GJI Geodesy, potential field and applied geophysics
Geophys. J. Int. (2007) 170, 534–544
doi: 10.1111/j.1365-246X.2007.03467.x
Wavelet modelling of broad-band receiver functions Qingju Wu, Yonghua Li, Ruiqing Zhang and Rongsheng Zeng Institute of Geophysics, China Earthquake Administration, No. 5, Minzuxueyuan Nanlu, Haidian District, Beijing 100081, China. E-mail:
[email protected]
Accepted 2007 April 11 Received 2007 April 11; in original form 2005 November 7
SUMMARY We present a wavelet modelling approach to invert for S-wave velocities from broad-band receiver functions. Taking spline function as the basic wavelet, the broad-band receiver function is decomposed into five resolution scales by Mallat’s pyramid algorithm. The linearized least-squares inversion procedure is applied to every resolution scale. The fifth-scale approximation of receiver function is first inverted to recover the slowly varying background velocity variations with respect to a reference model. This solution is then taken as the initial model for fitting the fourth-scale wavelet coefficients of receiver function to further tune the solution to resolve sharper variations. This procedure is iteratively carried out up to the first-scale wavelet coefficients of receiver function. In this manner, the model neighbourhood containing the global minimum is first searched from the coarsest-scale receiver function, and the search gradually focuses on the global minimum by introducing finer-scale information of receiver function. Noise-free synthetic receiver function tests show that wavelet modelling of receiver functions can guide a certain range of initial models to converge to the true velocity distribution. Tests on actual data indicate that wavelet modelling can provide results very similar to those inferred by joint inversion of receiver function and surface wave dispersion. Key words: inversion, receiver function, wavelet. 1 I N T RO D U C T I O N Since Phinney (1964) first put forward modelling the spectral amplitude ratios of teleseismic P waveforms, receiver function analysis has been extensively developed to study the S-wave velocity of the crust and upper mantle. Burdick & Langston (1977) extended the spectral modelling into the time domain. Langston (1979) put forward a source equalization procedure, and further improvements in methodology expanded the amount of data available for receiver function study. Owens et al. (1984, 1987) developed a time domain waveform inversion procedure to recover the detailed crustal and upper-mantle structure information present in broadband P waveforms. Randall (1989) implemented an efficient calculation of differential seismograms for lithospheric receiver function. Ammon et al. (1990) discussed the non-uniqueness of receiver function inversion in detail. Ammon (1991) made a modification to the equalization technique that preserves amplitude information. Gurrola et al. (1995) and Bostock & Sacchi (1997) extended the frequency-domain deconvolution of receiver functions to the timedomain. Ligorria & Ammon (1999) presented an iterative deconvolution approach. Park & Levin (2000) put forward multiple-taper spectral estimation to increase the resolution of receiver function. Dueker & Sheehan (1997) presented the migration and stacking of receiver functions to image the upper-mantle discontinuities. Levin & Park (1997) and Savage (1998) applied the receiver functions to study crustal anisotropy. Receiver function inversion is a non-linear problem. Linearized inversion, implemented by linearization of non-linear functions
534
through first-order partial differential operators, is strongly dependant on the initial model, and will be trapped in local minima for those initial models far from the truth. It is well known that an intrinsic depth–velocity trade-off exists for receiver function inversion (Ammon et al. 1990; Last et al. 1997; Sandvol et al. 1998; Juli`a et al. 2000, 2003), that is, the so-called non-uniqueness of receiver function inversion. Ammon et al. (1990) pointed out that receiver functions are primarily sensitive to velocity contrasts and relative traveltimes, and concluded that the possible way to distinguish amongst several competing models that fit the data is to use independent information. A joint inversion procedure of receiver functions and surface wave dispersion was put forward to further constrain shear wave velocity structure (Last et al. 1997; Juli`a et al. 2000, 2003), in which surface wave dispersion recovers the largescale background velocity and the receiver function resolves smallscale velocity contrasts (Juli`a et al. 2003). A particularly useful approach would be to invert the data with respect to scale, in other words, beginning at large scale and using this information to tune the solution to finer detail. Wavelet transformation provides us such an approach. Since Grossman & Morlet (1984) first put forward continuous wavelet transformation, wavelet transformation has developed rapidly and has been applied successfully in many different fields, such as image compression and processing, optimum signal smoothing and speech analysis. The review of the overall theory can be found in Daubechies (1992) and Meyer (1993). Recent applications in geophysics include, data compression (Luo & Schuster 1992; Bosman & Reiter 1993), tube-wave filtering (Schuster & Sun 1993), seismic signal analysis C
2007 Institute of Geophysics, China Earthquake Administration C 2007 RAS Journal compilation
Wavelet modelling of receiver functions (Goupillaud et al. 1984; Yomogida 1984; Lilly & Park 1995) and seismic traveltime inversion and tomographic reconstruction (Bhatia et al. 1994; Li 1996). Based on the multiresolution feature of wavelet transformation (Mallat 1989; Daubechies 1992; Meyer 1993), we put forward a dyadic discrete wavelet transformation method to address the nonuniqueness of receiver function inversion. The broad-band receiver function is decomposed into five scales with different resolutions. A linearized least-squares inversion is adopted for each resolution scale. The fifth-scale approximation of receiver function is first inverted, for a given initial model. The inversion result is then taken as the initial model for fitting the fourth-scale wavelet coefficients of receiver function, and such a procedure is iteratively carried up to the final first-scale wavelet coefficients of receiver function. In this paper, we discuss linearized inversion of receiver functions, some fundamental principles of the wavelet transform, and introduce our new wavelet modelling approach. We use synthetic and observed receiver functions to compare the inversion results between the traditional linearized inversions and those based on wavelet transformation.
