Selection of Co-Belonging Ceramic Fragments from Archaeological Excavations and Their Location in Vase Bodies from Thermoremanent Magnetization

Abstract

Selection of co-belonging fragments from the numerous ceramic findings of an archaeological excavation based exclusively on the experience and patience of the conservators remains a difficult process of questionable effectiveness. While the screening of the fragments is a central prerequisite and the most important stage of the process of vase reconstruction, established methods based on scientific criteria and guaranteed efficiency for the detection of co-belonging ceramic fragments suggested in the bibliography do not exist. On the contrary, for methods dealing with the assembly of vases from co-belonging fragments, which is a secondary process that can be done more easily and effectively in an empirical way, there exist numerous studies based on fragment morphology. However, even these are also not implemented because of the time requirements, sheer volume and complexity of the proposed methods, in order for them to be applicable in practice. The proposed methods in this paper are based on thermoremanent magnetization (A/m), which is calculated from the weak magnetic field measurements by a fluxgate-sensor/magnet apparatus forming a three-dimensional orthogonal system. Experimental measurements from fragments of six vases show that the magnetization magnitude of co-belonging fragments display similar values, despite the magnetic anisotropy of the ceramic material, since these belong to vases made of the same clay and fired under the same conditions. This is the criterion for finding ceramic fragments of the same vase from archaeological excavations. The thermoremanent magnetism directionality of fragments, which is aligned along the geomagnetic field at the same place and time during the vase firing process, as it is configured by their rotational symmetry, defines the position of the fragments on the body of the six vases. The shape of the original vase can be reconstructed when only a few non adjacent fragments are available. The proposed measurement apparatus can be used for the construction of a useable portable magnetometer specialized for ceramic surface measurements to achieve the above objectives.

1. Introduction

While the assembly of excavation fragments from their remaining thermal magnetism is suggested as a theoretically sound method in the bibliography [1], there are only a few older (1975) references [2] regarding corresponding research. Research on thermoremanent magnetism in the archaeometry field since the 1980s mainly focused on the chronological dating of fired clay walls of kilns. At the same time, as reported in numerous publications [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33], the reconstruction of vases is attempted almost solely using algorithms for optical data processing based on fragment morphology. Despite the intense scientific interest in resolving the matter using the capabilities of the new digital technology, it remains an exclusively empirical process due to the time requirements and sheer volume and complexity of the proposed methods, in order for them to be applicable in practice.

A central prerequisite before fragment assembly is the selection of the fragments that belong to the same vase from among the numerous ceramic findings of an archaeological excavation. This is performed empirically and with great difficulty, especially in excavations conducted over long periods of time or ones with disturbed excavation layers.

However, the criterion for finding excavation fragments of the same vase should be the similarity of the values of the remanent magnetism, since these belong to vases that are made of the same clay, fired under the same conditions. Therefore, the natural way of assembling ceramic shards is not the processing of their imagery but rather the study of the directionality of their oriented thermoremanent magnetism along the geomagnetic field, which happened at the same place and time, and was configured by their rotational symmetry. In the present work, magnetic field measurements were performed using fluxgate sensors. All experiments were conducted in the Experimental Physics Laboratory, School of Electrical and Computer Engineering, Technical University of Crete. Application of the proposed method to six vases from the last two centuries fired in traditional kilns gave excellent results.

We describe the detection methodology of the fragments’ field by a fluxgate sensor in Section 2. The easy and unexpected assembly of base fragments and neighboring body fragments in six vases from their magnetic field directivity determined by a single fluxgate sensor, despite the irregular shape of fragments and the magnetic anisotropy of the fired clay, was the motivation for further investigation. For the systematic investigation and the comparison of the magnetic field, base and body fragment measurements were taken at the vertices of a lattice formed by the grooves and the vertical lines to them, with a specific orientation of the sensor. The measurements in the above positions were repeated in a less tedious and more accurate way, by using a sensor/magnet apparatus forming a three-dimensional orthogonal system for the magnetic field, which is described in Section 3.

