ap_1997_4-4b

Quantitative Interpretation of MagneticData over Settlement Structures byInverse Modelling

G. DITTRICH{ AND U. KOPPELTUniversitat Leipzig, Institut fur Geophysiki, Talstrasse 35, 04103 Leipzig, Germany

ABSTRACT A robust and stable inversion algorithm for the joint reconstruction of different archaeologicalfeatures, such as pit houses, ditch systems or single pits, from surface magnetic data wasdeveloped. In a first step a simplified model is assumed to estimate the mean magnetizationdirection. During the second step the shape of the features is estimated automatically. Thealgorithm is based upon an evolutionary strategy and the Marquardt Levenberg method. It wastested on real and synthetic data. The influence of inhomogeneous pit fillings on depth estimateswas studied. *c 1997 John Wiley & Sons, Ltd.

Archaeol. Prospect. 4: 165–177, 1997.

Key words: archaeometric prospecting; magnetic survey; inverse modelling; evolutionarystrategies.

Introduction

Correct information about the shape of archaeo-logical features can be obtained by excavationsonly. However, once an archaeological featureis excavated, it is destroyed. Therefore, non-destructive investigation methods that allow anestimation of the approximate shape of archaeo-logical features become more and more import-ant. Magnetic surveying has proved to be aneffective tool to locate archaeological features(Scollar et al, 1990; Becker, 1993), and to estimateshape and physical parameters of particularfeatures on site (Boucher, 1996; Eder-Hinterleit-ner et al, 1996).

Once a settlement, consisting for instance of pithouses, wells, clay-producing pits and a defenseditch system around it, was abandoned, differ-ences in the surface topography become levelledout due to wind and water induced erosion. Thusmost of the clinal structures become filled, mainly

with topsoil material. Usually this material hasa higher susceptibility than the soil-forminggeological minerals placed in the environment(LeBorgne, 1955; Thompson and Oldfield, 1986,p. 72). Due to different processes, such as heatingor the influence of magnetic bacteria, the fillingmaterial may carry remanent magnetization(Faûbinder et al, 1990) of the same order ofmagnitude as the induced part.

Disregarding particular features with a strongremanent magnetization, e.g. ovens, iron objectsor slag bodies, one expects the overall appearanceof a settlement to behave like a layer withpositive magnetization contrast with respect tothe underlying half space. Therefore a layer withan undulating base and an unknown but con-stant magnetization was chosen as the physico-archaeological model (PhAM) of the site inquestion. As different features are not necessarilyconnected the layer may vanish in the spacebetween them, or alternatively, a zero thicknessmay be assigned.

The inverse problem for this class of modelsis non-linear, because it falls into the class of

{Correspondence to: G. Dittrich, UniversitaÈt Leipzig, Tal-strasse 35, 04103 Leipzig, Germany.

CCC 1075±2196/97/040165±13$17.50 Received 30 September 1997# 1997 John Wiley & Sons, Ltd. Accepted 26 January 1998

Archaeological Prospection, Vol. 4, 165±177 (1997)

shape-determining problems (Blakely, 1995,p. 228). It is unique in the case of a depth func-tion with limited spectrum width, i.e. a suffi-ciently smooth undulation, but is unstablebecause of noise corrupted data.

A forward modelling algorithm for a magnet-ized undulating layer was developed by Parker(1972). Based on Parker's approach many differ-ent inversion schemes for estimating the depth ofan undulating interface as a function of positionwere developed (Oldenburg, 1974; Xia andSprowl, 1992), and therefore, the problem iswell understood today. All of these proceduressuffer from the drawback of utilizing analyticalspectral expressions for the depth function whileperforming discrete numerical modelling. For adiscussion see Koppelt and Rojas (1994).

