Archaeological Prospection Archaeol. Prospect. (2013) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/arp.1468
Geophysical Observations at Archaeological Sites: Estimating Informational Content LEV V. EPPELBAUM* Department of Geophysical, Atmospheric and Planetary Sciences, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel
ABSTRACT
The application of geophysical methods to archaeological sites is limited by physical, environmental, economic and time considerations. The presence of numerous kinds of noise means that in many cases the archaeological targets and surrounding media are best approached as probabilistic objects, such that the amount of information potentially available from different geophysical methods can be estimated by probabilistic and statistical methods, including the risks associated with this decision-making. Here it is shown that simple informational and probabilistic criteria can be applied to formalize the information that can be obtained by applying different geophysical methods. To assess their relative value, geophysical methods, geophysical information and cost and time factors are convoluted in order to generate integrated parameters. This theoretical presentation of the information parameters is illustrated by the calculation of actual results. The solution to this ‘four colour’ mathematical problem shows that two independent geophysical methods are sufficient to characterize the archaeological potential of a site. Copyright © 2013 John Wiley & Sons, Ltd. Key words: Information and probabilistic approaches; convolution of information; geophysical indicators; archaeological target; physical archaeological model; logical heuristic model
Introduction It is well-known that the majority of inverse-problem solutions in geophysics are ill-posed (e.g. Tikhonov and Arsenin, 1977; 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 and probabilistic methodologies in applied geophysics. Geophysical observations at archaeological sites are notoriously complicated by numerous factors (Eppelbaum et al., 2001a). The most common forms of noise affecting archaeogeophysical investigations are depicted in Figure 1, and to eliminate many of these disturbances modern interpretational methodology has been developed (e.g. Eppelbaum et al., 2001a, 2001b, 2010; Eppelbaum, 2010, 2011). However, * Correspondence to: L. V. Eppelbaum, Department of Geophysical, Atmospheric and Planetary Sciences, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel. E-mail:
[email protected]
Copyright © 2013 John Wiley & Sons, Ltd.
at times the complexity of the geological environment (extreme variability in lateral and vertical physical properties), the presence of several archaeological targets (AT) in close proximity and additional disturbances (see Figure 1) make it impossible or unfeasible to apply this methodology. In such cases information–probabilistic methods are effective tools to recognize and classify targets, estimate the potential information value of geophysical methods and decide upon a workable solution. The objective of geophysical surveys at archaeological sites is to obtain qualitative and quantitative information about the geometric and physical characteristics of buried archaeological remains; for example, to develop physical archaeological models (PAMs) of target objects. Physical archaeological models of varying degrees of complexity (the simplest PAMs are target identification and complex PAMs can be three-dimensional models of archaeological remains) can be used to justify direct excavation in defined areas (or prohibit industrial activity) and generate future strategies for archaeological investigations at sites where ancient remains are known to exist. A very short description of some of the key features of the informational approach to geophysical methods Received 22 March 2013 Accepted 9 October 2013
L. V. Eppelbaum
Figure 1. A generalized scheme of noise in archaeogeophysical investigations. This figure is available in colour online at wileyonlinelibrary.com/ journal/arp
at archaeological sites was presented in Eppelbaum et al. (2001a). Here it is shown that simple informational and probabilistic criteria can be applied to formalize the relative value of geophysical methods, geophysical information and cost and time factors in order to generate integrated parameters. Estimating the information value of geophysical and other means can be formalized on the basis of the following criteria (after Eppelbaum et al., 2003) (Figure 2): (i) ’Informativeness’ of the application (informational criterion Γ); (ii) Cost of implementing the method (cost criterion C); (iii) Time required to implement the method (time criterion T). Criteria C and T are easy to calculate directly, but criterion Γ is a non-trivial research problem. A simplified algorithm can be written as: Ω ¼ Γ ∪C∪T
(1)
where ∪ is the symbol of unification. All the available archaeological and geological information can be represented as the classic three-level model (Figure 2): (i) syntactic – quantity of information;
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(ii) semantic – content of information; and (iii) pragmatic – value of information. The logical heuristic model for describing environmental information thus takes the following form (Eppelbaum et al., 2003): Γ ¼ I∪R∪V
(2)
where I is the quantitative estimation of information, R is the estimation of informational reliability corresponding to the semantic criterion and V is the estimation of informational value in terms of feasibility according to the pragmatic criterion. A semantic-based platform for the digital analysis of architectural heritage was described in detail in a recent publication by De Luca et al. (2011). Algorithm (2) is based on the fundamental terms of information theory and is combined with the structural (hierarchical) approach. This approach defines each indicator as a structure reflecting a set of typical situations and is then used to calculate the value of each estimator using the informational measure. Parameters V and R should be estimated archaeologically but this is beyond the scope of this paper. Here parameters V and R will be neglected, and it is assumed that Γ = I.