A brief review of clay magnetic compounds is provided in Appendix A, which is suggested background reading material for non-experts in mineralogy. A series of experiments, mainly in base specimens of vases 4, 5 and 6 were carried out (see Appendix B.1, Appendix B.2, Appendix B.3, Appendix B.4 and Appendix B.5) to investigate how the sensors on the fragments’ surface were excited and the region of the fragment that is detected by the sensors was identified, for the calculation of thermoremanent magnetization (Appendix B.6) from the sensor readings. We propose the experiments to be read in order to provide a clear focus on their results and the theoretical approach to determine the sample area detected by the sensor, which are summarised in Section 4, and which were used for calculation of magnetization (Appendix B.7) in irregular fragments of the base vases 1, 2 and 3, with the described methodology presented in Section 5.

The aggregate results and conclusions of calculating magnetization from magnetic field measurements on samples and fragments of the base of vases 1–6 in Section 6, were applied for the calculation of magnetization (see Supplementary Files) in irregular fragments of the body vase 1–6, using the methodology described in Section 7.

In the 8th section, the mathematical relationships of the calculated magnetization changes in the fragments’ reference systems in vases with cylindrical (Section 8.1) and arbitrary rotational (Section 8.2) symmetry are described. By using rotational transformations of magnetization components in the reference systems where field measurements were taken (Section 8.3), the location of fragments was identified in the bodies of the six vases. Finally, Section 9 contains some proposals for future research.

2. Detection of the Weak Magnetic Field of Ceramic Fragments and Its Direction Using a Fluxgate Sensor

The most important problem in detecting the weak field of fragments, on the order of a few tens of nT, is its overlay with the Earth magnetic field, which is about 1000 times stronger, as well as the lab magnetic background.

For the detection of the weak magnetic signal [34,35,36] of fragments within the geomagnetic field, a fluxgate sensor (model MAG 03 IE three independent axes fluxgate magnetometer, Bartington, Witney, Oxon, UK) [37] at a fixed horizontal position (Figure 1) is oriented perpendicularly to the vertical line (B_vert) and to the horizontal component (B_horz) of the Earth’s magnetic field (B_earth), such that its reading is zero before the fragment placement. When the sensor distance from the fragment surface is greater than 1.5–2 cm, its reading becomes zero.

Accuracy of measurements at a practical level depends: a) on the ability of aligning the x-sensor with respect to the vessel body grooves, which can not be performed with an error less than 1nT. For this reason, measurements on the fragments of the vase body are obtained with a 1nT error (See Supplementary Files), while the maximum sensor accuracy is 0.1–0.2 nT. Therefore, sensors of greater accuracy (eg SQUIDsensors), would not give better results regarding the computation of κ,ω angles, which determine the position of the fragments in the vase bodies. b) on he contact of the fluxgate sensors (of length 3 cm) with the shell surface (see Section 9: Conclusions and Proposals for Future Research), which decreases as their curvature increases.. Thus the precision of the method could be improved or extended to vase fragments of smaller size, using shorter sensors.

For the weak magnetic field extraction from the unwanted lab magnetic background noise the Phase Sensitive Detection (PSD) method [38] was used. The electric signal from the disk rotation, at a certain frequency, f, is used as a reference signal to the phase detector [39]. The direct electric current (1 mV/7 nT) from each fragment, detected by the magnetometer sensor, is converted during the disk rotation into alternating current and used as an input to the phase detector.

The electric signal from the horizontal rotating component, B_horz, of each fragment is differentiated from the magnetic interferences because it is detected by the phase detector exclusively at the disk rotation frequency, f. If the magnetic field of each fragment displays constant directionality, then its magnitude and initial angular phase difference (δ-φ), are recorded, after the processing of the sinusoidal signals at the phase detector. During the disk rotation, the ceramic material induced magnetization, M*, is oriented perpendicularly to the sensor along the direction of B_Earth. Using the PSD method, constant phase differences and weak magnetic fields, with constant directionality and values of a few tens of nT can be found.

