Measurement of interstation phase velocity by wavelet transformation

425 Earthq Sci (2009)22: 425−429 Doi: 10.1007/s11589-009-0425-3 Measurement of interstation phase velocity by wavelet transformation∗ Qingju Wu Xiufen Zheng Jiatie Pan Fengxue Zhang and Guangcheng Zhang Institute of Geophysics, China Earthquake Administration, Beijing 100081, China Abstract In this paper, we present wavelet transformation method to measure interstation phase velocity. We use Morlet wavelet function as mother wavelet to filter two seismograms at various period of interest, and correlate the wavelet filtered seismograms to form cross-correlogram. If both wavelet filtered signals are in phase at that period, the phase of the cross-correlogram is a minimum. Using 3-spline interpolation to transform cross-correlation matrix to a phase velocity verse period image, it is convenient for us to measure interstation phase velocity. Key words: dispesion, phase velocity, wavelet transform CLC number: P315.63 Document code: A 1 Introduction Analysis of surface wave dispersion information is a useful means to determine crust and upper mantle structure, this information include group arrival time, phase angle, and amplitude as a function of period. Since its introduction by Dziewonski et al (1969), frequency-time analysis has been widely used in seismology to measure surface wave dispersion. Determinations of surface wave dispersion have commonly done using either single-station or two-station methods, a number of different approaches have been developed for the measurement of group and phase velocities and attenuation (Kovach, 1978; Nakanishi, 1979; Taylor and Toksoz, 1982; Hwang and Mitchell, 1986). Single-station methods assume that either the initial phase of the earthquake source is known or its effects are small enough to be ignored when they are used to determine surface wave dispersion. However, Knopoff and Schwab (1968) have shown that corrections to these measurements are necessary whenever the seismic source is not oriented in either a purely horizontal or a purely vertical direction. Two-station methods avoid the necessity of knowing the earthquake source information, but they require ∗ Received 2 June 2009; accepted in revised form 12 July 2009; published 10 August 2009. Corresponding author. e-mail: that the two stations should lie on a common great circle path with the earthquakes. Bloch and Hales (1968) provided to pass both seismograms through a narrow bandpass digital filter and form cross-correlogram to measure interstation phase velocity. Landisman et al (1969) noticed that considerable reduction of the noise level and stabilization of measured phase velocity can be achieved by windowing a cross-correlogram, since the cross -correlation is an approximation of the interstation impulse response. Narrow band-pass cross-correlogram is a commonly used method to determine interstation phase velocity. A different approach to the problem of calculating the interstation phase velocity have been given by Taylor and Toksoz (1982) and Hwang and Mitchell (1986), which are based on the use of time domain and a frequency domain Wiener filters respectively. Either single-station method or two-station method is implemented by frequency-time analysis, Gabor-Heisenberg inequality demonstrates that there is a trade-off between time resolution and frequency resolution. In order to give frequency and time information equal accuracy for phase velocity measurement, we present wavelet transformation method to determine interstation phase velocity. In this paper, we firstly summarize the commonly used method to measure interstation phase velocity, then present wavelet analysis method, and give some synthetic and real data examples to show the application. 426 Earthq Sci (2009)22: 425−429 2 Interstation phasevelocity measurement Interstation phase velocity is calculated from the phase spectra of Green functions using the formula c( f ) = fΔ , f t0 + [ϕ ( f ) ± N ] (1) where f is the frequency, Δ is the interstation distance, t0 is the first time point of the Green function, and ϕ is the phase of the Green function in cycles, N is an integer. In order to reduce noise and multimode interference, and stabilize phase velocity measurement, narrow-band pass cross-correlogram is extensively used to measure interstation phase velocity. This method is to pass both seismograms through a narrow bandpass digital filter centered at group-delay time corresponding to the various periods of interest and form the cross product of the filtered seismograms, after time shifting. The average of the resultant time series is a maximum when the two signals are in phase. This process is approximately equivalent to time-variant filtering in that the limited portion of the seismograms used will eliminate the unwanted noise and non-least time path arrivals outside each window. A digital bandpass boxcar shaped filter is commonly used to do narrow band pass filtering, given by bt = sin 2 πht cos 2 πf 0t , πt (2) where 2h is the bandwidth and f0 is the center frequency. Because bt is symmetric, the filter introduces no phase shift. The practical situation requires that the length of the digital filter be finite, windowing should be applied to the infinite filter to produce finite digital filter, instead of merely truncating. The coefficients of the finite digital filter is usually taken to be sin(2πhndt ) ⋅ πnΔt πn cos(2 πf 0 ndt ) cos( ). 2N Cn = (3) Where dt is the sampling rate, both n and N are positive integer. Noted from the above formula, a cosin window is used to windowing data. 3 Wavelet transform Wavelet analysis, also known as multi-resolution analysis, uses a series of dilated and delayed oscillatory functions to decompose a time-varying signal into its nonstationary spectral components. There are two key advantages of wavelet analysis over traditional Fouries analysis and windowed Fourier methods, one is that the wavelet analysis retains information on how the spectral content varies with time delay, and another is that the accuracy of the time and frequency remain constant over the entire time-frequency domain. We present wavelet analysis to measure interstation phase velocity. In our method, we use a non-orthonormal Morlet wavelet g(t), composed of a harmonic wave modulated by a Gaussian envelope, as mother function g (t ) = exp( − t2 i 2πt )exp( ), 2 2σ T0 (4) where T0 is the central period. The mother wavelet is dilated and delayed to produce a set of daughter wavelet according to g( ⎡ i2π(t − τ ) ⎤ ⎡ − (t − τ ) 2 ⎤ t−τ exp⎢ ） = exp ⎢ ⎥, 2 ⎥ α ⎣ 2(σα ) ⎦ ⎣ αT0 ⎦ (5) where α is the dilating scale, and τ is the time delay. The central frequency of the daughter wavelet is thus α/T0. The wavelet transform of a signal is obtained by integrating the time varying signal S(t) over the daughter wavelet as a function of dilating scale α and time delay τ. W (α , τ ) = 1 ∞ t−τ ∫ g ( α )S (t )dt , α −∞ * (6) where g* is the complex conjugate of the daughter wa (...truncated)