1The full recovery of magnetic structures from potential anomalies measured at the surface is an ill-conditioned inverse problem (Tarantola, 198;, Parker, 1994; Blakely, 1995). Any practical method has to be limited to the estimation of a few parameters. The purpose of this paper is to introduce a new method of tomography for potential field, that we call Multi-Scale Tomography, where sources are not assumed to be homogeneous, but complex. The method is quite general; here it is applied to magnetic fields. The method allows us to obtain estimates of spatial location, depth, inclination and “effective degree” of buried structures from above-ground measurements of the total magnetic field at a fixed height from a cesium vapor magnetometer (Saracco et al., 2007).
2Before giving a rough description of the method, we recall a few basic points:
A stationary magnetic field is the gradient of a scalar magnetic potential. This potential satisfies the Poisson equation, which becomes the Laplace equation in regions without sources (i.e. in the half-space z>0 in which we take measures).
If we know the potential in any plane with z=z0 and z0>0, we can calculate the potential in the whole half-space z>0, without making any assumption about the source. This is done by taking the convolution product of the known potential measured at a fixed height z0 by the dilated Poisson kernel, where the dilation operator acts on the depth. The calculation of the potential field in the whole half-space z>0 is analogous to the continuous wavelet transform of the potential measured at the fixed height z0, where the translation operator acts on the horizontal space (x,y), while the dilation parameter represents here, |z-z0|, the depth or height. The continuous wavelet theory associated with potential theory can be used.
3The Multi-Scale Tomography (MST) method developed here, is based on the dilation property of a complex Poisson kernel (solution of the Laplace equation for complex potential fields) (Saracco et al., 2004, 2007) and on the properties of the wavelet theory which analyses local singularities (Grossmann et al., 1987: Saracco et al., 1990; Moreau et al., 1997). The presence of a cone line structure of the potential field anomalies is a necessary condition to estimate the depth and other parameters of arbitrary sources (inclination, degree of structure), that the Poisson kernel be complex or real (Saracco et al., 2004, 2007). If the sources are homogeneous tempered distributions, the “effective degree” is then the “homogeneous degree” defined in Grossmann et al. 1987, Moreau et al. 1997, 1999 or in Euler Method. The fact that many magnetic sources can be considered in first approximation, as magnetic dipoles, makes it interesting to define an extension of Patella method, the Complex Dipolar Occurrence Tomography (CDOT) (Saracco et al., 2004). This method is based on phase and modulus, as the complex continuous wavelet transform. The CDOT method allows us to obtain rapid information about the position and inclination of buried dipolar archaeological objects.
4We recall briefly the geomagnetic field equations, and present the map of magnetic anomalies obtained from the magnetic survey at Fox-Amphoux (Fig. 1) conducted on cultivated fields. We applied both methods, conjointly to an electrical resistivity tomography. The estimates are then compared to excavation results.
Figure 1: 2D map of magnetic anomalies (Fox-Amphoux, Var, France).
5In absence of external sources (external current density due to solar wind, solar cycles…, are negligible) and in the quasi-static limit, the equation for magnetic induction B of declination D, and inclination I, is: ▽∧B=0. It follows that B= -▽Φ, where Φ is the scalar magnetic potential. Moreover B is the sum of the ambient magnetic field H, modulated by the diurnal variations, and the total magnetization M (induced Mi, plus remanent Mr magnetization), modulated by ambient noise and topographic effects. Mi is parallel and proportional to H, while Mr depends on the ferromagnetic composition of the material and H at the original time. Let ε be the set of fluctuations we have:
6B=µo(H+M)+ε=µo(1+K)H+ε, where µo is the vacuum permeability, K is the magnetic susceptibility of the ground. Mr is here principally due to the natural thermoremanent magnetization acquired during the cooling of archaeological structures to a temperature below the Curie temperature.
7In our experiments, the magnetic data were obtained with a cesium vapor probe with a sensitivity of 0.01nT, where Mr was 20-50 times higher than Mi.The present magnetic field given by the IGRF at site coordinates was 45983nT with D≈0.14o and I≈59o. Data acquisition was performed with a magnetic receiver located at 0.8m above ground, every 1m along profiles oriented North 145°at a distance of 2.5m of each other. The temporal drift of the magnetic intensity was 4nT/hr, and the error due to moving position of the receiver during data acquisition was less than 5nT. A linear correction for diurnal variations was made. Three magnetic anomalies corresponding Roman kilns are detected by the probe on a cultivated field of 70x110m. Results are mapped Fig. 1. A kringing extrapolation and a linear correction for diurnal variations were made.
8Assuming that the volume of a Roman kiln is a cube of edge r1, and the magnetization is induced in the present-day field, the magnetic anomaly Φ due to a dipolar source of susceptibility K, located in an homogeneous medium of susceptibility k0, is Φ=(k-k0)F(r1/r)3, where F is the intensity of the total magnetic field and r the distance between the probe and the center of the dipole. A rough estimation of the mean depth of structures can be obtain, considering a fixed distance r1 (1<r1<1.5m), F=45983nT and K=(k-k0)=10-2SI. The mean depths corrected of the height of the probe are respectively for the three anomalies 0.54, 0.82 and 1m. This empirical method easily used in first approximation to estimate the depth of simple structures, requires some ‘a priori’ information on the size of object due to the non-uniqueness of the solution. To reduce these limitations, it is necessary to formulate this problem in terms of an inverse problem in potential theory.
9The magnetic field measured at the altitude z>0 (see Section 2) and generated by a buried source σ(x,z), (x=(x,y)), located in z<0, satisfies: B(x,z)=-▽Φ(x,z), Φ denoting the scalar potential. We have:
10For z>0, ∆Φ(x,z) = 0; For z=0, Φ(x,0) =Φmes(x); For z<0, ∆Φ(x,z)=-σ(x,z).