After the detection of the fragments magnetic field (Figure 2), the directionality of the base and body in co-belonging fragments is compared. The sensor remains in the same horizontal position, perpendicular to the vertical component and the horizontal component of the geomagnetic field.

During the vase formation at the pottery wheel, circular grooves remain on the fragment surface. Parallel straight lines at an arbitrary direction (σ) are marked on the base fragments (Figure 3a), which suggests their joining direction. Regarding the fragments from the vase bodies (Figure 3b), their direction (σ) is selected along the surface grooves of the pottery wheel.

The fragments are positioned on a leveled angle-measuring apparatus, with the sensor axis aligned along the direction (σ), at a position where B_horz diverges at an arbitrary angle, φ, from the sensor direction, while the disk is rotated manually until its reading is zeroed. At this position, where B_horz is vertical to the sensor direction, its direction is marked.

Neighboring fragments are oriented in a joining manner because their magnetic field, acquired before their fracturing, is due to the thermoremanent magnetization along the direction geomagnetic field during their cooling in the kiln.

While finding the joining orientation of the base excavation fragments is not of essential practical value, because it is more easily performed empirically, it can be concluded from the above that neither the magnetic anisotropy of the material nor the irregular fragment shape prevent their joining from the orientation of their magnetic field; in fact this happens with remarkable precision. This is not the case for the distant body fragments, which come in a variety of shapes and are collected in large numbers. This is to be expected, because the thermomagnetic directionality varies depending on the orientation of the vase regions with respect to the geomagnetic field, and it is formed by the rotational symmetry of the ceramic vases.

Fragments from 6 ceramic vases (Figure 4a) are placed on the horizontal angle-measuring apparatus disk, with the marked reference direction (σ) on their surface matching the fixed sensor direction. The component parallel to the fragment surfaces (B^//) and the angle of divergence (φ) of the reference direction (σ) with the fixed sensor direction (Figure 5a) are measured at the rotational position where the reading displays its maximal positive value. Representative measurements of the angle φ in body (a) and base fragments (b) of vase 1 are presented in Figure 6.

The vertical component (B^˪) and its angle (θ_B) of divergence from the fixed sensor direction (Figure 5b) are measured at the same turning fragments position, albeit on the vertical disk. Representative measurements of the angle θ_B in body (a) and base fragments (b) of vase 1 are presented in Figure 7.

The computed magnitude (Chart 1c), B, and the measured angle, φ (Figure 6b), display similar values only in the co-belonging base fragments of each vase. For the co-belonging body fragments, the magnitude B differs (Chart 1a,b), while the angles φ (Figure 6a), change in a systematic way.

The computed angle θ_B (Figure 7), displays similar values only for the base fragments of the vases, while it varies in a systematic way for the body fragments.

The measured magnetic field of the co-belonging base fragments by a fluxgate sensor shows similar magnitude and directionality. This is the result of thermoremanent magnetization, which despite the ceramic material magnetic anisotropy, maintains similar magnitude and directionality, because the co-belonging base fragments are composed of the same clay, were fired under the same conditions, at the same time and with a fixed orientation with respect to the earth magnetic field. In any case, local differences are not improbable, which can be due to the incomplete mixture homogenization of clays or eutectic additive materials, temperature variations or kiln aeration and later magnetizations.

The method by which measurements are taken, by fragments placed on the disc in horizontal and vertical positions, due to the time requirements and the required accuracy, is difficult to be used for wide application. The dependence of thermoremanent magnetization in vases shaped in pottery wheels was systematically studied using precise and less tedious magnetic field measurements, by the three fluxgate magnetometer sensors in an orthogonal coordinate system in three dimensions.

