Theoretical array response

The theoretical frequency-wavenumber response of an array is a semblance map that would have been obtained for a single vertically incident plane wave ( $(k_x^{(1)},k_y^{(1)})$ equal to $(0,0)$ in equation 1.2). It is also called the array transfer function because the array output is the convolution of the wavefield and of the theoretical frequency-wavenumber response. The normalized theoretical array response in the ($k_x$ , $k_y$ ) plane is given by

$$R_{th}(k_x,k_y)=\frac{1}{n^2}\left\vert \sum_{i=1}^n e^{-j(k_x x_i+k_y y_i)}\right\vert^2$$ (1.1)

where n is the number of sensors in the array, and $(x_i,y_i)$ are their coordinates. For one single plane wave $S_i(f)=A(f)e^{j(x_i k_x^{(1)}+y_i k_y^{(1)}-2\pi f t+\phi)}$ crossing the array at wavenumber $(k_x^{(1)},k_y^{(1)})$ an at frequency $f$ , recorded at sensor $i$ , at time $t$ and with a phase $\phi$ , the array output is

$$R(k_x,k_y,f)=\left\vert \sum_{i=1}^n S_i(f) e^{-j(k_x x_i+k_y y_i)}\right\vert^2 = n^2A^2(f)R_{th}(k_x-k_x^{(1)},k_y-k_y^{(1)})$$ (1.2)

where $A(f)$ is the amplitude spectrum. The array output is equal to the theoretical response translated by vector $(k_x^{(1)},k_y^{(1)})$ and multiplied by the square of the amplitude. For multiple plane waves travelling across the array, $S^{(1)}$ to $S^{(m)}$ , the array output is

$$R(k_x,k_y,f)=\left\vert \sum_{i=1}^n \left(\sum_{l=1}^m S_i^{(l)}... ...t) e^{-j(k_x x_i+k_y y_i)}\right\vert^2 \leq n^2\sum_{l=1}^m R^{(l)}(k_x,k_y,f)$$ (1.3)

where $R^{(l)}$ are the array outputs for single plane waves defined by equation (1.2), and $S_i^{(l)}$ the wave $l$ recorded at station $i$ . In this case, the array output is always lower than the sum of translated theoretical responses, the maximum being reached when all waves are in phase.

Figure 1.1: Theoretical array responses for 25 sensors. Array geometries: (a) circle, (d) Cartesian grid, and (g) spiral. (b), (e), and (h) Theoretical array responses in the plane $(k_x,k_y)$ . (c), (f), and (i) Sections across theoretical array responses for various propagation azimuths (628 values between 0 and $2\pi$ ).

Figure 1.2: Theoretical array responses for 10 sensors. (a), (d), and (g) Array geometries. (b), (e), and (h) Theoretical array responses in the plane $(k_x,k_y)$ . (c), (f), and (i) Sections across theoretical array responses for various propagation azimuths (628 values between 0 and $2\pi$ ).

From equation (1.1), $R_{th}$ always exhibits a central peak the value of which is one ($k_x$ and $k_y = 0$ ) and lateral aliasing peaks the amplitude of which is less than one. Beyond a certain limit which is called the theoretical aliasing wavenumber, this pattern is repeated due to the periodic nature of $e^{jx}$ . Below this theoretical limit, equation (1.2) shows that the position of the highest peak of the array output is directly linked to the apparent velocity and the azimuth of the propagating wave. For a complex wavefield described by equation (1.3), and assuming that all contributing waves are in phase to get equality, aliasing is likely to occur for lower wavenumbers due to the summation of the lateral peaks of $R_{th}$ . Hence, $R_{th}$ is of prime importance to define the potential aliasing limits ($k_{max}$ ) of the chosen array geometry. On the other hand, it is obvious that the thinner the central peak is, the more capable is the array to distinguish two waves travelling at close wavenumbers. The resolution limit ($k_{min}$ ) is controlled by the width of the central peak. For simple array geometries, for instance a cartesian grid, $k_{min}$ and $k_{max}$ are linked to the minimum and maximum distance between sensors. For usual irregular array geometries, $R_{th}$ is necessary for the definition of objective wavenumber limits.

We define practical rules for the aliasing and resolution limits from $R_{th}$ , setting $k_{max}$ at the first peak exceeding 0.5 (or -3 dB) and $k_{min}$ being measured at the mid-height of the central peak (Gaffet (1998); Asten and Henstridge (1984); Woods and Lintz (1973)). If the aliasing peaks are less than the central peak, and if a single source is acting, $k_{max}$ does not effectively limit the power of the array. However for multiple sources, even if the aliasing peaks are less than the central peak, the superposition may create artefacts leading to the confusion of aliasing peaks with the main one. If the aliasing peaks are of the same order of magnitude as the main peak, the wavenumber limit is always $k_{max}/2$ . In a safe approach, it is better to limit the valid array output to $k_{max}/2$ in all cases. These rules are compared to frequency-wavenumber output in chapter 6.

Table 1.1: Properties of the array geometries. For each array, the minimum and maximum wavenumbers (rad/m) deduced from the theoretical frequency-wavenumber responses in figures 1.1 and 1.2.

Array geometry	Number of sensors	$k_{min}$	$k_{max}$
Perfect circle	25	0.024	1.00
Cartesian grid	25	0.022	0.25
Spiral	25	0.036	2.75
Perfect circle	10	0.024	0.40
Three triangles	10	0.038	0.36
Irregular circle	10	0.026	0.15

The theoretical array response is calculated for various array geometries containing 25 and 10 sensors in figures 1.1 and 1.2, respectively: a perfect circle (figures 1.1(a) to 1.1(c), and 1.2(a) to 1.2(c)), a Cartesian grid (figures 1.1(d) to 1.1(f)), a perfect spiral (figures 1.1(g) to 1.1(i)), an ensemble of three triangles rotated by 40$^{\circ}$ (figures 1.2(d) to 1.2(f)), and an irregular circle (figures 1.2(g) to 1.2(i)). The aperture^1.1 is always around 100 m. The grey curve of plots (c), (f) and (i) are sections across the theoretical array response for various propagating azimuths. The $k_{min}$ and the $k_{max}$ are estimated in table 1.1. The width of the central peak at its mid height presents small variations versus the geometries. For instance, the perfect circle in figure 1.1(a) has an aperture of exactly 100 m and a $k_{min}$ around 0.024 rad/s. On the other hand, the spiral array in figure 1.1(g) has an aperture of 98.5 m and a $k_{min}$ around 0.036 rad/s. Hence, $k_{min}$ cannot be deduced from the aperture by a simple linear relationship. On another hand, $k_{max}$ is strongly dependent upon the number of sensors and their geometries. When multiple waves are travelling across the array at the same time, the performances of an array depend also upon the "ground" level of its theoretical array response. For example, the rectangular array is almost flat between 0.05 and 0.2 compared to the circular array, which means that two semblance peaks separated by $k_{min}$ are not affected by each other in the summation of equation (1.3).

The arrays of figure 1.2 with 10 sensors are more common than arrays with 25 sensors, but their available wavenumber range between $k_{min}$ and $k_{max}$ is usually not large enough to obtain a complete dispersion curve. Hence, various array apertures and geometries with overlapping wavenumber ranges must be planned before any experiment. It can be based on a first guess of the dispersion curve calculated with common properties for the expected geology. The wavenumber ranges must cover the whole dispersion curve down to the expected resonance frequency. This limit only applies to arrays for which the vertical components are processed. Extention towards lower frequencies might be necessary if horizontal components are planned to be processed (chapter 6).