| Geophysical methods are prompt, non-invasive and low-cost tool for quantitative delineation
of buried archaeological targets. However, taking into account the complexity of
geological-archaeological media, some unfavourable environments and known ambiguity of
geophysical data analysis, a single geophysical method examination might be insufficient
(Khesin and Eppelbaum, 1997). Besides this, it is well-known that the majority of
inverse-problem solutions in geophysics are ill-posed (e.g., Zhdanov, 2002), which means,
according to Hadamard (1902), that the solution does not exist, or is not unique, or is not a
continuous function of observed geophysical data (when small perturbations in the
observations will cause arbitrary mistakes in the solution). This fact has a wide application
for informational, probabilistic and wavelet methodologies in archaeological geophysics
(Eppelbaum, 2014a).
The goal of the modern geophysical data examination is to detect the geophysical
signatures of buried targets at noisy areas via the analysis of some physical parameters with a
minimal number of false alarms and miss-detections (Eppelbaum et al., 2011; Eppelbaum,
2014b). The proposed wavelet approach to recognition of archaeological targets
(AT) by the examination of geophysical method integration consists of advanced
processing of each geophysical method and nonconventional integration of different
geophysical methods between themselves. The recently developed technique of
diffusion clustering combined with the abovementioned wavelet methods was utilized
to integrate the geophysical data and detect existing irregularities. The approach
is based on the wavelet packet techniques applied as to the geophysical images
(or graphs) versus coordinates. For such an analysis may be utilized practically
all geophysical methods (magnetic, gravity, seismic, GPR, ERT, self-potential,
etc.).
On the first stage of the proposed investigation a few tens of typical physical-archaeological
models (PAM) (e.g., Eppelbaum et al., 2010; Eppelbaum, 2011) of the targets under study for
the concrete area (region) are developed. These PAM are composed on the basis of the
known archaeological and geological data, results of previous archaeogeophysical
investigations and 3D modeling of geophysical data. It should be underlined that the
PAMs must differ (by depth, size, shape and physical properties of AT as well as
peculiarities of the host archaeological-geological media). The PAMs must include also
noise components of different orders (corresponding to the archaeogeophysical
conditions of the area under study). The same models are computed and without the
AT.
Introducing complex PAMs (for example, situated in the vicinity of electric power lines,
some objects of infrastructure, etc. (Eppelbaum et al., 2001)) will reflect some real class of
AT occurring in such unfavorable for geophysical searching conditions. Anomalous effects
from such complex PAMs will significantly disturb the geophysical anomalies from AT and
impede the wavelet methodology employment. At the same time, the “self-learning”
procedure laid in this methodology will help further to recognize the AT even in the cases of
unfavorable S/N ratio.
Modern developments in the wavelet theory and data mining are utilized for
the analysis of the integrated data. Wavelet approach is applied for derivation of
enhanced (e.g., coherence portraits) and combined images of geophysical fields. The
modern methodologies based on the matching pursuit with wavelet packet dictionaries
enables to extract desired signals even from strongly noised data (Averbuch et al.,
2014).
Researchers usually met the problem of extraction of essential features from available
data contaminated by a random noise and by a non-relevant background (Averbuch et al.,
2014). If the essential structure of a signal consists of several sine waves then we may
represent it via trigonometric basis (Fourier analysis). In this case one can compare the
signal with a set of sinusoids and extract consistent ones. An indicator of presence a
wave in a signal f(t) is the Fourier coefficient /«
f(t) sinwt dt. Wavelet analysis
provides a rich library of waveforms available and fast, computationally efficient
procedures of representation of signals and of selection of relevant waveforms. The basic
assumption justifying an application of wavelet analysis is that the essential structure of a
signal analyzed consists of not a large number of various waveforms. The best way
to reveal this structure is representation of the signal by a set of basic elements
containing waveforms coherent to the signal. For structures of the signal coherent
to the basis, large coefficients are attributed to a few basic waveforms, whereas
we expect small coefficients for the noise and structures incoherent to all basic
waveforms.
Wavelets are a family of functions ranging from functions of arbitrary smoothness to
fractal ones. Wavelet procedure involves two aspects. The first one is a decomposition, i.e.
breaking up a signal to obtain the wavelet coefficients and the 2nd one is a reconstruction,
which consists of a reassembling the signal from coefficients
There are many modifications of the WA. Note, first of all, so-called Continuous WA in
whichsignal f(t) is tested for presence of waveforms Ï(t-b)
a. Here, a is scaling
parameter (dilation), bdetermines location of the wavelet Ï(t-b)
a in a signal f(t). The
integral
( )
/« t–b
(W Ïf) (b,a) = f (t) Ï a dt
is the Continuous Wavelet Transform.When parameters a,b in Ï( )
t-ab take some discrete
values, we have the Discrete Wavelet Transform. A general scheme of the Wavelet
Decomposition Tree is shown, for instance, in (Averbuch et al., 2014; Eppelbaum et al.,
2014).
The signal is compared with the testing signal on each scale. It is estimated wavelet
coefficients which enable to reconstruct the 1st approximation of the signal and details. On
the next level, wavelet transform is applied to the approximation. Then, we can
present A1 as A2 + D2, etc. So, if S – Signal, A – Approximation, D – Details,
then
S = A1 + D1 = A2 + D2 + D1 = A3 + D3 + D2 + D1.
Wavelet packet transform is applied to both low pass results (approximations) and high pass
results (Details).
For analyzing the geophysical data, we used a technique based on the algorithm to
characterize a geophysical image by a limited number of parameters (Eppelbaum et al.,
2012). This set of parameters serves as a signature of the image and is utilized for
discrimination of images (a) containing AT from the images (b) non-containing AT (let will
designate these images as N). The constructed algorithm consists of the following
main phases: (a) collection of the database, (b) characterization of geophysical
images, (c) and dimensionality reduction. Then, each image is characterized by
the histogram of the coherency directions (Alperovich et al., 2013). As a result
of the previous steps we obtain two sets: containing AT and N of the signatures
vectors for geophysical images. The obtained 3D set of the data representatives
can be used as a reference set for the classification of newly arriving geophysical
data.
The obtained data sets are reduced by embedding features vectors into the 3D Euclidean
space using the so-called diffusion map. This map enables to reveal the internal
structure of the datasets AT and N and to distinctly separate them. For this, a matrix of
the diffusion distances for the combined feature matrix F = FN /ª FC of size
60 x C is constructed (Coifman and Lafon, 2006; Averbuch et al., 2010). Then,
each row of the matrices FN and FC is projected onto three first eigenvectors of
the matrix D(F ). As a result, each data curve is represented by a 3D point in the
Euclidean space formed by eigenvectors of D(F ). The Euclidean distances between
these 3D points reflect the similarity of the data curves. The scattered projections
of the data curves onto the diffusion eigenvectors will be composed. Finally we
observe that as a result of the above operations we embedded the original data into
3-dimensional space where data related to the AT subsurface are well separated
from the N data. This 3D set of the data representatives can be used as a reference
set for the classification of newly arriving data. Geophysically it means a reliable
division of the studied areas for the AT-containing and not containing (N) these
objects.
Testing this methodology for delineation of archaeological cavities by magnetic and
gravity data analysis displayed an effectiveness of this approach.
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