2 LINEARIZED INVERSION Based on the source-equalization scheme for three-component teleseismic P waveforms presented by Langston (1979), a receiver function can be expressed by D H (ω) iωt R(t) = e dω, (1) DV (ω) whereR(t) is the receiver function as a function of time, t; D(ω) is the spectrum of three-component teleseismic P waveforms, subscripts H and V , respectively represent the horizontal (radial and transverse) and vertical components of three-component teleseismic P waveforms. Various methods are used to estimate the receiver function (Gurrola et al. 1995; Bostock & Sacchi 1997; Ligorria & Ammon 1999; Park & Levin 2000). The receiver function forward problem can be expressed in the functional form (Owens et al. 1984; Ammon et al. 1990), d = F[m],
d ∼ = D m,
(3)
where d is the residual vector, given by d = F[m] − F[m0 ],
C
3 WAV E L E T T R A N S F O R M AT I O N The continuous wavelet transform (Mallat 1989; Chui 1992; Daubechies 1992; Meyer 1993) consists of moving and stretching a basis wavelet, an oscillating function satisfying certain zeromoment properties, and then to integrate the product between the basis wavelet and the signal along the time axis, thereby obtaining a time–frequency variable spectrum of the signal. In wavelet analysis, the stretch factor or scale is introduced as an alternative to frequency, leading to the term timescale in 1-D or space-scale in 1-D, 2-D or 3-D representation. Unlike the Fourier transform, an orthogonal wavelet basis consists of a family of localized, compactly supported functions. The basis functions are similar in shape but dilated at different scales. Of particular significance is that the energy of a signal in the timescale or space-scale space (where the energy is defined as the squared modulus of the wavelet transform) is extracted over long intervals at coarse scales, but over short intervals at finer scales. Assuming the Fourier spectrum of a function φ(t) to be (ω), and satisfying the admissibility condition ∞ |(ω)2 | cϕ = dω < ∞, (6) |ω| −∞
∞ |ϕ(t)| dt < ∞,
(7)
−∞
the function φ(t) can be regarded as a basis wavelet. The continuous wavelet transformation of a function f (t) using the wavelet basis φ(t) is defined as ∞ 1 t −b (wϕ f )(a, b) = √ dt, (8) f (t)ϕ¯ a |a| −∞
where a is a dilation variable, b is a time-shift variable. The inverse wavelet transformation is expressed ∞ ∞ 1 1 1 f (t) = (wϕ f )(a, b) √ ϕ(a, b)dadb. cϕ a2 |a|
(9)
−∞ −∞
(4)
and m is the disturbed model vector, also called model correction vector, D is the partial differential matrix of functional F evaluated at m 0 , which can be calculated by Randall’s efficient differential seismogram method (Randall 1989). Eq. (3) can be adjusted to Dm ∼ = d + Dm0 .
In contrast to (3), (5) is called a ‘jumping’ method (Shaw & Orcutt 1985; Constable et al. 1987). Both eqs (3) and (5) can be iteratively solved by singular-value-decomposition under least-squares meaning. It is well known that linearized inversion is strongly dependent on the initial model, thus the initial model should be carefully selected, or, the solution will be far from true model. Ammon et al. (1990) have pointed out that linearized inversion of receiver function cannot uniquely determine the depth to and the average velocity above a contrast in elastic properties. Through a number of synthetic tests, Ammon et al. (1990) showed that the best-fitting solution models actually cluster into three families of low, middle and high crustal velocities.