Archaeol. Prospect. (2013) DOI: 10.1002/arp
Estimating Informational Content
Figure 2. A scheme of archaeogeophysical target recognition and physical archaeological models construction with elements of information theory (on the basis of Eppelbaum et al. (2003), with modifications). This figure is available in colour online at wileyonlinelibrary.com/journal/arp
Evaluating the efficiency of geophysical methods with informational–statistical procedures General considerations Choosing the right method (or number of methods) can be based on a quantitative estimate (Figure 2). For this purpose, reliable informational and statistical criteria are needed. The first issue is the quantity of information that can be obtained by a single method or a set of methods. The second is to define a criterion to express the decision-making risk as a function of the geophysical data. Nevertheless, informational criteria are preferable, because geophysical prospecting is a permanent process of acquisition and analysis of information. The classic work on information theory by Shannon (1948; and later by Brillouin, 1962) prompted Khalfin (1958) to apply these criteria to geophysics in his pioneering studies. It has been shown (Khesin et al., 1996; Eppelbaum et al., 2001a) that informational and statistical approaches represent two aspects of a shared approach. For instance, the solution to an identification problem using the criterion of minimal average risk or that of maximum information (minimal residual uncertainty)
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under certain conditions results in the same expressions (Svetov, 1992; Nikitin, 1993). A review of the archaeogeophysical literature shows that there are several approaches to integrated geophysical analysis and calculation of the relationship between geophysical surveys, time and cost factors. Ellwood (1994) put forward a nomogram to evaluate time, cost and geophysical grid size and survey intervals in archaeogeophysical investigations. Schurr (1997) suggested using a ‘learning curve’ for optimization of geophysical expenditures at archaeological sites (reducing time and financial expenditure, correct selection of geophysical instruments, etc.). Marchisio and Ranier (2000) used a ‘decision tree’ graph to show how geophysical surveys can actually decrease the costs of exploration and increase the effectiveness of searching for AT. Piro et al. (2000) suggested applying an ‘indicator of source occurrence’ based on a spatial comparison of geophysical methods. A preliminary informational and probabilistic approach to archaeogeophysical investigation appeared in Eppelbaum et al. (2001a, 2001b). Kvamme (2006) suggested a number of statistical techniques for geophysical method integration. The need for a tight integration of archaeology, geophysics and chemistry was illustrated with clarity and wit by Pollard and Bray (2007). Baxter (2008) presented a very useful review on different ways to apply mathematics and statistics in archaeology. Ernenwein (2009) applied supervised and unsupervised image classification for integration of multidimensional geophysical data. McCoy and Ladefoged (2009) suggested applying a spatial technology to integrate geophysical surveys and archaeologists’ finds, and discussed how to store, analyse and interpret results in digital datasets. Markov randomfield image processing was successfully employed for magnetic field analysis of the ruins of the Hittite Empire in Turkey (Ucan and Albora, 2009); this method has great potential for geophysical method integration. An enhanced wavelet approach to integration of geophysical data in environmental and archaeogeophysical surveys was proposed by Eppelbaum et al. (2011). Bonsall et al. (2013) reported on increasing the role of the informational content in the archaeological geophysics. Schmidt and Tsetskhladze (2013) have shown, by example of GPR measurements at a necropolis in central Anatolia (Turkey), that a vector-processing scheme could significantly increase the quantity of information obtained. This lays the groundwork for presenting some of the key elements of the theory of information and statistical probabilistic approaches in archaeogeophysics. Illustrating how these fairly straightforward methods can yield valuable results should highlight commonalities in archaeological and geophysical theory and
Archaeol. Prospect. (2013) DOI: 10.1002/arp
L. V. Eppelbaum where P(Bi) is the probability of a Bi outcome (B is the range of outcomes of experiment β). In this situation the H(β) value (log2N) is maximum possible uncertainty. Experiment α (geophysical observation) provides additional information. The A value is a range of outcomes for experiment α. The difference between uncertainties in the β results before and after experiment α serves to estimate the information in α as related to β:
hence optimize PAM development and successful archaeological digs. As shown in countless publications on archaeological prospection, there are typically no more than two or three geophysical methods employed to solve problems of detection, contouring and the development of three-dimensional models of archaeological sites (e.g. Linford, 1998; Darnet et al., 2004; Di Fiore and Chianese, 2008). A simple model showing the results of the application of two methods (e.g. magnetic and direct current surveys) is described below. These methods are labelled in Table 1 as follows: 1 = weak negative anomaly, 2 = weak positive anomaly, 3 = negative anomaly, 4 = positive anomaly, 5 = high-gradient field, and 6 = roughly zero field. The values of the physical properties are assumed to be consistent with published data on the magnetic and electric properties of AT and host rocks in archaeological sites in Israel. The four classes of targets can be represented by four combinations of two parameters, each ranging from 1 to 6. The number of A possible combinations of two parameters divided into six categories is 36. Each class of archaeological features can be characterized by one of these 36 combinations. The number of combinations can be increased at the expense of secondary parameters related to certain transformations of the fields (downward and upward continuation, derivatives of various orders, etc.). Assume that there is only one AT in the area and that the area is divided into N equal cells. For simplicity each cell is assumed to have the same probability of containing the AT. Thus, the probability of finding the AT is equal to P = 1/N in each cell. Hence, the entropy of experiment β (discovery of AT) is log2N. The entropy is determined using the following expression: N
HðβÞ ¼ ∑ PðBi Þ log2 PðBi Þ
I ðα; βÞ ¼ HðβÞ HðβjαÞ
(4)
where H(β|α) is the conditional entropy for experiment β (provided that experiment α has been conducted). The conditional entropy is the average value of a random variable taking a H ðβi jαi Þ value with a probability of P(Ai): N
Hðβi jαi Þ ¼ ∑ PðAi ÞHðBi jAi Þ
(5)
i¼1
Estimating the efficiency of individual methods When selecting methods for integration, it makes sense to evaluate the amount of information provided by each. Starting from a well-investigated site (with an equal distance between observation points) typical of the area under study, it is assumed that 1/50 of it contains an AT. It is known that in the AT part the magnetic field is always positive, whereas in the empty part of the area it may be either positive or negative with equal probability. In other words, it is known a priori that 2% and 98% of the area are target-containing and empty, respectively, and in 49% and 51% of this area the magnetic fields, respectively, are negative and positive. The results of experiment β can be designated as follows: B – AT occupying part of the area, B – empty part of the area. Thus, PðBÞ ¼ 0:02; P B ¼ 0:98: According to expression 3 H(β) ≅ 0.14.
(3)
i¼1
Table 1. An example of geophysical data combination over archaeological remains. Typical field combination for magnetic and resistivity fields 1
2
M
R
3
4
6 Limestone constructions
R M
5
Ratio ‘target/ surrounding rocks’
Class of targets
R
M R M
Iron objects (kilns, furnaces) Basaltic constructions Zones of garbage accumulation
J1/J2 ≈ 1/(10 ÷ 30) ρ1/ρ2 ≈ (3 ÷ 7)/1 J1/J2 ≈ (100 ÷ 1000)/1 ρ1/ρ2 ≈ 1/(100 ÷ 300) J1/J2 ≈ (20 ÷ 50)/1 ρ1/ρ2 ≈ (5 ÷ 15)/1 J1/J2 ≈ (3 ÷ 8)/1 ρ1/ρ2 ≈ 1/(5 ÷ 12)
M, magnetization; R, resistivity; J1 and J2 are the magnetization of target and surrounding rocks, respectively; ρ1 and ρ2 are the resistivity of target and surrounding rocks, respectively
Copyright © 2013 John Wiley & Sons, Ltd.
Archaeol. Prospect. (2013) DOI: 10.1002/arp
Estimating Informational Content The result of experiment α is expressed as follows: A is a positive field, A is a negative field. The relative partial entropy (after recording the positive magnetic field at the measurement point) can be calculated in the following way: P(A) = 0.51; P A ¼ 0:49; P(B|A) = 2/51, P BjA ¼ 1 PðBjAÞ ¼ 49=51: Thus, PðβjαÞ ≅0:24 . According to the recorded negative magnetic field, the area is certainly empty: H B A ¼ 0: Consequently, the partial entropy for experiment β under the conditions of α, as stated in expression 5, is PðβjαÞ≅ 0:12 . Thus, the uncertainty of the determination of AT decreased after magnetic field measurement from 0.14 to 0.12. By taking into account the cost of each measurement, the effectiveness of respective methods can be estimated (e.g. Eppelbaum et al., 2003). For this purpose it is sufficient to compare the values (costs) of the information units provided by each method. On this basis, a valid solution for a reasonable integration of the methods can be made.
