Figure 1. WAM conception of target acquisition and engagement

Source Location and Tracking Capability of a Small Seismic Array

BEAMFORMING PRINCIPLES

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Warhead Deployment Phase

Figure 1. WAM conception of target acquisition and engagement.

explosions. The application of the Capon maxi-

beamformers as a function of SNR under the as-

mum-likelihood beamformer to teleseismic events

sumption that the spatial correlation matrix is

using the Large Aperture Seismic Array in Mon-

known exactly and the sources are ideal stationary

tana was first reported by Capon et al. (1967). In

plane waves. For two or more sources they show

1969 Capon derived the beamformer and noise

that the bias in the wavenumber estimate is a

suppression characteristics. Beamforming is also

function of the source separation and the degree of

applied to sonar problems where passive arrays of

coherence between the sources, the SNR, the rela-

hydrophones locate and track submarines and

tive source power, and the number of observation

surface vehicles (Hahn 1975). Aircraft tracking

array elements. When the spatial correlation ma-

with both acoustic and radar technologies and

trix must be estimated from a finite amount of

stellar mapping of distant radio-wave emissions

data, De Graaf and Johnson (1985) show how the

are other prominent areas of application for

maximum-likelihood estimator becomes increas-

beamforming technology.

ingly unstable as the data set is reduced.

The use of seismic and acoustic arrays to track

Knapp and Carter (1977) and Carter (1977) dis-

and locate targets of military interest dates from

cussed the impact of source motion on bias in

the Vietnam War. Beginning in the late 1960s,

bearing estimation from a two- or three-element

military efforts focused, with limited success, on

array using a time-delay estimation procedure.

locating artillery positions. In the later 1970s,

Gragg (1990) showed that there is a time depen-

targets such as surface vehicles were investi-

dence in the Doppler-shifted spectra observed by

gated (Cress 1976). Recently, bearing estimation of

a single receiver when an acoustic source has a

slow-moving military aircraft using acoustic and

velocity vector that does not pass through the

seismic array processing technologies has been

receiver location. He showed that this time depen-

actively investigated. Of particular note are the

dence makes the Doppler shift difficult to predict.

efforts of Nawab et al. (1985) and Lacoss et al.

Unless source motion effects can be corrected,

(1991), who have successfully tracked both air-

there will be a phase bias in the Fourier estimate of

borne and ground targets at distances of a few

the time series. The impact of the phase bias from

kilometers.

source motion on wavenumber estimates remains

Several researchers have investigated the im-

unquantified.

pact of nonideal source attributes on bearing esti-

In the following sections we discuss the

mates. De Graaf and Johnson (1985) discuss the

beamforming equations and explicitly define the

bias of Bartlett and Capon maximum-likelihood

frequency domain estimation process used to gen-

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