(2)
where d is the data vector of the receiver function, F is the functional operator producing a receiver function d from a model m and m is the parametrized model space of S-wave velocity. We use Kennett’s reflectivity method to calculate the synthetic receiver function (Kennett 1983). In linearized form, the inversion problem of receiver function analysis can be iteratively solved by iterative ‘creeping’ method (Shaw & Orcutt 1985; Constable et al. 1987), which solve for updates from the initial model or the previous iteration:
535
(5)
The discrete wavelet transformation is usually applied with the following expressions a = a0m , a0 > 1,
m ∈ Z,
b = na0m b0 ,
n ∈ Z.
The correspondent discrete wavelet transformation is ∞ 1 t f (t)ϕ¯ (wϕ f )(m, n) = m m − nb0 dt. a0 a0
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
−∞
(10)
(11)
536
Q. Wu et al.
Figure 1. S-wave velocity profile and the synthetic receiver function. A thin (2 km) low-velocity zone and the Moho interface are, respectively, located at the depth of 20 and 34 km; the synthetic receiver function is calculated with the slowness of 0.06.
Taking a 0 = 2, b 0 = 1, if there is a special basis wavelet φ(t), which stratifies ∞ 1 t t ϕ − n ϕ ¯ − k dt = δm, j δn,k (12) √ 2m 2j 2m+ j −∞
and constitutes a orthogonal basis of L2 (R), such a basis wavelet is called a dyadic orthogonal basis wavelet; the correspondent wavelet transformation is the so-called dyadic wavelet transformation, and the transformation formula is ∞ √ √ (wϕ f )(m, n) = 2m f (t)ϕ( ¯ 2m t − n)dt. (13) −∞
In our approach, we choose spline function as basis wavelet due to its symmetric and small compact support, and dyadic wavelet transformation is implemented by Mallat’s pyramid algorithm (Mallat 1989). In the pyramid algorithm, the wavelet transform is performed by convolving with a pair of quadrature mirror filters at different scales. The algorithm can be extended easily to other orthogonal wavelet bases.
4 WAV E L E T M O D E L L I N G P R O C E D U R E A useful view of complicated velocity structure of crust and upper mantle is a sudden velocity variation superimposed upon a slowly varying velocity background. On the other hand, broad-band receiver functions may be viewed as the summation of various responses caused by receiver structures at different spatial scales, C
their low-frequency content contributed by the slowly varying background velocity and their high-frequency information sensitive to the sudden velocity variation. Cassidy (1992) noted that the amplitude and width of the multiples of a receiver function are frequency dependent, which aids in resolving sharp discontinuities versus gradational transitions. Ligorr´ıa (2000) used this property to extract the crust–mantle transition (CMT) sharpness in North America, and Juli`a et al. (2003) simultaneously inverted receiver functions in two overlapping frequency bands, weighted to give more importance to the first-order velocity structure. If we extend this approach to decompose the broad-band receiver function into different frequency bands, we may resolve velocity structures present in receiver function at various scales, such a waveform fitting procedure has been applied to multimode inversion of surface wave and S waveforms (Lebedev et al. 2005). The concept of scale can be incorporated easily into the inversion scheme when the problem is posed in the wavelet domain. The wavelet transformation is carried out on both sides of eq. (2), denoting the wavelet operator to be w w(Dm) ∼ = w(d + Dm0 ).
(14)
The wavelet operator on the right side of eq. (14) is linear and may operate on the differential operator, the model vector, or their product. The aim of waveform inversion is to search the solution in the global model space that can reasonably simulate the data. We have the wavelet operator act on the differential operator, and the following wavelet transformation inversion is obtained (wD)m = wd + (wD)m0 .
(15)
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
Wavelet modelling of receiver functions
537
Figure 2. Wavelet approximation of the synthetic receiver function. From top to bottom, first to fifth scale. The wavelet approximation of the broad-band receiver function can be regarded as its low-pass filtered versions of the signal. The fifth-scale receiver function takes very simple waveform. From the fourth scale to the first scale, the receiver functions shows gradually more detailed.