3. Magnetic Field Measurements of Ceramic Fragments by 3 Fluxgate Sensors in 3 Orthogonal Axes

The horizontal x-sensor is oriented vertically to the horizontal component and to the vertical component of the earth magnetic field (Figure 8), measured by the y and z-sensors, respectively.

The sensor reading (1 mV/7 nT) is zeroed with a maximum precision of 0.01 mV or 0.1 nT by adjusting their axial distance from the Al-Ni-Co type cylindrical magnets (400 KA/m), by turning their mounting screws. The excitation area of each sensor is located on the axis that passes through the center of its square cross section of side 2d = 6.5 mm and at a distance of a = 6.5 mm from its curved edge.

The best, simplest and cost-effective solution is to attach a magnetic field and possible laboratory interference neutralization system to the three-dimensional sensor apparatus. Active shielding could be used, but is not the best choice, from a practical point of view. Within the area of zero magnetic field, the fragments of various sizes should be placed, beside the sensors, and the lifting, leveling and rotation apparatus should be attached. This would require a particularly bulky and complex construction compared to the proposed layout.

Each magnet generates, besides its axial field at the sensor direction, also side fields, which excitate the perpendicular sensors.

The measured axial fields B_z^z B_y^y,B_x^x (Figure 9) of the cylindrical magnets, with magnetization M, radius b, and height H_μ, with dipole moment m = M.π.b².H_μ and at axial distances ξ_μ^α from the sensors, are approximated (

Table 1. Computation of the fields at the sensors at the distances x,y,z that zero their readings. The magnetic field is computed at those distances, inside the measurement area. The magnet fields at the sensors and in the measurement area in the Figure 9 and Figure 10 above and in the table are marked with the same color.

Magnets		Fields (nT) at the Sensors			Fields (μΤ) Inside the Measurement Area
Distances (mm)		z	y	x	z	y	x
Z		−61,830.98			−22.2
	z = 44.91841		60,880.61
				60,880.61
Y		17,197.76			−4.3	−19.9
	y = 45.63295		−112,658.65
				35,669.95
X		12,633.18			−3.2		−13.2
	x = 41.33643		24,378.01
				−96,550.59
Earth’s field		32,000			32.0	27.4
			27,400
Total fields		0.0	0.0	0.0
					2.3	7.5	−13.2

) by the formula given in [40] (pp. 394–399):

$${\mathrm{B}}_{\mathsf{\mu }}^{\mathsf{\alpha }=\mathsf{\mu }}=\frac{{\mathsf{\mu }}_{\mathsf{o}}.\mathrm{M}{.\mathrm{b}}^2{.\mathrm{H}}_{\mathsf{\mu }}}{{2\mathsf{\xi }}_{\mathsf{\mu }}^3},(\mathsf{\mu }=\mathrm{x},\mathrm{y},\mathrm{z})$$

(1)

The magnet side fields, B_μ^α≠μ, at axial distances g_μ^α≠μ and side distances from the sensors, h_μ^α≠μ, are approximated by the formula:

$${\mathrm{B}}_{\mathsf{\mu }}^{\mathsf{\alpha }\ne \mathsf{\mu }}=\frac{{3\mathsf{\mu }}_{\mathsf{o}}.\mathrm{M}{.\mathrm{b}}^2{.\mathrm{H}}_{\mathsf{\mu }}}{4}.\frac{{\mathrm{g}}_{\mathsf{\mu }}^{\mathsf{\alpha }}{.\mathrm{h}}_{\mathsf{\mu }}^{\mathsf{\alpha }}}{{{[(\mathrm{g}}_{\mathsf{\mu }}^{\mathsf{\alpha }}{)}^2{+(\mathrm{h}}_{\mathsf{\mu }}^{\mathsf{\alpha }}{)}^2]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},(\mathsf{\mu },\mathsf{\alpha }=\mathrm{x},\mathrm{y},\mathrm{z})$$

(2)