A different model, which consists of a two-dimensional regular grid of columns equi-distantly filled with dipoles from the earthsurface down to the depth of the layer, wasused by Eder-Hinterleitner et al (1996). Theforward problem was solved by calculating theeffect of each dipole and summing up theindividual effects, while the inversion was con-trolled by a simulated anealing algorithm. Aweighted least-squares criterion with regular-ization was applied to overcome the instabilityintroduced by measurement errors and highfrequency content of the depth function spectrumdue to steeply striking side walls of the ditches.

The main drawback of this particular numeri-cal approach is the modelling of continuouslydistributed material by dipole sources situated ineach column. This is equivalent to numericalintegration of the formula for the magneticanomaly due to a vertical prism using an openone-point Newton±Cotes formula for bothhorizontal integrations, and an extended trape-zoidal rule for the integration along the verticalaxis (Bezvoda et al, 1992). In this case, differentweights should be assigned to the dipoles atboth ends of every column, as follows from theextended trapezoidal rule (see Press et al, 1990,p. 116). Therefore, the forward modelling isnot correct. Moreover, because the distancebetween dipoles is comparable to the distancebetween observation points and dipoles,discretization effects may affect computed data(see Ku, 1977).

These problems were overcome by theapproach of Herwanger et al (1997). Theseauthors used a two-dimensional regular grid ofvertical prisms with varying bottom depth tomodel the layer. The depth estimations werecomputed by means of linearized least-squarestechniques applying positivity constraints onprism depth and Laplacian smoothing to regular-ize the inversion. Although the mathematicalconcept is reasonable, the assumption of homo-geneously and purely induced magnetizedprisms must be regarded as being rather poor.

Susceptibility measurements on ditch fillingmaterial showed a decrease of susceptibility withdepth (Eder-Hinterleitner and Neubauer, 1997).As will be discussed below, the same effect wasdiscovered on pit-fill material from the archae-ological site of Zwenkau, Saxony. Therefore, ouraim was to develop an inversion algorithm thattakes into account possible changes of magneti-zation with depth and that does not suffer fromthe assumption of purely induced magnetization.

Our model is close to the model chosen byHerwanger et al (1997), except for a free magnet-ization vector instead of purely induced magnet-ization. Both topography and magnetization willbe estimated during the inversion process.However, as has been shown by different authors(see e.g. Blakely, 1995, p. 292), it is impossible toestimate magnetization and the bottom topogra-phy of the layer simultaneously. Therefore, a two-step inversion procedure was developed. At thefirst stage the mean magnetization is estimatedfor a simplified model, then the topography ofthe undulating interface between the layer andthe underlying half-space is calculated. Althoughthe inversion procedure developed determinessource parameters directly, it is not fully auto-matic. The experiences of the operator are of greatimportance when preparing the data, e.g. trendestimation or stripping of unwanted features inthe data.

Methods

Archaeometric prospecting

The target of the magnetic prospecting was aBronze Age settlement structure in Altranstaedtnear Leipzig. The survey was carried out using aGSM-19 Overhauser Gradiometer. In gradient

166 G. Dittrich and U. Koppelt

# 1997 John Wiley & Sons, Ltd. Archaeological Prospection, Vol. 4, 165±177 (1997)

mode, the instrument measures the total mag-netic flux density at two sensors simultaneously.The result is a high quality difference measure-ment, compared with difference data derivedfrom two consecutive total field measurements.The data are independent of diurnal variations ofthe Earth's magnetic field. We refer to thesedifferences as vertical differences, V, because thesensors are alligned vertically. The term verticalgradient appears not to be correct in this set up(Koppelt et al, 1996).

From the aerial photograph we expectedfeatures such as pit alignments and ditch systemsof linear dimensions ranging from 1 m to 3 m. Agrid spacing D� 0.5 m was chosen to achieve amean coverage of 5 to 10 points per anomaly.