In eq. (15), (wD) is the wavelet transformation of the differential operator and wd is the wavelet transformation of the data residuals, eq. (15) can be iteratively solved by singular-value-decomposition under least-squares meaning. Based on multiresolution analysis, it is well known that L2 (R) space composed by all square-integrable functions can be continuously decomposed into the orthogonal direct sum of scale space V j and wavelet space W j , and can be expressed as L2 (R) = V0 ⊕ W0 ⊕ W1 ⊕ · · ·
(16)
Thus, for any function f ∈ L (R), it can be uniquely expressed as 2
f = f 0 ⊕ w0 ⊕ w1 ⊕ · · · ,
(17)
where f 0 ∈ V0 , w j ∈ W j . According to (17), we can decompose receiver function into the orthogonal direct sum of an approximation at the coarse scale and a series of wavelet transforms at successive resolution scales. We then can apply (15) iteratively in the inversion of the receiver function at each scale. The inverse problem is thus posed into five different scales, and the inversion result of the Nth-scale receiver function is taken as the initial model for the (N – 1)th-scale inversion (high scale numbers indicate coarseness of structure). By broadening the band of the receiver function, high wavenumber information, that is, sharp variations of S-wave velocity, is gradually introduced into the model. The model is gradually tuned to fit receiver function in detail. Actually, there are two levels of iteration, one is for the lin C
earized problem in every scale, and another is for scale itself, the result of the coarse-scale inversion is taken as the initial model for the next more detailed level. In the fifth-scale, the approximation of receiver function in the coarser scale space is linearized and solved for the coarser model. In the remaining scales, the wavelet coefficient of receiver function in its wavelet domain is linearized and iteratively solved for the finer model. Singular-value-decomposition under least-squares meaning is used to iteratively update solution model. The procedure of wavelet modelling of the receiver function is divided into following three steps. (1) The broad-band receiver function is first decomposed into five-scales by the dyadic wavelet transformation. (2) An initial model is then selected to iteratively fit the approximation of receiver function in the fifth-scale space, and the coarserscale velocity structure is first inverted for and used as the initial model for the fourth-scale inversion. (3) From the fourth-scale to the first-scale, we use wavelet coefficients to iteratively update the solution model in wavelet domain, that is, the solution model after iteration for fitting the fourth-scale wavelet coefficient of receiver function is taken as the initial model for matching wavelet coefficients of receiver function at third-scale, and this procedure is iteratively conducted up to the first-scale, that is, the finest scale. At each scale, the receiver function is reconstructed, which represents the approximation of receiver
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
538
Q. Wu et al.
function at that scale and can be intuitively understood and easily explained. 5 NUMERICAL EXPERIMENTS To investigate the resolution and uniqueness of receiver function, we follow the procedure discussed by Ammon et al. (1990). We first performed a series of inversions with noise-free synthetic receiver functions, by the new wavelet modelling and the classic least-squares method, respectively. For classic least-squares method, we iteratively solve eq. (5) by singular-valve-decomposition under least-squares meaning. Although actual S-wave velocity distributions in the Earth are much more complicated, it will be useful to use simplified synthetic models to study the resolving power of receiver function inversions. Such numerical experiments can compare the performance of different inversion techniques. In a last instance, we also use an actual receiver function data set to further check the feasibility of wavelet modelling. Fig. 1 presents the synthetic S-wave velocity profile and the corresponding receiver function. The model contains a 10 per cent velocity reversal over a very small depth range (2 km) at a depth of 20 km, which is responsible for the strong peaks and troughs observed in the 8–12 s time window in the receiver function. The wavelet transformation result for the synthetic receiver function is presented in Fig. 2. We note that the wavelet representation of broadband receiver functions can be regarded as their low-pass filtered
versions. The waveforms being built up from simple at low-scales to complicated at high-scales, individual phases present in the broadband receiver function are gradually recovered by the addition of successively finer scales. In our inversion experiment, we take the same model generation approach as presented by Ammon et al. (1990) to produce initial model clusters. It may be better and to use different model parameter sets at different scales, that is, the coarse temporal scale, the coarse spatial parametrization, vise verse. Presently, we just use the same parameter set for all scale levels, and different model parametrization at every scale need to be further investigated. We first select four reference models (two of them are simple layer-over half-space models, the Moho interface shifted 6 km up and down, respectively, from its true location; another two are identical to the true model, except the low-velocity zone has been shifted 4 km up and down, respectively), and then add perturbation vectors onto each reference model, to form a cluster of initial models, which are significantly different from each other and span a broad range. Each perturbation vector consists of two separate quantities: one is defined by a cubic polynomial in depth to produce broad differences over depth between the initial models, and another is a random velocity change to impose fine-scale differences onto the initial models. In all of our inversion tests, a maximum amplitude of 1.0 km s–1 for the cubic perturbation vector and a variance of 0.2 km s–1 for the random perturbation vector are taken. Each reference model produces 36 perturbation models and the total 144 models are produced as
