Plane Waves at Small Arrays

Abstract

The resolving power of a seismic array is defined in terms of the array response function and via the classical uncertainty principle. Using the theory of maximum likelihood wavenumber spectra (Capon, 1969), we show for the case of two correlated plane waves that arbitrarily high resolution is achievable in the limit as the background white noise tends to zero. This extends Barnard’s (1969) result to the case of correlated plane waves. The increased resolution arises from the additional assumption that the data are plane waves over all space, and not zero off the array as the classical result assumes. It is found that a sample rate (in time) large compared to the Nyquist rate, is needed in the case of a short time gate at a small array. Cross‐power spectral matrices are estimated at 4 hz from 1 sec of computer generated data consisting of two correlated plane waves in white noise. These spectral matrices are then used to generate maximum likelihood wavenumber spectra. The two plane waves are resolved at various signal‐to‐noise ratios and at correlations up to ρ=0.8. The need for using a high sampling rate is demonstrated. Results are compared with conventional wavenumber spectra, where the classical resolution results hold. The use of a 1‐sec window provides improved resolution of the wavenumber structure as it changes in time, resulting in better separation of any time‐overlapping phases and multipathed waves that arise from one event.

Array performances for ambient vibrations on a shallow structure and consequences over V s inversion

“…The high resolution frequency-wavenumber technique is theoretically able to distinguish two waves travelling at close wavenumbers in a better way than the f-k method (Capon [1969], Horike [1985]). Empirical results (Woods and Lintz [1973]) show that the resolving power of the high-resolution method is three to six times greater than the one of the f-k method. However, as outlined by Asten and Henstridge [1984], the performance of the method is very dependent upon signal to noise ratio and array design.…”

“…More recently, Kind et al [2005] used the common rules of thumb to quantify the low frequency limit of the deduced dispersion curve, and a manual interpretation to identify the aliasing limits. After Woods and Lintz [1973], the resolving power of an array depends not only on the diameter of the array but also on the spatial distribution of the sensors and on the correlation between the events to be resolved. They proposed to estimate this resolving power by using the theoretical array response function.…”

“…They proposed to estimate this resolving power by using the theoretical array response function. This latter function, which we also use for defining two wavenumber limits k min and k max , is given in the (k x , k y ) plane by (Woods and Lintz [1973], Asten and Henstridge [1984]):…”

“…Resolution and aliasing limits are then directly derived from the R th map. Following Woods and Lintz [1973], the resolution limit (k min /2) is taken as the radius of the central peak of R th measured at the mid-height (0.5). We define the aliasing limit (k max ) as the lowest k value (greater than k min /2) obtained at the intersection of R th with the horizontal line at 0.5, considering all directions.…”

Estimations of the S-wave velocity structures in Chia-Yi City, Taiwan, using the array records of microtremors

“…This first is the frequency domain Beam-Forming Method (BFM) developed by Lacoss et al (1969), while the second is the Maximum Likelihood Method (MLM) developed by Capon (1969). According to a number of researchers (Capon, 1969;Mack and Filnn, 1971;Woods and Lintz, 1973;Huang and Yeh, 1990), the resolving power of the MLM is higher than that of the BFM; however, these two methods do agree in the case of perfectly uncorrected array data.…”