The height of the lower sensor, h, and thesensor separation, s, determines the frequencycontent of the measured signal. Both must be setaccording to local noise characteristics of themagnetic field (Koppelt et al, 1996). To achieve amaximum resolution, but to avoid aliasing of thesignal due to finite grid spacing, we made testmeasurements using different sensor configur-ations. A sensor separation s� 1.5 m and a heighth� 0.36 m of the lower sensor were found toprovide an acceptable signal-to-noise ratio.

Susceptibility measurements

To obtain information about the vertical suscepti-bility distribution within different kinds of pits,samples of pit fills from a nearby archaeologicalsite, situated in a comparable environment, weretaken. We sampled vertical profiles starting fromthe topsoil layer to the bottom of the pit, with10 cm point spacing. Bulk susceptibility wasmeasured with a Bartington MS2 susceptibilitymeter. In addition, for some probes the Koenigs-berger Q-ratio, as the ratio of remanent andinduced magnetization, was measured.

Forward and inverse modelling

As a basic physico-archaeological model (PhAM)representing single pits as well as ditch systemsand pit houses situated at a particular site, arectangular grid of vertical prisms with varyingdepth was chosen. We assumed the magnetiza-tion contrast of all prisms to be constant withrespect to the environment, but its direction and

magnitude were included as free parameters intothe first step of the inversion procedure. For thispurpose a distinct anomaly was chosen andmodelled with a single prismatic body by esti-mating its geometry and magnetization numeric-ally. The anomaly was chosen with respect tothree criteria:

(i) the shape of the anomaly should resemblean anomaly due to a homogeneouslymagnetized prism;

(ii) the maximum amplitude of the anomalyshould be within the range defined by otheranomalies resulting from similar archaeo-logical features at that particular site;

(iii) the anomaly must not interfere with otheranomalies.

The goal of the second inverse modelling stepwas the estimation of the bottom depth for allprisms, where the previously estimated magnet-ization and depth to the top remained un-changed.

For the overlaying algorithm controlling para-meter estimates, an evolutionary strategy (ES)was chosen. Evolutionary strategies are probabil-istic methods that are based upon principles ofnatural selection and adaptation. Their mainscheme will be outlined briefly, but for a detaileddescription we refer to BaÈck and Schwefel (1993)and for applications in archaeometry to Dittrich(1996).

A number, m, of starting models of n-dimen-sional parameter vectors are regarded as parentelements. From every parent an offspring of lmodels is generated by adding different Gaussianrandom vectors with zero mean and standarddeviation, s. In parameter space, the offspringforms a cloud with mean radius s around theparent; thus exploring the space in a neighbour-hood whose linear dimensions are controlled bys. The quality of every model is quantified by itsweighted Euclidian distance from the observa-tions measured in data space

C�i� � �~do ÿ ~d�i�c �TCÿ1�~do ÿ ~d�i�c �;i � 1; . . . ; m�1 � l� �1�

where ~do, ~dc, C denotes the observed andcomputed data and the covariance matrix of

Magnetic Data over Settlement Structures 167


the observed data, respectively. The index i,running from 1 up to m(1 � l) means that allparent and child elements take part in thecompetition. According to this quality a numberof m0 new parent elements is selected. Usually ism0 � m. For this selection there are two mainstrategies. For the first, all elements (parentand child) of the generation take part in theselection. This strategy (so-called plus strategy)has the advantage that no deterioration of thequality of the parent-pool is possible. In case ofmultimodal error-functionals the m � l-strategymay have problems finding an acceptablesolution.

A modification of the selection process may beperformed in such a way that only the (m� l)child elements take part in the competition (so-called colon strategy). Because such a selectionallows a deterioration of the goodness of onegeneration, the convergence speed of the pro-cedure will decrease. There is, however, the greatadvantage that it is possible to overcome localoptima for problems with many local solutions,or to make this event easier.

The process of reproduction and selection isrepeated until a fit of the observations withintheir error bounds is achieved.