Figure 3. Examples of the initial models used in wavelet modelling. Each initial model is calculated by adding two perturbation vectors onto the reference model (Ammon et al. 1990). The left-hand panel shows the models generated by a layer-over a half-space model, with the Moho interface shifted 6 km down. The right-hand panel presents the models produced by the true model shown in Fig. 1 (left-hand panel), with the low-velocity zone shifted up 4 km. Note that the initial models sample a broad zone. C
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
Wavelet modelling of receiver functions
539
(A)
(B)
Figure 4. Wavelet modelling for the fifth-scale, third-scale and firth-scale receiver functions. We just show the wavelet-reconstructed approximation of receiver function for the three scales, instead of wavelet coefficients, due to the former is much more intuitively understood than the latter. In total, 144 inversion results are presented. Red lines represent true S-wave velocity distribution (left-hand panels) and receiver functions (right-hand panels). Black lines indicate inversion models (left-hand panels) and the synthetic receiver functions (right-hand panels). (a) Solutions for the fifth-scale receiver function. The Moho interface is resolved to be stepwise; the broad initial model zone converges to a narrow zone, centred around the true velocity profile, but no sign of low-velocity zone is indicated. (b) Solutions for the third-scale receiver function. A thin low-velocity is resolved nearby the true depth, the Moho interface becomes sharp, and the velocity distribution further converges to the true model. (c) Solution models for the first-scale receiver function. Both of the low-velocity zone and the Moho interface are resolved to their true locations, the velocity profile is very close to the true model.
shown in Fig. 3. We notice that the initial models sample a broad zone, and such a broad model space is favourable to investigate the non-uniqueness of receiver function inversions. We then carry out wavelet modelling for all of the 144 initial models. The wavelet modelling processes are shown in Figs 4(a–c) for the fifth-, third- and first-scale receiver functions, respectively, and all of the 144 results are kept. We just show the wavelet-reconstructed C
approximation of receiver function at each scale, both for synthetic and real data, instead of the wavelet coefficients themselves, because the former is much more intuitively understood and easily explained than the latter. From Fig. 4(a) we notice that by modelling the three broad pulses of the fifth-scale receiver function, the Moho interface is resolved to be stepwise; the broad initial model space converges to a
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
540
Q. Wu et al. (C)
Figure 4. (Continued.)
narrow zone centred about the true velocity profile; and that there is no indication of a low-velocity zone. This shows that the fifth-scale receiver function, although it is only characterized by three broad pulses, can actually tell us the average background velocity. Shown in Fig. 4(b), the phase converted from the Moho interface and the two multiple phases from the low-velocity zone are recovered in the third-scale receiver function. By matching the third-scale receiver function, sharp velocity variations are partially recovered. Note that the thin low-velocity is resolved nearby the true depth, with stepwise upper and lower boundaries and a wave speed contrast smaller than the true one. The Moho interface sharpens, and the velocity distribution further converges to the true model. Sharp velocity
variations are gradually added onto the slowly varying background velocity. The first-scale receiver function is actually a broadband one, with all of the seismic phases restored (Fig. 4c). By modelling its peaks and troughs in detail, the solution models are fine-tuned. Note that both the low-velocity zone and the Moho interface are completely resolved at their true locations, and all of the 144 velocity profiles are very close to the true model. Wavelet modelling can greatly improve the convergence of broad-band receiver function inversion. Receiver functions are insensitive to half-space velocity, thus the recovered half-space velocities span a broad zone centred around the true value. Independent information (Ammon et al. 1990), such
Figure 5. Solutions inverted by traditional least-squares method. All of the models are kept to compare with those obtained by wavelet modelling. C
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
Wavelet modelling of receiver functions
541
(A)
(B)
Figure 6. Solutions inverted by low-pass filtering and fitting method with the Gaussian parameter of 0.5 (a) and 2.5 (b), respectively. During each iterative procedure form lower-frequency band to final broadband, those models mismatching waveforms are discarded, the final models converge to two clusters.