Estimation of information by indicator (field) gradations The geophysical fields applied in archaeology usually have maximal and minimal intensity within studied areas. The difference between the maximal and minimal intensities may be subdivided into some intervals (gradations). Gradations of indicators can be also used to obtain information about the types of AT. Physical fields, geochemical analyses, archaeological features, etc., can serve as indicators. Let PðAi jBÞ denote the posterior probability of finding the ith gradation of indicator A (e.g. magnetic field intensity) over the targets B, where P(Ai) is the prior probability of finding the same gradation in a survey area. Thus the partial information on the presence of target IAi →B contained in the recorded Ai gradation takes the following form: P ðA i j B Þ (6a) IA i →B ¼ log2 PðAi Þ as the uncertainty in the recording of Ai before the survey was log P(Ai) and after the survey – log P(Ai|B). Similarly, " # P A i B (6b) I A i →B ¼ log2 P ðA i Þ Statistically, the probabilities (or, to be more precise, relative frequencies) are expressed by the following ratios: S is the total number of values, Si is the number of values occupied by the ith interval of the Ai values, Sp is the total number of points reflecting the AT
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projections to the earth’s surface, and Spi is the number of points common for interval Ai and the AT projections. Then: PðAi Þ ¼
Si ; S
P ðA i j B Þ ¼
Spi ; Sp
Si Spi P A i B ¼ : S Sp
The increments in information contained about the presence of the object "
ΔIi ¼ I Ai →B IAi →B
P ðA i j B Þ ¼ log2 P Ai B
# (7)
are summed up in each elementary cell of the archaeological site studied. At the same time is necessary to take into consideration that inclined magnetization (polarization) can distort the projection of anomalous targets to the Earth’s surface (e.g. Eppelbaum, 2010); hence a buried target roof position and its vertical projection to the Earth’s surface in this case could not coincide. These computations are based on the results of an integrated survey at the Halutsa site in southern Israel (Eppelbaum et al., 2001b) (Table 2). The Halutsa site is located 20 km southwest of the city of Be’er-Sheva. It was the central city of southern Palestine in the Roman and Byzantine periods and was founded as a way-station for Nabatean (seventh to second centuries BC) traders travelling between Petra (Jordan) and Gaza and was occupied throughout the Byzantine period (fourth to seventh centuries AD) (Kenyon, 1979; Kempinski and Reich, 1992). Magnetic and self-potential measurements (Eppelbaum et al., 2001b) were conducted in a 20 × 10 m area with a 1 × 1 m grid (Figure 3A and B). The buried targets (ancient Roman limestone constructions) induced negative anomalies in both fields (Figure 3C and D). Figure 3C and D shows initial PAM of these concrete AT. To estimate the informational significance, the following expression can be used: I’A→B ¼ ∑½PðAi ÞIAi →B
(8)
i
The results using expression 8 appear in the second column of Table 3. The estimates served to substantiate the ratio between informational significance, cost and time. To normalize the results, the following semi-empirical expressions (obtained on the basis of the informational approach and long-term experience in the field of applied geophysics) can be used:
Archaeol. Prospect. (2013) DOI: 10.1002/arp
L. V. Eppelbaum Table 2. Example of information parameter calculation (archaeological site Halutsa, southern Israel). Field
Interval values
Self-potential (S = 210, Sp = 26; milliVolt)
Magnetic (S = 217, Sp = 16; nanoTesla)
ΩSP
51 to 40 39 to 30 29 to 20 19 to 10 9 to 0 1 to 13.6 14 to 11 10 to 8 7 to 5 4 to 2 1 to 1 2–4 5–8
I’A→BðSPÞ ¼ CSP =CMagn T SP =T Magn
ΩMagn ¼
Indicator
I’A→BðMagnÞ CMagn =CSP T Magn =T SP
(9a)
(9b)
The data compiled in the last column of Table 3 show that magnetic prospecting is characterized by more optimal parameters.