11It follows: (1): Φ(x,z)=(DzP*Φmes)(x),
12Φ is called the harmonic extension of Φmes (the measured magnetic potential at a fixed z0) in z>0.
13* represents the convolution product; D the dilation operator acting on the depth z: DzP(x)=(1/z)P(x/z).
14P, the Poisson Kernel, is defined in R2, P(x)=Cst(1+|x|2)-5/2 (Courant & Hilbert, 1990). The Fourier transform of P is (2): P(u)=exp(-2π|u|), where u is the dual variable of x. P satisfies the crucial equation (3): DzP*Dz’P=Dz+z’P, which can be easily verified in Fourier space. This property is the starting point on the method and in (Moreau et al., 1997), on the use of wavelet theory in the study of potential fields.
15Since the wavelet transform L(b,a) of a signal F∈Rn is defined as the scalar product between F and the dilated and translated analyzing wavelet we have
16(4): L(b,a)F=a-n∫g[(x-b)/a]F(x)dxn. The translation parameter has the dimension of the space variable x, while the dilation parameter (a>0) is dimensionless and plays the role of a zoom in the frequency space (kx). Small dilations are related to high wave numbers and large dilations to low wave numbers (Saracco, 1994) We have analogous formulas between the calculation (1) of the magnetic potential Φ(x,z) in the half-space z>0, and the wavelet transform of F (4), where dilations in (1) are analogous to the depth (Saracco et al., 1990, 2007). The analyzing wavelet g well localized in space and frequency domain, must verifies some admissibility conditions (Saracco, 1994).
17The presence of dilation operator in both formulas allows us to use wavelet theory to characterize and localize buried structures generating magnetic anomalies above ground (Saracco et al., 2007).
18A characterization of the local regularity of a signal can be obtained from the modulus of the CMST (complex MST) along the lines of constant phase or from the lines of extrema extracted from the RMST (real MST) (Saracco et al., 2004, 2007). Singularities or structures are generally given by the local Lipschitz exponents (Grossmann, 1986). The exponent, or degree of homogeneity is estimated from the evolution across scales of the wavelet transform in log-log representation and characterizes the local structures of s. The source depth is estimated from the intersection of lines of extrema (Moreau et al., 1997), or of constant phase (Grossmann et al., 1987; Saracco, 1994). Inclination is estimated from the phase. If s(p) is a tempered distribution in Rn, ∀λ in R, s is homogeneous of degree α, if s verifies s(λp)=λnαs(p)⇒L(b,a)s=λαn L(b/a,1)s.
19Let β be the degree of the magnetic field B which differs by one degree from the potential field Φ:
20B= −▽Φ and ∆Φ(x, z)=-α(x, z) ⇒β=α+1. It follows (Saracco et al 2007): L(x,z)Φ=(a/a+z)γ−2−αL[x(a+z)/a,1]Φ. Accordingly, β=1-γ+α (γ is the degree of the complex wavelet gγdefined from P(u) (2)). gγ(u)= P(u)+iTH[P(u)], (TH: Hilbert transform). We obtain gγ=uγ-1P(u)(i2πγ(u+i|u|), (γ=2).
21To obtain a better estimation of depths, magnetic data were interpolated with a sample rate of 0.25m. In order to localize and characterize buried sources σ in the earth (z<0), we use the property of harmonic extension of the solution Φ(x,z) (1), and the dilation property (3). It is not necessary to use all the information of the half-plane z>0, that is, all wavelet coefficients L(x,z)Φ. Only the restriction of the wavelet coefficients along the lines of extrema presenting a cone line structure is necessary. If the lines of extrema do not present a cone-line structure no solution exists for the localization. The convergence of the lines (extrema or constant phase) through the intersection point zs<0 gives the depth of magnetic buried sources. This extension or extrapolation is similar to a contraction D-z in the half-plane z < 0. The degree α of buried structures can be extracted from the slope β of lines of extrema in log–log representation (Fig. 2, top-right)). The phase (Fig. 2, middle-right) gives an estimation of the local structure (or inclination). Moreover we obtained information on the localization of lithological structure (limestone substratum), and on archeological fragment objects of the same century (fragments of kiln, tiles, potteries, etc.).
Figure 2: Complex Multi-Scale Tomography (CMST), Characterization and depth anomaly T2.
22Excavations were made at Fox-Amphoux (Var) by archeologists (Fig. 3). They found Roman kilns on magnetic anomalies locations (Cf. Fig.1). The dimensions of kilns were: 2m wide by 2–3m high. The depths found were in agreement with results obtained from MST. Only the top of anomaly T3 was excavated. The strong and wide anomaly observed in this place using both electrical and magnetic surveys is due to the presence of a depot of blows. The kiln (anomaly T1) was partially broken down, and was discovered after the first 0.7m of clearing, corresponding to the estimated depth values. Each structure possesses its own magnetic field, because the overheated earth has a different magnetization from the natural magnetic environment rock. Potteries (overheated earth) contain magnetic minerals, principally iron oxides, responsible for remanent magnetization. When the temperature reaches the Curie or Neel temperature, during the firing this remanent magnetization disappears. During the cooling below this critical temperature, a new remanent magnetization is acquired, guided by the surrounding magnetic field of blows. This new remanent magnetization is then a record of the field during the cooling. T2 and T3 anomalies are in agreement with the depth obtained respectively, by archaeologists after excavation (0.9 and 1.2 m) and (1.3-1.5 m).
Figure 3: Excavation of anomaly T2.
We want to thank particularly J.-C. Michel (Archaeologist, Toulon) for his help and collaboration.