as surface wave dispersion (Last et al. 1997; Juli`a et al. 2000; Juli`a et al. 2003), should be supplied to constrain the half-space velocity. In order to further compare the performances between our new wavelet modelling technique and the traditional least-square method, we use the same initial models to carry out least-squares inversions of the same synthetic broad-band receiver function. To show the inversion convergence between two methods, those unreasonable models without good waveform match are not discarded for traditional least-squares method, and the total 144 inversion results are presented in Fig. 5. As expected, the fit to the receiver function is excellent in most cases, but the resulting velocity models are strongly influenced by the starting models used, and span a broad C
range. Discarding solutions without a good match to the waveform, the acceptable solutions from the traditional least-square method converge to three clusters: the first cluster with lower velocity and deeper interface; the second cluster with higher velocity and shallower interface and the third cluster with fitful velocity valve and interface depth. As pointed out Ammon et al. (1990), a trade-off between velocity and depth exits for receiver function inversion: the lower the velocity above an interface, the deeper the interface and the higher the velocity, the shallower the interface. The classic least-squares method is a special case of the wavelet transformation approach obtained when all the scales are treated simultaneously. We should point out that some models match the
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
542
Q. Wu et al.
Figure 7. Solutions inverted by wavelet modelling, for receiver function at AFIF station, Arabian Shield (Juli`a et al. 2003). Red lines indicate the model inferred by the joint inversion of receiver function and surface wave dispersion (left-hand panel) and the observed receiver function (right-hand panel). Black lines indicate the inversion models (left-hand panel) and the correspondent synthetic receiver functions (right-hand panel). All of the models are kept. Note that all of the solutions approximate to those inferred by joint inversion of receiver function and surface wave dispersion.
data at the finer scale, but do not fit the data at coarse scales, and vice versa. It shows that traditional least-squares inversion emphasizes high-frequency over low-frequency information. In contrast, wavelet modelling of broad-band receiver function deals with unequal scales separately, thus the average background velocity and sharp velocity variations can be resolved by large- and small-scale receiver functions, respectively. The sharp velocity change across an interface is smeared out to a broad gradient zone at large scales, and this broad zone is gradually tuned to the sharp features at the small scales. Thus, unlike traditional least-squares method, wavelet modelling can make all of the initial models focus nearby the true velocity distribution. We also use low-pass filtering method commonly used in waveform fitting to compare with our wavelet inversion method. The receiver function is low-pass filtered by the Gaussian filter with parameters of 0.5, 1.0, 1.5, 2.0 and 2.5 respectively. We first invert the receiver function within the lowest frequency band filtered by the Gaussian parameter of 0.5 to get coarser models, the coarser modes are then iteratively used as the initial models for fitting the receiver function filtered by the Gaussian parameter of 1.0, up to 2.5, respectively. We only show the inversion results for the receiver function with Gaussian parameter of 0.5 and 2.5 in Fig. 6. We should point out that in each inversion with increasing frequency band, we discard those models without good waveform match, and the final models converge to two groups as shown in Fig. 6. It demonstrates that low-pass filtering and fitting method perform superior to traditional linearized least-squares method, but inferior to wavelet modelling. The final test of the wavelet modelling method is an application to real data. We use the receiver function data measured at AFIF station, on the Arabian Shield (Juli`a et al. 2003). Juli`a et al. (2003) have published the results of a joint inversion of the receiver function and surface wave dispersion, and their model is compatible with the P-wave model of Mooney et al. (1985). C