Estimates of the efficiency of geophysical integration based on the probability of type I and type II errors Classification efficiency can be estimated quantitatively not only for separate methods, but also for geophysical integration by calculating the reliability of revealing AT. At four archaeological sites in central Israel more than 20 anomalies have been contoured and localized by the integration of detailed magnetic and self-potential (SP) surveys. The anomalies revealed were divided into three groups with various degrees of desired AT discovery. Under the assumption that the results of direct excavations are absolutely reliable, classification reliability can be assessed by calculating the probability of type I and type II errors. The probability of a type II error (M2) is expressed as the relative frequency of an erroneous diagnosis for objects from sampling B (AT). The probability of a type I error (M1) is expressed as the relative frequency of an erroneous diagnosis for objects from the sampling B (the remainder of the objects). These errors are used to determine the total unconditional error of separation q between the classes B and B (i.e. the risk of an erroneous solution): q ¼ M 2 P ðB Þ þ M 1 P B
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(10)
Si
PðAi Þ ¼
27 38 29 17 64 35 3 9 13 47 36 74 35
0.1286 0.1810 0.1381 0.0810 0.3048 0.1667 0.0138 0.0415 0.0600 0.2166 0.1659 0.3410 0.1613
Si S
PðAi jBÞ ¼
Spi Sp
Si Spi P Ai B ¼ SS p
ΔIi, bit
0.8579 0.8489 0.8563 0.8563 0.8220 0.8514 0.9851 0.9552 0.9353 0.7662 0.8209 0.6318 0.8259
1.2485 1.5118 1.2769 0.8120 2.0562 1.4370 0.2696 0.7010 0.9545 2.3615 1.9852 3.1543 1.9484
2.0385 2.4615 2.1154 1.6538 3.4615 2.3462 1.1875 1.5625 1.8125 3.9375 3.2500 5.6250 3.1875
where P(B) and P B are the prior probabilities of the appearance of objects of the first and second classes, respectively. If P(B) = P B = 0.5, then the q value corresponds to the intersection area of distribution densities PðXjBÞ and P XB . Here X is the separation index. It can represent geophysical field amplitude or gradient, the value of integrated indicator, etc. The separation reliability (γ) is: γ¼1q
(11)
The total empirical error should be compared with the theoretical error. The approximation of the errors can confirm a correct assumption and provide high reliability of identification. Using logical informational methods (Khesin et al., 1996), the classification reliability is estimated solely from empirical errors. The errors due to assigning observation results to a class (with or without the AT) can be determined as follows. The absence of anomalies for the complex indicator in a known target-containing area is a type II error, or ‘omission of target’. The presence of these anomalies in the empty part of this area is a type I error, or a ‘false alarm’ (Khesin and Eppelbaum, 1997). Linford and David (2001) suggested another designations for the I and II type error: “false positive” and “false negative”, respectively. Comparing these rapid results with those of a more complex integration also can be used to estimate the respective errors and the reliability of classification. In this way, the archaeological nature of eight out of nine recognized magnetic anomalies at three prehistoric sites in Israel (Eppelbaum and Itkis, 2000; Eppelbaum et al., 2010) was confirmed. New archaeological objects were not revealed by direct excavation at the sites where geophysical methods were applied.
Archaeol. Prospect. (2013) DOI: 10.1002/arp
Estimating Informational Content
Figure 3. Maps of (A) magnetic and (B) self-potential (SP) fields at the Halutsa site (Negev Desert, Israel). Results of the quantitative interpretation for (C) profile I–I and (D) profile II–II. (+) Position of the middle of upper edge of anomalous body for magnetic anomaly. ⊙ Upper edge of anomalous body for the SP anomaly. This figure is available in colour online at wileyonlinelibrary.com/journal/arp
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Archaeol. Prospect. (2013) DOI: 10.1002/arp
L. V. Eppelbaum Table 3. Informational significance of geophysical method application at the site of Halutsa. Indicator
I ’ A→B
C
T
Ω
Self-potential Magnetic field
1.3265 2.1374
2 5
4 2
1.6581 1.7100
C, expenditure; T, time. Expenditure and time units are for geophysical survey in an area of 10 × 10 m with a grid of 1 × 1 m. The calculation of variables C and T includes geophysicists’ salaries, amortization (rent) of magnetometric and self-potential equipment and time.
Thus, M1 = 0, M2 = 1/9. Assuming that P(B) = P B = 0.5, and taking into account expressions 10 and 11, then q = 0.0556 and γ = 0.9444. Thus, parameter γ shows a high reliability of performed magnetic data analysis.