Particular implementations of an ES maydiffer, e.g. in the way the selection is performedor the choice of the scattering parameter s. Thesedifferences may be important with regard to thespeed of convergence. Evolutionary strategies donot require derivative determination or any kindof linearization of the inverse problem, but theforward problem must be solved repeatedly. Thismakes them well suited for non-linear inverseproblems, although they are much more timeconsuming than standard gradient algorithms.

Fast forward modelling algorithms for ahomogeneously magnetized rectangular prismwere published by Bhattacharyya (1964),Kunaratnam (1981) and Ivan (1996). Because themagnitude of remanent magnetization cannotbe neglected, we followed the concept of totalmagnetization as a sum of remanent and inducedmagnetization, rather than considering inducedmagnetization only. The forward modellingalgorithm we applied to compute the totalmagnetic field anomaly due to a single prismdiffers only slightly from Kunaratnam (1981).

Results

Vertical changes of susceptibility

The results of five vertical susceptibility profilestaken from different pits are shown in Figure 1(aand b). Figure 1a clearly shows the decrease ofsusceptibility with depth for three particular pits.A different pattern is shown in Figure 1b. Thesusceptibility of these pits is almost constant,except for a strong positive contrast at the bottomof the pit.

Estimation of the Koenigsberger Q-ratio fortwo different samples showed a value between1.4 and 1.5, indicating that remanent magnetiza-tion must be considered as significant. Becausesamples were not oriented, no conclusions aboutthe direction of the remanent magnetization canbe drawn. The direction of remanent magnetiza-tion is not constant within the filling and doesnot carry any archaeomagnetic record (H. Becker,pers. comm.). Thus it appears to be desirable toincorporate as little supposition as possible aboutthe direction of total magnetization into theinversion procedure.

Inversion of synthetic data

Based on the results of susceptibility measure-ments on excavated pits, data sets for fivedifferent pit models were calculated. The pitswere assumed to be prismatic bodies that weresplit into a number of horizontal layers. Suscepti-bility values from the measured vertical profileswere assigned to corresponding layers. The totalmagnetic field anomalies of all layers werecomputed separately and summed.

Because the algorithm utilizes a homo-geneously magnetized single layer model, theeffects of heterogeneous pit filling and magneticremanence are projected on to a single magnet-ization vector. To test the ability of the algorithmto handle the effect of different types of magnet-ization, estimates of the body's lateral extensionsand its depth were computed under two differentassumptions. At first geometry and a scalarsusceptibility had to be estimated, thus assuminginduced magnetization only. In the second run,geometry and total magnetization vector werechosen as free parameters, and therefore, a



Figure 1. Vertical susceptibility distribution for five different pits situated at the archaeological site of Zwenkau, Saxony.(a) The predominant behaviour is a decrease of susceptibility with depth. (b) Some pits show almost constant susceptibilityexcept for a sharp increase at the bottom of the pit.



greater degree of freedom was given to the algor-ithm to handle different types of magnetization.

A comparison of given and estimated modelparameters is represented in Table 1, showingsignificantly better results in the case of free totalmagnetization. It can be seen that percentageerrors of estimates for the depth to the top are onaverage less for the free magnetization model

than for the model of a homogeneously magnet-ized body with purely induced magnetization.These results were independent of the actualsusceptibility distribution within the pit.

For a second test of the algorithm a homo-geneously magnetized pit model with anundulating bottom was assumed (see Figure 2a).As stated above, the inverse problem is non-

Table 1. Relative errors for depth-to-top and depth-to-bottom estimates computed for the free magnetization and inducedmagnetization model.