We take two reference models, the P-wave model of Mooney et al. (1985), and the S-wave model of Juli`a et al. (2003). Taking the same perturbation approach as the above, each reference model produces 36 initial models. The final results of the wavelet modelling are shown in Fig. 7. In spite of the uncertainty of the half-space velocity, the wavelet modelling result is very close to that of the joint inversion of the receiver function and surface wave dispersion. It shows that, even without surface wave dispersion constraints, wavelet modelling of broad-band receiver functions can reveal models similar to that obtained by joint inversion. We note there is a minor difference between the models inferred by wavelet modelling and classic least-square methods, especially in the lower crust and uppermost mantle. Despite the fact that noise-free synthetic receiver function inversions can guide the initial models spanning a wide range to converge to real model, for noise-contaminated data, the wavelet filtering and inversion may be simultaneously applied to receiver function, and need to be further studied. Moreover, independent information, such as surface wave dispersion, can put further constraint on crustal and upper-mantle structure. 6 D I S C U S S I O N A N D C O N C LU S I O N We put forward a new wavelet modelling method to invert receiver function. Through comparison with traditional linerized leastsquares method and low-pass filtering and fitting method, we note that for a given initial model space spanning a wide range, traditional linearized least-squares method can’t make all of the initial models converge to the truce velocity distribution, if discarding those unreasonable models with unexpected waveform match, the remaining models with good waveform fitting converge to three clusters; for low-pass filtering and fitting method, after discarding those unreasonable models with bad waveform match at each frequency band,
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
Wavelet modelling of receiver functions the final models converge to two group; in contrast, our new method can guide all of the initial models to converge to the true velocity distribution. Wavelet representations of broad-band receiver functions can be regarded as the their bandpassed versions. The fifth-scale receiver function has a very simple waveform consisting predominantly of low frequencies, effectively combining the effect of several converted and multiple phases into a few simple, broad and pulses. From the fourth-scale to the first-scale, the receiver function becomes more and more intricate and the broader pulses are gradually decomposed into individual phases, approximating to the broad-band receiver function in increasing detail. Thus, the wavelet decomposition of receiver functions provides us with an effective approach to gradually focus on individual seismic phases, from the simple to complicated, and it may help us to resolve detailed structure under scale meaning. Unlike traditional least-squares inversion method, wavelet modelling of broad-band receiver function deals with unequal scales separately, thus the average background velocity and sharp velocity variations can be resolved by large- and small-scale receiver functions, respectively. By projecting the solution onto a subspace of increasing resolution, and zooming in from the coarse to the fine scale, wavelet modelling of receiver function can converge to global misfit minimum by successive refinement, without getting trapped in local minimum too early. AC K N OW L E D G M E N T S We thank Dr J. Juli`a for providing us their receiver function data and velocity models at AFIF station, Arabian Shield. We are also grateful to Dr J. Juli`a for reviewing the manuscript and providing helpful suggestions. We greatly appreciate many constructive comments and detailed modification by two anonymous reviewers and the Editor, Dr Bruce Buffett, for improving the manuscript. This work is supported by the National Science Foundation of China under contracts 40274029 and 40574040.
REFERENCES Ammon, C.G., Randall, G.E. & Zandt, G., 1990. On the nonuniqueness of receiver function inversions, J. Geophys. Res., 95, 15 303–15 318. Ammon, C.J., 1991. The isolation of receiver effects from teleseismic P waveforms, Bull. Sesimol. Soc. Am., 81, 2 504–2 502. Bhatia, M., Karl, W.C. & Willsky, A.S., 1994. A wavelet-based method for multi-scale tomographic reconstruction: MIT Tech. Rep. LIDS-P2182. Bosman, C. & Reiter, E., 1993. Seismic data compression using wavelet transforms: 63th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp. 1261–1264. Bostock, M.G. & Sacchi, M.D., 1997. Deconvolution of teleseismic recordings for mantle structure, Geophys. J. Int., 129, 143–152. Burdick, L.J. & Langston, C.A., 1977. Modeling crustal-structure through the use of converted phases in teleseismic body-waveforms, Bull. Seismol. Soc. Am., 67, 677–691. Cassidy, J.F., 1992. Numerical experiments in broadband receiver function analysis, Bull. Seismol. Soc. Am., 82, 1 453–1 474. Chui, C.K., 1992. An Introduction to Wavelets, Academic Press, New York. Clayton, R.W. & Wiggins, R.A., 1976. Source shape estimation and deconvolution of teleseismic body waves, Geophys. J. R. Astron. Soc., 47, 151–177. Constable, S.C., Parker, R.L. & Constable, C.G., 1987. Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data, Geophysics, 52, 289–300. C
543