Calculation of information parameters Typically, the first step in the qualitative interpretation involves visual inspection of profile observations or compiled maps. However, this simple analysis is often not sufficient to locate buried objects. Detecting magnetic anomalies caused by AT is often difficult because the effect may be masked by the influence of surrounding inhomogeneties of different sizes and intensity. Various transformation and filtering procedures have been developed for more effective localization of finite objects (e.g. Telford et al., 1990; Parasnis, 1997; Piro et al., 2000). We have shown elsewhere (Khesin et al., 1996; Khesin and Eppelbaum, 1997) that applying information parameters often overcomes these problems. The method involves considering that the noise component – the combined effect of the surrounding inhomogeneties that are not associated with the target objects – is random. The amount of information (Ii) at each observation point by application of the ith method is I i ¼ log2 Pj or
Ii ≅ log2
Hi ΔHi
(12a)
(12b)
where Pj is the frequency rate of the jth interval of the ith indicator (field) in the histogram (more precisely, Bayes’ evaluation of the probability that the results will end up in the jth interval) and Hi and ΔHi are the amplitude and determination error of this indicator, respectively. Eppelbaum et al. (2003a) proposed the following expression: " 1 1 # n n K (12c) Ii ¼ Ui ∑ Ui log10 Ui ∑ U i i
i
where Ui is the geophysical observation at the ith point
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(Ui > 0) in the area under study, n is the total number of observations and K is a coefficient. After summing the information elements that suggest a priori that a target object is present, the random noise and components caused by dissimilar geological features are suppressed. In order to avoid singling out fictitious objects by the plot of n1 ∑ni¼1 I i , which can occur when a large amount of information is contained in the data of only one or two methods, an additional complex criterion that depends on the number of significant indicators can be calculated while adjusting for their relative influence: nðn1Þ 2 Ip k (13) Iinteg ¼ ∑ k¼1 I p max where Ip is determined from the formula I1 Ip ¼ ðI1 þ I2 Þ ; ðI 1 ≤I2 Þ I2 by using pairwise combinations of the n methods employed. The results of the parameter calculation may be compiled as maps of Ii and Iintegr. To avoid missing deeply embedded objects, in some cases it is better to use frequency rates of average values or average field estimates on a sliding scale instead of the Pj and Hi values, respectively. The correlation of Iintegr with the sum of information elements makes it possible to avoid missing an object, which for these or other reasons, was not revealed by other indicators. The combination of indices permits certain interpretative conclusions. In practice Ji is usually replaced by the relative amount of information, also known as the coefficient of informativity (Khesin et al., 1996): J (14) Ki ¼ i Ji The value of J i determines the information obtained when the result of Uj falls into the xj interval of the histogram with an equal probability of falling into any of the R intervals. According to probability principles (Ventsel, 1969; Daston, 1988), it is equal to the average (complete) information obtained when using a single method: J i ¼ log2 R (15) The application of Ki takes differences in the ranges of different fields into account. However, the application of expressions 11 and (13)–(15) may not be effective for sparse sampling. Consider converting the magnetic survey data observed at the Munhata site into informational parameter Ii. The Munhata site is located on a terrace at the outlet of Nahal Tabor to the Jordan Valley, some
Archaeol. Prospect. (2013) DOI: 10.1002/arp
Estimating Informational Content 11 km south of the Sea of Galilee in northern Israel, 215 m below sea level. The depth of the archaeological remains at the Munhata site is between 2 and 3 metres. The magnetic survey at this site was expected to be successful by taking into account the considerable contrast between features of interest (built mainly of different kinds of basalts) and the dark brown surrounding soil. The magnetic survey was carried out to the north of the excavation area. However, in the compiled magnetic map (Figure 4A) it is difficult to see the anomalous objects, but this map when processed using Equation 12c (Figure 4B) clearly displays the target objects (they are outlined by white rectangular and circular frames). The known archaeological objects excavated several tens of metres to the south of the surveyed area are presented in Figure 4C. Clearly the AT shown in Figure 4C (discovered objects) and the informational images presented in Figure 4B (proposed objects) are highly consistent. Thus, Figure 4B represents the simplest qualitative PAM. The site of Nahal Hagit is situated 20 km of coastal plain in northern Israel 30 km to south of the city of Haifa. Here non-magnetic remains of Roman constructions (limestone) occur in the low-magnetic medium. The characteristic peculiarity of the site is an extremely low level of useful signal – several nanoTesla (Figure 5A). For improving the ratio useful signal/noise here, entropy estimation based on Equation 12a was applied (Figure 5B). The specific peculiarity of this procedure was employment of adaptive moving window. White lines were drawn to indicate the results of qualitative interpretation in the right lower part of this area (see Figure 5B). This qualitative PAM is in good agreement with the available results of excavations. Obviously, the anomalies marked in the left upper part of Figure 5B may be used for planning future excavations.
