 12 Original Ux component and the resulting receiver function seismogram in the t − x domain. The P-PS separation step is performedin the τ − p domain but the source function removal step can also be performed in the t − x domain by deconvolving the UPS(t, x) seismogram,after inverse τ − p transformation. The receiver function signal is composed of a serie of Dirac delta functions representing 95% of the UPS(t, x)seismogram. When compared with the Ux(t, x) component, all P waves have disappeared and the source signature is removed.

Figure 13 (a): input velocity model used to compute 47 shot gathers. (b): results from the stack of the horizontal Ux component, which containsmixed P and S arrivals. (c): the stacked receiver function section contains only S waves with sharp resolution.

according the dominant frequency f 0 of the signal (a = β/4f 0).Figure 13(b) represents the original Ux stacked section andFig. 13(c) is the resulting receiver function stacked section.Compared to the stack of the Ux component, our processclearly improves the quality of mapping shear waves structure,by attenuating contaminated P wave energy and producing asharp resolution of the impedance contrasts. Despite the factthat the tau-p transform relies on the assumption of isotropiclaterally homogeneous medium, we obtain a satisfactory re-

sult. However, if local anisotropy is significant, or if there isstrong topographic effect, the events may map onto slightlyerronious p values in the slowness domain and the free sur-face correction we apply would not be appropriate (we woulddistort the estimates of the amplitudes of the different wavetypes). In addition, care should be taken to ensure that neitherprocessing nor acquisition affects the relative ratio between thecomponents. Relative horizontal and vertical geophone cou-pling, sensitivity and calibration all affect the performance.



The separation quality may degrade if the above conditionsare not respected. The overriding requirement is high vectorfidelity recording.

C O N C L U S I O N

In this paper, we introduce the receiver function concept to re-flection seismology and propose an equivalent scheme adaptedto the reflection array geometry. We focus on land data acqui-sition. The conventional receiver fucntion technique consistsin deconvolving the Ux by the Uz recording of nearly verticallyemergent earthquakes in order to estimate the transmitted PSwave’s impulse response of the target below the receiver. Itacts as a decomposition, a source side multiple removal anda source signature removal tool in one pass. In reflection seis-mology, because the illumination of the target comes fromabove, the conventional process cannot be applied. We havedeveloped a scheme that reproduces the receiver function ef-fect, but a multi-step procedure is required. In the first stage,the data are sorted in the τ − p domain to ensure a constantemergent angle of wave types in a given trace. This transfor-mation enables us to extend the process to wide-angle data. Inthe second stage, we determine the S wave velocity just belowthe free surface, giving the partition rate of the P waves be-tween the two components. The contaminated P wave energyrecorded on the Ux component can then be eliminated. Thefinal stage consists in removing the source signature, yieldingthe pure PS wave reflectivity response estimation. Our schemedoes not require any a priori knowledge of the subsurface norvelocity discrimination between wave types. The separationstep can be automatically performed and the near-surface Swave velocity is given as an additional output. The 2D syn-thetic example demonstrates the potential for structural PSwave imaging. Our scheme results in a sharp resolution ofpure PS wave structure, with additional possibilities for AVSor AVO studies.

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