Daubechies, I., 1992. Ten Lectures on Wavelets, SIAM Press, Philadelphia. Dueker, K.D. & Sheehan, A.F., 1997. Mantle discontinuity structure form midpoint stacks of converted P to S wave across the Yellow-stone hotspot tracks, J. Geophys. Res., 102, 8 313–8 327. Goupillaud, P., Grossmann, P. & Morlet, J., 1984/85. Cycle-octave and related transforms in seismic signal analysis, Geoexpl., 23, 85– 102. Grossman, A. & Morlet, J. 1984. Decomposition of Hardy function into square integral wavelets of constant shape, J. Math. Anal., 15, 723– 736. Gurrola, H., Baker, F.G. & Minster, J.B., 1995. Simultaneous time-domain deconvolution with application to the computation of receiver functions, Geophys. J. Int., 120, 537–543. Juli`a, J., Ammon, C.J., Herrmann, R.B. & Correig, A.M., 2000. Joint inversion of receiver function and surface wave dispersion observations, Geophys. J. Int., 143, 99–112. Juli`a, J., Ammon, C.J. & Herrmann, R.B., 2003. Lithospheric structure of the Arabian Shield from the joint inversion of receiver functions and surfacewave group velocities, Tectonophysics, 371, 1–21. Kennett, B.L.N., 1983. Seismic Wave Propagation In Stratified Media, Cambridge University Press, New York. Langston, C.A., 1979. Structure under Mount Rainier, Washington, inferred from teleseismic body waves, J. Geophys, Res., 84, 4 749–4 762. Last, R.J., Nyblade, A.A. & Langston, C.A., 1997. Crustal structure of the East African Plateau from receiver functions and Rayleigh wave phase velocities, J. Geophys. Res., 102, 24 469–24 483. Lebedev, S., Nolet, G., Meier, T. & van der Hilst, R.B., 2005. Automated multimode inversion of surface and S waveforms, Geophys. J. Int., 162, 951–964. Levin, V. & Park, J., 1997. Crustal anisotropy beneath the Ural mtns foredeep from teleseismic receiver functions, Geophys. Res. Lett., 24, 1283– 1286. Li, X.G., Sacchi, M.D. & Ulrych, T.J., 1996. Wavelet transform inversion with prior scale information, Geophysics, 61, 1 379–1 385. Ligorria, J.P. & Ammon, C. J., 1999. Iterative deconvolution and receiverfunction estimation, Bull. Seismol. Soc. Am., 89, 1 395–1 400. Ligorr´ıa, J.P., 2000. An investigation of the mantle-crust transition beneath North America and Poisson’s ratio of the North American crust, Ph.D. thesis, Saint Louis University, Saint Louis. Lilly, J.M. & Park, J., 1995. Multiwavelet spectral and polarization analyses of seismic records, Geophys. J. Int., 122, 1 001–1 021. Liu, Q.Y., Kind, R. & Li, S.C., 1996. The maximum likelihood estimate and nonlinear inversion of complex spectrum ratio for receiver function, Acta Geophys. Sin., 39, 502–513. Luo, Y. & Schuster, G.T., 1992. Wave packet transform and data compression, 62th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp. 1187–1190. Mallat, S.G., 1989. A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern anal. Machine intell., 11, 674–693. Meyer, Y., 1993. Wavelets: Algorithm and Applications, SIAM Press, Philadelphia. Mooney, W.D., Gettings, M.E., Blank, H.R. & Hearly, J.H., 1985. Saudi Arabian seismic-refraction profile: a traveltime interpretation of crustal and upper mantle structure, Tectonophysics, 111, 173–246. Owens, T.J., Zandt, G. & Taylor, S.R., 1984. Seismic evidence for an ancient rift beneath the Cumberland Plateau, Tennessee: A detailed analysis of broadband teleseismic P waveforms, J. Geophys. Res., 89, 7 783– 7 795. Owens, T.J., Taylor, S.R. & Zandt, G., 1987. Crustal structure at regional seismic test network station determined from inversion of broadband teleseismic P waveforms, Bull. Seismol. Soc. Am., 77, 631–632. Park, J. & Levin, V., 2000. Receiver functions from Multiple-Taper Spectral Correlation estimates, J. Geophys. Res., 103, 26 899–26 917. Randall, G.E.,1989. Efficient calculation of differential seismograms for lithospheirc receiver functions, Geophys. J. Int., 99, 469–481. Phinney, R.A., 1964. Structure of the Earth’s crust from spectral behavior of long-period body waves, J. Geophys. Res., 69, 2 997–3 107.
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
544
Q. Wu et al.
Sandvol, E., Seber, D., Calvert, A. & Barazangi, M., 1998. Grid search modeling of receiver functions: Implications for crustal structure in the Middle East and North Africa, J. Geophys. Res., 103, 26 899–26 917. Savage, M.K., 1998. Lower crustal anisotropy or dipping boundaries? Effects on receiver functions and a case study in New Zealand, J. Geophys. Res., 103, 15 069–15 087. Schuster, G.T. & Sun, Y., 1993. Wavelet filtering of tube and surface
C
waves: 63th Ann. Internat. Mtg., Soc. Expl. Geophys. Expanded Abstracts, pp. 25–28. Shaw, P.R. & Orcutt, J.A., 1985. Waveform inversion of seismic refraction data and applications to young Pacific crust, Geophys. J. R. Astron. Soc., 82, 375–414. Yomogida, K., 1994. Detection of anomalous seismic phases by the wavelet transform, Geophys. J. Int., 116, 119–130.
2007 Institute of Geophysics, China Earthquake Administration, GJI, 170, 534–544 C 2007 RAS Journal compilation