The summed components of the field B_μ^δ of the μ=x,y,z magnets and the earth field components at the axial distances δ=x,y,z where the sensor readings become zero (Figure 10a), are approximated [40], at the measurement area, by the formulas:

$${\mathrm{B}}_{\mathrm{z}}^{\mathrm{z}}=-\frac{{\mathsf{\mu }}_{\mathsf{o}}.\mathrm{M}{.\mathrm{b}}^2{.\mathrm{H}}_{\mathrm{z}}}{{2\mathrm{z}}^3}$$

(3a)

$${\mathrm{B}}_{\mathsf{\mu }=\mathrm{x},\mathrm{y}}^{\mathsf{\delta }=\mathsf{\mu }}=-\frac{{\mathsf{\mu }}_{\mathsf{o}}.\mathrm{M}{.\mathrm{b}}^2{.\mathrm{H}}_{\mathsf{\mu }}}{4}.\frac{{2\mathsf{\delta }}^2{-\mathrm{d}}^2}{{{(\mathsf{\delta }}^2{+\mathrm{d}}^2)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(3b)

$${\mathrm{B}}_{\mathsf{\mu }=\mathrm{x},\mathrm{y}}^{\mathsf{\delta }=\mathrm{z}}=-\frac{{3\mathsf{\mu }}_{\mathsf{o}}.\mathrm{M}{.\mathrm{b}}^2{.\mathrm{H}}_{\mathsf{\mu }}}{4}.\frac{\mathsf{\mu }.\mathrm{d}}{{{(\mathsf{\mu }}^2{+\mathrm{d}}^2)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(3c)

All the following measurements are taken at the same fixed position with respect to the geomagnetic field by the sensor/magnet apparatus forming a three-dimensional orthogonal system. The induced-magnetization contribution of the ceramic material along M^*(Figure 10b) to the sensor readings is checked in the following experiment by comparing the measurements (Appendix B.2) taken inside a null magnetic field.

4. Experimental Results: Modelling the Ceramic Region generating the Magnetic Field detected by the Sensors for the Calculation of Magnetization in Base Vase Specimens

The remanent magnetization dependence and computation on the magnetic sensor measurements, is studied in Appendix B through a series of experiments (Appendix B.1, Appendix B.2, Appendix B.3, Appendix B.4 and Appendix B.5) in irregular base fragments (Appendix B.1) and mainly in cylindrical specimens (Appendix B.2, Appendix B.3, Appendix B.4 and Appendix B.5) from the base of vases 4–6 (Figure 4a).

The angles, θ_B, γ > θ_B, between the field, B, and the magnetization, M, increase as the distance to the measurement position from the edge of the fragment decreases.

From the experimental results of the measurements on specimens and fragments from the base vases 4–6, it is found that:

• The sensors detect the axial field B at the edge of a cylindrical region of ceramic material (Appendix B.1, Appendix B.2 and Appendix B.5) in the direction of the remanent magnetization, M.
• The measured magnetic field is only due to the remanent magnetization of the ceramic material (Appendix B.2), and it is not altered by its induced magnetization.
• The height, ℓ, of the magnetically detecting area of the ceramic material by the sensors (Appendix B.2, Appendix B.3, Appendix B.4, Appendix B.5 and Appendix B.7), depends of the angle, γ, between the magnetization M and the vertical z-axis on specimens’ surface, each specimen thickness, L, and the distance, D, between the sensors and the fragment edge along B_xy, when the vertical z-sensor reading is doubled:
  • When D > d = L.tanθ_B the height, ℓ = L/cosθ_B, depends on the thickness, L, of the fragment.
  • When D < d the height, ℓ =D/sinθ_B is dependent on the distance D.
• The radii (Appendix B.5), rⁱ = 2αⁱ.cosγ (i = ˪,//), of the ceramic cylindrical areas depends on the directionality (γ) of magnetization and of the parameters α^˪,α^//, which, geometrically, are the sensitivity radius of sensors around the measurement position of the vertical and the parallel sensors, respectively.