Code of feature Relative error (%)

Depth to top Depth to bottom

Free magnetization Induced magnetization Free magnetization Induced magnetization

7-505 11 20 5 127-13 11 25 6 18-324 11 15 5 18-97 5 30 2 328-313 11 37 13 19

Figure 2. Test of the inversion algorithm. (a) Depth function of the model pit.



unique in the case of a parameterization includ-ing magnetization and interface undulationsimultaneously. Moreover, from the principle ofsuperposition it follows that it becomes ambig-uous in the case of a multilayered Earth model aswell. As, for a homogenously magnetized prism,magnetization and geometrical parameters canbe calculated simultaneously (Rao and Babu,1991), such a model was used to estimatethickness of the top soil and magnetization.Herein the depth to top of the prism served asthe estimate of the topsoil thickness. These resultswere used as fixed parameters for the shapereconstruction (Figure 2). Estimated maximumdepth and three-dimensional shape of the bodyagreed well with the original model. Misfit indepth estimation of the interface was between5 per cent and 10 per cent. The maximum depthhad an error of less than 3 per cent.

The investigation of Bronze Agesettlement structures

The archaeological site Altranstadt was knownfrom aerial photographs, but no further infor-mation about extent, structures or its age wereavailable. The task of the magnetic survey was toprovide high precision mapping as well as amodel reconstruction of main features.

The results of the geomagnetic survey togetherwith the interpretation are given in Figure 3. Therectangular structure (A) could be interpreted asa ditch system surrounding an inner circular ditch(B) and a central anomaly that was interpreted asa grave (C). A second characteristic feature is alinear axis of magnetic anomalies that appears tobe a pit alignment known to be Bronze Age.Another group of anomalies occupying a largerarea (H, I, J) was interpreted in terms of filled

Figure 2(b). Calculated total field anomaly.



pits, whereas anomalies with large amplitudesappear to be due to strongly magnetized sourcessuch as modern iron objects or magnetic rocks.All of these anomalies correspond to darkenedareas in the aerial photograph. The interpretationof less distinct anomalies has proved to be morecomplicated. They could be related to geologicalfeatures, e.g. boulders with slightly highermagnetization than the top soil. Some of themcould be related to archaeological features aswell. If the main grave surrounded by two ditchsystems proves to be a main barrow, gravessituated around this structure can be expected.

One of the pits from the pit alignment (Figure 3,anomaly marked as X) was used for estimation ofmagnetization and topsoil thickness (40 cm) byusing a single prism model. Prior to shapereconstruction, two anomalies that are mostprobably related to geological features, e.g.

strongly magnetized boulders, or modern ironobjects had to be stripped from the data. Theselection criterion was an amplitude much higherthan the mean amplitude of pit-related anomaliesor an anomaly orientation that significantlydeflects from today's N±S direction.

The stripping was done by data windowingand inversion using a dipole model. Once thedipole related to a particular anomaly wascalculated, its total magnetic field was computedand subtracted from the measured data. Afterthat, a three-dimensional inversion was com-puted (Figure 4). The maximum depth for thepit alignment was estimated to be about 1.6 m,the two ditch systems appear to have a meanbottom depth of only a few centimetres (innerring ditch) and 1.2 m (rectangular ditch system).As a cut-off depth of 10 cm was applied, i.e. allprisms with a depth less than the cut-off value

Figure 2(c). Estimated depth function.



were set to zero, the inner ring ditch becomesinvisible.

Discussion

The computational results showed that it maybe advantageous to use an inversion methodutilizing a free magnetization vector rather thanthe concept of purely induced magnetization.The benefit increases with the degree of hetero-geneity of the deposed material and with thedeflection of the total magnetization vector fromtoday's main field direction, e.g. due to remanentmagnetization.

As the evolutionary strategies are probabilisticmethods, it may happen that a single inversioncalculation using the free magnetization vectoryields worse results than the calculation with a

fixed magnetization direction. This is a commonproblem of probabilistic methods. The solution ofa single run may not be representive for itsoverall performance. Therefore, many differentruns should be performed and a statisticalanalysis of the results should be carried out. Inthis sense the method relying on the concept offree magnetization vector leads to betterparameter estimates.