Estimating integration efficiency by localization of weak anomalies If a set of methods is focused on investigating independent indicators of equal value, the anomaly detection reliability γ can be described by an error function (probability integral) (Khesin and Eppelbaum, 1997) as: 0qffiffiffiffiffiffiffiffiffi1 ∑ υi i A γ ¼ F@ 2
(16)
where υ is the ratio of the anomaly squared to the noise dispersion for each ith geophysical field, and F is the probability integral of type
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1 t FðtÞ ¼ pffiffiffiffiffi ∫ e 2π ∞
2 x 2
dx
Now let us assume that the anomaly is indicated by three points and that the mean square of the anomaly for each field is equal to the noise dispersion. For a single method, the reliability of the detection of an anomaly of a known form and intensity by Kotelnikov’s
pffiffiffiffiffiffiffiffiffi criterion (e.g. Borda, 2011) is expressed by FðtÞ υi =2 . Hence the reliability for individual methods is 0.61 and 0.77 and 0.87 for a set of two or three methods respectively. This means that the q value (risk of an erroneous solution) when integrating two or three methods decreases (according to Equation 11) by factors of 1.7 and 3, respectively. A comparison of the risk with costs C yields an optimum set of methods.
Some practical notes When using quantitative estimations, it should be recalled that they are based on definite assumptions and simplifications. Thus, if the probabilities cannot be determined (the probabilities can vary from 0 to 1), they are usually expressed as 0.5 (which is the most plausible). Geological conditions also need to be taken into account. For example, in Table 1 the combination ‘5/5’ might be obtained for iron objects instead of basaltic constructions (as a rule, basalts are characterized by the combination ‘4/5’). This may lead to the erroneous conclusion that other objects are located at this point. Therefore, a certain redundancy in the set of methods is highly advisable. The investigation of probability models has shown that the number of methods and their informativity influence integration efficiency, and also the success of their application.
Minimizing the number of combined methods by the ‘four colours theorem’ How many geophysical methods should be applied at an archaeological site? Extending a set of methods could be considered to be at variance with its economic efficiency, and complicated from both an organizational and a technical point of view. In addition, there is a basic limitation imposed on the number of methods. As noted in (Duda and Hart, 1973), a growing number of target indicators require larger amounts of standard information. However, sufficient standards are available only in well-explored provinces, where quantitative prediction is obviously less urgent. Therefore, a survey set should involve the minimum number of methods.
Archaeol. Prospect. (2013) DOI: 10.1002/arp
L. V. Eppelbaum
Figure 4. Magnetic maps of the northern part of Munhata site: (A) observed magnetic map (after Eppelbaum et al., 2001b); (B) magnetic map from (A) transformed using an informational parameter; (C) excavated part of the Munhata site, some 10 m south of the site studied (after Commenge, 1996). This figure is available in colour online at wileyonlinelibrary.com/journal/arp
The intuitive use of a small number of integration elements in practice can be theoretically substantiated applying the well-known mathematical and cartographic
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‘four colours’ solution (Appel and Haken, 1977; Appel et al., 1977; Robertson et al., 1997) for integrating geophysical methods by solving different environmental problems.
Archaeol. Prospect. (2013) DOI: 10.1002/arp
Estimating Informational Content
Figure 5. Archaeological site Nahal Hagit (northern Israel): (A) observed magnetic map; (B) magnetic map from (A) transformed using an entropy parameter. This figure is available in colour online at wileyonlinelibrary.com/journal/arp
Copyright © 2013 John Wiley & Sons, Ltd.