The field flow in the excitation coil of the vertical sensor, B_i^˪ (i = xy,z), is smaller than that of the parallel (B_i^// ≈ λ.B_i^˪) sensor, for the measurement of the same component B_i of the field. The normalization factor λ ≈ 2, for the correction of their readings (Appendix B.5), has a similar value in non co-belonging fragments with different magnetization M due to the different orientation of the excitation coils on the specimen surface. The radii α^˪, α^// increase with the magnetization magnitude, M, increase and display similar values in co-belonging shells.

•

The sensor readings display similar values when the ratios r/ℓ, of the cylindrical detecting areas of the ceramic material by the sensors, as does the measured axial field at the end of the solenoids with the same current and number of spirals, where the ratio of the radius, r, to their height, ℓ, is kept constant. The measured magnetic field B of the ceramic cylindrical areas is approximated by the magnetic field of a solenoid with the same dimensions.

-

If D ≥ d = L.tanγ (Figure 11a), the height, ℓ = L/cosγ, depends on the thickness, L, of the fragment and the components, M_i (i = xy,z), are approximated by the sufficient length relationships:

$${\mathrm{M}}_{\mathrm{i}}=\frac{{2\mathrm{B}}_{\mathrm{i}}}{{\mathsf{\mu }}_{\mathsf{o}}}.\sqrt{1+(\frac{{\mathsf{\alpha }}_i^{\perp ,//}{.\cos }^2\mathsf{\gamma }}{\mathrm{L}}{)}^2}$$

(4)

-

If D ≤ d (Figure 11b), the height ℓ=D/sinθ depends on the distance, D, of the measurement position from the fragment edge in the magnetization direction, M. The components M_i (i = xy,z) are approximated by the insufficient length relationships:

$${\mathrm{M}}_{\mathrm{i}}=\frac{{2\mathrm{B}}_{\mathrm{i}}}{{\mathsf{\mu }}_{\mathsf{o}}}.\sqrt{1+(\frac{{\mathsf{\alpha }}_i^{\perp ,//}{.\sin }^{2}2\mathsf{\gamma }}{2\mathrm{D}}{)}^2}$$

(5)

Since α_xy^//>α_z^˪, the computed deviation, θ_B, of the field from the vertical (Figure 11c) at the measurement is less than the angle γ of the magnetization M.

The horizontal components, B_xy, M_xy, of the measured field, B, and magnetization, M, diverge at the same angle φ from the horizontal direction of the x-sensor.

The quantities λ, α^˪, α^//, γ, M, M_i, (i = xy,z) were calculated with remarkable accuracy using the above relationships from the sensor readings and also by the thickness, L, and distance, D, measurements (Appendix B.6) in base specimens of vases 4 and 6.

5. Computation of Magnetization in Irregular Fragments of the Vase Base 1, 2 & 3

The components, M_z, M_xy, angle γ of the magnetization M, and sensitivity radii of the sensors, α^˪, α^//, in fragments from the bases of vases 1, 2 and 3 (Figure 4a) are computed by the same using way (Appendix B.7) in specimens from base of vases 4, 5 and 6, without any need for cutting specimens.

The three-dimensional orthogonal system apparatus of sensors/magnets remains in the same position with respect to the earth magnetic field (Figure 12), with the x-sensor oriented perpendicularly (B_y = 0, B_x = B_xy > 0) to the horizontal and vertical component of the geomagnetic field.

The measurements of B_xy, B_z are taken on B_xy in the direction of the x-sensor from the measurement position where the systematic reduction in the magnetic field readings begins at measured distances D_n < d = L.tanγ from the fragments’ edge.