Some parameters, such as topsoil depth andmagnetization, have to be fixed for thereconstruction of the three-dimensional shape.Otherwise the inverse problem becomes ambig-uous. As they are usually unknown, they weredetermined in the first inversion step. This stepshould be carried out for one or more distinct andwell developed anomalies according to the rulesstated above. The appropriate choice of theanomaly is a crucial point of the algorithm,

Figure 2(d) Data misfit.



because parameter estimates derived throughoutthis step will remain fixed in any followingcalculation. Therefore, the initial choice should bedone by the operator according to their experi-ence. We doubt that any kind of automatizationcould be useful here.

The main drawback of any automated inver-sion procedure is its incapacity to discriminateproblem-related anomalies from unwantedfeatures still inherent in the data. Therefore,anomalies unrelated to pits and ditches, such asanomalies due to kilns, iron-slag bodies as wellas anomalies of modern iron objects or of geologyshould be identified and removed prior to theinverse modelling. A kind of stripping algorithmwas applied to clear out these anomalies. After

identifying them as being related to unwantedfeatures, the anomalies can be separated by datawindowing. A simple idealized body, e.g. adipole, could be used to model the anomaly.Once the parameters of the idealized body areestimated, its effect could be subtracted from thedata.

The choice of which anomaly to delete must bemade by the operator. Therefore, it depends onhis or her a priori knowledge about the anomalypatterns of unwanted features at this particularsite. Modern iron objects for example are com-monly represented by small-scale bipolaranomalies, with amplitudes much higher thanthose produced by archaeological features. Thepositive and negative part of such an anomaly

Figure 3. Measured vertical different data over part of the archaeological site of Altranstaedt, Saxony.



may have approximately the same amplitude,and the axis connecting the two parts may deflectseriously from the N±S direction.

Anomalies due to large-scale geological fea-tures may be recognized by their frequencycontent. The long wavelength part of the spec-trum is determined by these anomalies. Onemethod to remove these features from the data ishigh-pass filtering in the frequency domain.Although simple, the method has a seriousdrawback. If the filter is not constructed properly,i.e. sufficiently smooth, it may corrupt localanomalies as well (Clement, 1973). Therefore,we prefer polynomial trend estimation andsubtraction instead (Skeels, 1967).

The inversion method for the first step may bean evolution strategy. For data sets of somehundred points (which may be enough for singlefeatures) these calculations can be carried out

with good computation times. This result, or themean of the calculated parameters of somedifferent calculations, can then serve as a fixedvalue for the three-dimensional reconstruction oflarge data sets. This reconstruction is carried outunder the assumption of the model approxi-mation by means of a number of vertical prisms.The inversion method for the problem of thebody depth calculation for each of these prismsshould be as fast an inversion algorithm as theMarquard Levenberg method, with regard to thecomputation times.

Both the reconstructions for the analyticallycalculated models and the measured data setsyielded a good correspondence with the expectedvalues. The calculated maximum depth for theanalytical model had an error of less than 5 percent. The reconstruction result for the measureddata set from Altranstaedt was compared with

Figure 4. Inversion results. Estimated depth function for the part of the site investigated (contour interval: 0.3 m).



neighbouring excavation, which yielded amaximum depth for a rectangular ditch systemof about 1 m. This is a good correlation to thereconstruction results because the calculateddepth is within the range for a neighbouringequivalent structure.

Conclusion

A robust inversion algorithm for shape recon-struction of settlement structures was developed.It works in two steps under the assumption of afree magnetization vector with a minimum ofa priori information. As suggested by real andsynthetic data examples, our algorithm is a goodtool for inverting magnetic field data over arch-aeological sites. Different tests have proved thestability and the robustness of the algorithm. Theimpact of heterogeneous pit filling on depthestimates was studied. It was shown that betterresults were obtained for a free magnetizationvector. The algorithm was applied to magneticdata from the archaeological site Altranstaedt. Thecomputed depth estimates are within the rangeexpected from an archaeological point of view.

Acknowledgements

We thank C. Reiûmann for helping us with thefield survey and R. Zergenyi from ETH ZuÈ rich formaking some of the petromagnetic measurements.

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