Archaeol. Prospect. (2013) DOI: 10.1002/arp
L. V. Eppelbaum Geophysical investigation is usually a multistage procedure and for simplicity it is assumed that the goal of each prospecting stage is the selection of an area for more detailed operations at the next stage. The result of prospecting is primarily a substantiated evaluation of the areas under investigation and their classification into two groups: those worthy and unworthy of further study. The objective of prospecting is to obtain the maximum information at a given cost. Let us now examine the ‘four colours theorem’ from this standpoint. Using elementary notions of graph theory the problem can be formulated as follows: prove that all vertices of an arbitrary planar graph can be coloured with four colours in such a way that no two vertices joined by a common edge are the same colour. It was proved as early as the middle of the nineteenth century that four colours suffice to colour different counties on the map of England. However, a solution to this theorem was only found more than 100 years later (Appel and Haken, 1977; Appel et al., 1977). The authors subdivided all possible maps into almost 2000 types and developed a computer program for their investigation. For each type the problem – whether a map that cannot be coloured with only four colours can be found among the variety of maps – was solved. After lengthy investigations, an answer of ‘no’ was obtained for all types, and this fact confirms the above solution. A new (general) proof of the theorem was put forward (Robertson et al., 1997). Any area under study can be divided into separate subareas according to certain indicators. The following system of prospect classification has been adopted in the USA (US Geological Survey, 1978): high (H), medium (M), low (L) and unknown (U). The objective is to single out promising areas (if any) from the entire set by an integrated geophysical survey. The colours refer to different combinations of geophysical methods. A positive conclusion for a certain prospecting method is labelled (+) and a negative () Hi ðx; y; z; tÞ ¼
jHi j≤φ þjHi j≥φ
(17)
where ϕ is some assumed value indicating the split between negative and positive values of geophysical field Hi, x, y, z are the space coordinates and t is the time. On the right-hand side of Equation 17 Hi is assigned an absolute value because often AT may be reflected by negative geophysical anomalies. Clearly a combination of at least two independent geophysical methods is necessary for the first three gradations (H, M and L): gradation U implies no application of the method set (on a scale or not at all)
Copyright © 2013 John Wiley & Sons, Ltd.
Table 4. Subdivision of an area according to geophysical survey results. Level of knowledge of the area High (H) Medium (M) Low (L) Unknown (U)
Geophysical method First
Second
+ + + – – – No necessary data
Combination number (colour) 1 2 3 4
in the area under investigation (Table 4). Interestingly, practically the same gradation was applied by Marshall (2001) for a subdivision of archaeological sites using magnetic anomaly gradations. The geophysical methods employed are a priori assumed to be of equal significance. The threshold field values (the split between plus and minus) and specific types of geophysical investigations are determined according to the prospecting results for similar objects investigated previously and other geological and geophysical considerations. The split refers to the threshold for field values representing specified physical characteristics. These physical characteristics may, for example, include values of observed fields, field gradients or indicators of field variability. However, it is assumed that the geophysical anomalies are produced by the same targets. It can be concluded from Table 4 that an optimum geophysical set consists of two independent geophysical methods. A map of geophysical results with four colours as determined by the above technique can serve as a basis for more detailed investigation. In this context, this division of the theory of graphs can be attributed to information theory. It should, however, be kept in mind that the geophysical set is usually oriented toward a particular problem and is substantiated by a corresponding physical–geological model of the medium. Any change in the problem (e.g. an increase in the necessary depth of investigation) or in the geological and geophysical pattern of the area can lead to a change in the set of methods. This may, in turn, affect the ‘colouring’ of the area under study. For this reason a certain redundancy of the set is needed.
Conclusion The information value of applying geophysical methods in archaeological sites can be estimated using various probabilistic statistical procedures. The components of these procedures detailed in this article can serve to
Archaeol. Prospect. (2013) DOI: 10.1002/arp
Estimating Informational Content assess the informativity of not only a single geophysical method but also of any geophysical integration. Elements of information theory can be easily applied in archaeogeophysics to optimize the process of target recognition and help reveal anomalous effects from AT against the noise background. The simple convolution of geophysical information, cost and time suggested here serves to compare the parameters as a whole. The solution to the ‘four colours problem’ theoretically supports the conclusion that two geophysical methods can be integrated to successfully classify an area according to its prospects. Thus overall, the concepts presented here should contribute to greater integrated archaeologicalgeophysical thinking.
Acknowledgement The author would like to thank two anonymous reviewers and Dr. Chris Gaffney, Editor of Archaeological Prospection, whose critical comments and valuable suggestions were helpful in the revision of this paper.
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Archaeol. Prospect. (2013) DOI: 10.1002/arp
