Processing approach

2.2.1 Seismic sensor array

Three-component broadband seismic array data is suitable for multidimensional signal processing techniques, where the size, shape, aperture of seismic antenna is designed in order to avoid spatial and temporal aliasing (Mars et al., 2004). We consider the scenario of an arbitrarily spaced array of triaxial sensors identically oriented and with identical instrument responses (Figure 2.3).

In order to explain the processing, we suppose P sources traveling from different directions, and arriving at the antenna with N triaxial sensors (P < N). Equation (2.1) represents wn(t) the signal recorded by one component of the the n-th sensor in the time-space domain.

Figure 2.3: Model of triaxial sensors array for one source. XYZ represent North, East and Vertical components, θ and φ are the azimuth and incidence angles

wn(t) =

∑

sp(t − τn,p) + bn(t); n = 0, 1, ..., N − 1 (2.1) where, s_p(t) is the p −th source signal, τ_n,pis the relative propagation time delay of the p −th source for the n−th sensor and b_n(t) represents the noise recorded of one component of the n−th sensor. The

noise b_n(t) is usually assumed to be uncorrelated with the sources and both temporally and spatially white with Gaussian variance σ_B². The corresponding relative propagation time delay is defined as

τ_n,p= d_n· u(θ_p, φ_p)

where, dnis the relative position vector of sensor n-th with respect to the first sensor located arbitrarely at (0,0,0) and u(θ_p, φ_p) is the slowness vector indicating the direction of the p-th source. The slowness vector is expressed in the equation (2.2), note that its module is the reciprocal of velocity vp of the uppermost layer beneath the array.

The wave number vector is related to the slowness vector and is defined as:

kp= 1

where λ_wis the wavelength for the frequency f_pand velocity v_p of the wavefield of the p-th source.

The frequency wave number (f-k) representation of equation (2.1) allows to express this signals as:

W_n( f_p) = respectively. The antenna output of the equation (2.1) for narrowband signals can be represented in matrix form as:

W(f) = A(θ , φ )S(f) + B(f) (2.4)

where, the N × P matrix A(f) is the “array response matrix” or “steering matrix” (Miron et al., 2006a) given in equation (2.5). The P sources are represented by a P × 1 matrix as:

S(f) = [S₁( f₁), ..., S_P( f_P)]^T

A(θ , φ ) = [a1(θ1, φ1), a2(θ2, φ2), .., aP(θP, φP)] (2.5) ap(θp, φp) = [1 exp(−2π j(d2· kp)) ... exp(−2π j(dN· kp))]^T

Note that the array manifold vector a_p(θ_p, φ_p) depends on the path direction of the signal through the wave number vector kp for frequency fp from the source p. Usually, A(f) is a full rank matrix assuming that the array manifolds a_p(θ_p, φ_p) with different path directions are independent. Using the group of Equations in (2.5) on Equations (2.3) and assuming that one-source wave plane impinging

the antenna with back-azimut θ an incidence angle φ at frequency f₀, the antenna output for a narrow band sense can be written as:

W ( f ) = matrix, it called manifold matrix which contains the phase delay information of the impinging signal to the array. In the same way, the final version of the model takes the following form in time domain as Equation (2.7):

w(t) = A(θ , φ )s(t) + b(t) (2.7)

The purpose of this work is to develop and test the MUSIC-3C algorithm and show that we obtain a more robust estimation of the azimuth angle θ and a reliable incidence angle φ related to the source depth determination. This implies the sensors spatial position are known.

2.2.2 Array Response Function

In this section, we examine the array response function (ARF) to an external plane wavefront (far field), in sensor outputs as a time-series waveform. The seismic antenna consists of a set of three component seismometers (isotropic sensor) located along the X-axis (Figure 2.3), in order to sam-ple the signal spatially at their relative positions. The ARF require knowledge on spatial parameters such as number of elements, inter-sensor distances in order to achieve a satisfactory performance.

The purpose of the array response function was to evaluate the sensitivity and resolution of seis-mic antenna recording wavefronts with different wavelengths. Classically, an antenna operates as a wavenumber filter, or wavelength filter. Assuming a plane wave propagating through the antenna with a back-azimuth θ and an incidence angle φ , the wavenumber vector (k) for this wave, in Cartesian coordinates, can be written as Equation (2.8) (Rost and Thomas, 2002):

u(θ , φ ) = 1

fk(θ , φ ) (2.8)

where f is the wavefront frequency. For an antenna composed by N seismometers, the wavefront s(t) recorded by the n-th seismometer with the relative position of rn, can be expressed as Equation (2.9):

w_n(t) = s(t − d_n· u) (2.9)

Consequently, the optimal beam of the N sensors in the antenna for any slowness u relative to slowness u₀can be represented by Equation (2.10):

b

The energy E of this beam can be computed by summing the square amplitude of bw(t). According to the Parseval theorem and Fourier shift properties, E can be defined as a function of waveforms k, using Equation 2.8, it can be written in the frequency domain as:

E(k − k₀) =

where k₀ is wevenumber related to k₀. Consequently, the array response function A(k) is expressed as:

In the case of surveying the array response function for the horizontal plane, the wavenumber should be equal k = (k_x, k_y). According to the spatial Nyquist wavenumber, a minimum of two grid points per wavelength are theoretically enough for the spatial sampling of the wavefront. For the WUBI antenna, with an X-north aperture of 321 m and a Y-east aperture of 315 m, the array response function for this antenna is shown in Figure 2.4. Analyzing this results gives the smallest wavenumber around 0.5 km⁻¹, and the maximum wavenumber without aliasing around 18 km⁻¹. With these characteristics, it is possible to discriminate the wavelengths associated with Ubinas Volcano. For example, for an explosive quake recorded by the WUBI antenna, the wavenumbers were identified as 5 km⁻¹ (f = 1.1 Hz; wave velocity, 1.4 km/s), or 6 km⁻¹ (f = 1.5 Hz; wave velocity, 1.5 km/s), which were in good

Figure 2.4: WUBI array response function A.