The computation of sensor magnetization components, M_i (i = xy,z), and sensitivity radii, α_z^˪,α_xy^//, from the field measurements, B_i (i = xy,z), is performed using the least squares method (Appendix A), from the equation:

$${\left(\frac{{\mathsf{\mu }}_{\mathsf{o}}}{{2\mathrm{B}}_{\mathrm{i}}}\right)}^2=\frac{a_{\mathrm{i}}^2{.\sin }^{2}2\mathsf{\gamma }}{4{.\mathrm{M}}_{\mathrm{i}}^2}.\frac{1}{{\mathrm{D}}_{\mathrm{n}}^2}+\frac{1}{{\mathrm{M}}_{\mathrm{i}}^2},\mathrm{i}=\mathrm{z},\mathrm{xy}$$

(6)

The measurement results are presented in Appendix B.7.

6. Aggregate Results and Conclusions of Calculating Magnetization from Magnetic Field Measurements on Samples and Fragments of the Base of Vases 1–6

The measured components of the magnetic field, B_i (i = xy,z), depend on the directionality (γ), magnetization magnitude M, and thickness, as well as the position and orientation of the sensors on the surface of the fragments. From the above perspective, the normalization constant, λ, the sensitivity radius of sensors α^˪, α^//, of the vertical and parallel sensors around measurement position, B_i, and the magnetization (M_i, M, γ) in irregular base fragments of the vases 1, 2 and 3 (Appendix B.6) and in base samples of vases 4, 5 and 6 (Appendix B.7) are computed.

The sensitivity radii, α^˪, α^//, increase with the increase of the magnetization magnitude (Chart 2) and differ (

Table 2. Computation of the magnitude, M, of the magnetization directionality (γ) and of the normalization constants, λ, α^˪, α^//, of the sensor readings in irregular base fragments of the vases 1, 2 and 3 and in base specimens of the vases 4, 5 and 6.

Vase	M(mA/m)	γ°	α˪(cm)	α//(cm)	λ
6	58.3 ± 0.2	43.3 ± 0.3	0,71 ± 0.03	1.07 ± 0.02	1.90 ± 0.01
4	84.1 ± 0.2	48.5 ± 0.3	0.67 ± 0.03	1.15 ± 0.02	1.97 ± 0.01
5	112.7 ± 0.3	61.3 ± 0.3	1.04 ± 0.03	1.41 ± 0.03	2.03 ± 0.01
1	131.3 ± 0.2	53.8 ± 0.1	0.82 ± 0.04	1.42 ± 0.02	2.0 ± 0.1
3	164.1 ± 0.4	46.6 ± 0.2	1.07 ± 0.05	1.61 ± 0.03	2.0 ± 0.1
2	169.3 ± 0.6	50.8 ± 0.2	1.15 ± 0.05	1.66 ± 0.05	2.0 ± 0.1

) in base fragments from different vases. Since α_xy^//>α_z^˪, according to the relationships (4,5), the computed deviation, θ_B, of the field from the vertical (Figure 13) at the measurement position is less than the angle γ of the magnetization M. Since α_x^//= α_y^//, the horizontal components, B_xy, M_xy, of the measured field, B, and magnetization, M, diverge at the same angle φ from the horizontal direction of the x-sensor. For this reason, the co-belonging base fragments or the adjacent body fragments of the vases are oriented in such a way that they join (Section 2), when the B_xy are placed in parallel directions.

With the proposed perspective of the sensors’ mode, the ceramic area that generates the magnetic field, which is detected by the sensors and the normalization constants (λ, α^˪, α^//) of their readings from measurements of the magnetic field in base vase fragments, the magnetization of co-belonging body fragments of vases 1–6 is computed.

7. Computation of Magnetization in Body Fragments from Vases 1–6

Measurements by the three-dimensional orthogonal system of sensors/magnet apparatus are taken in the same position, with the x-sensor oriented perpendicularly to the horizontal and vertical component of the geomagnetic field. The measurement positions are situated at the edges of a mesh formed from lines vertical to the traces of the grooves from the formation of the six vases (Figure 4b) on a pottery wheel. To exploit the rotational symmetry of vases (Figure 14a), the measurements are taken with respect to a right-hand coordinate system, with the x-axis along the groove direction, the y-axis perpendicular to the surface and pointing towards the inside of the fragments, and the z-axis pointing towards the base of the vases.

The components Mi (i = x, y, z) are computed from the measurements B_i of the field, in the rotating position where the grooves are oriented in the direction of the x-sensor.

The components of the measured field, B, and of the magnetization, M, in the considered coordinate system (Figure 14a) are computed by the relations:

B_x = B. sinθ_B. cosφ, B_y = B.sinθ_B.sinφ, B_z = B.cosθ_B, M_x = M. sinθ.cosφ, M_y = M.sinθ. sinφ, M_z = M.cosθ

To determine the deviation, θ, of the magnetization from the vertical (Figure 14b), measurements of the B_xz, B_y are taken, by rotating the fragments at an angle φ_B = φ and orienting B_xz in the x-direction, in the position where B_z = 0 and B_xz > 0.

The distance, D, between the measurement position and the fragment edge in the common direction of the M_xz,B_xz (Figure 14c) is measured in the negative or positive x-semi-axis, when B_y > 0 or B_y < 0.

The distance D (Figure 14d) may be sufficient (D > d = L.|tanθ|) or insufficient (D < d), for the measured fragment thickness, L, at the measurement position. In any case, the magnetization angles θ = θ_D or θ = θ_L are approximated [41] by the solution of the 3rd degree equations (using the Newton-Raphson method):

$$\begin{gathered} \mathrm{D}>\mathrm{d}:\frac{{\mathrm{B}}_{\mathrm{xz}}}{{\mathrm{B}}_y}{=\tan \mathsf{\theta }}_{\mathrm{B}}{=\tan \mathsf{\theta }}_{\mathrm{L}}.\sqrt{\frac{1+(\frac{{\mathsf{\alpha }}_y{.\cos }^2{\mathsf{\theta }}_{\mathrm{L}}}{\mathrm{L}}{)}^2}{1+(\frac{{\mathsf{\alpha }}_{\mathrm{x},\mathrm{z}}{.\cos }^2{\mathsf{\theta }}_{\mathrm{L}}}{\mathrm{L}}{)}^2}} \\ \mathrm{D}<\mathrm{d}:\frac{{\mathrm{B}}_{\mathrm{xz}}}{{\mathrm{B}}_y}{=\tan \mathsf{\theta }}_{\mathrm{B}}{=\tan \mathsf{\theta }}_{\mathrm{D}}.\sqrt{\frac{1+(\frac{{\mathsf{\alpha }}_y{.\sin 2\mathsf{\theta }}_{\mathrm{D}}}{2\mathrm{D}}{)}^2}{1+(\frac{{\mathsf{\alpha }}_{\mathrm{x},\mathrm{z}}{.\sin 2\mathsf{\theta }}_{\mathrm{D}}}{2\mathrm{D}}{)}^2}} \end{gathered}$$

(7)

The constants α_xz^//,α_y^˪, λ are computed (Table 2) from the field measurements in co-belonging base fragments of the vases:

-

If θ_L > θ_D, then D > d = L.tanθ_L and θ = θ_L. For the computation of the components of the magnetization, M, the sufficient length Equations (4) are used.

-

-If θ_L < θ_D, then D < d = L.tanθ_D and θ = θ_D. For the computation of the components of the magnetization, M, the insufficient length Equations (5) are used.

Experimental results (analytical results at each calculation stage of magnetization are presented in “Supplementary Files” for the body of vase 6. For the reproducibility of the results, the magnetic field measurements for the remaining vases in each position of their body are listed therein) show that the computed magnitude, M, displays similar values in the base and body fragments, which is a criterion for finding ceramic fragments of the same vase from archaeological excavations. The angles, θ, φ, change in a systematic way because their values are dictated by the rotational symmetry of the vase. This is confirmed in the next section, where the position of fragments in the bodies of 6 vases is identified by successive transformations of magnetization components.

8. Position Localization of Fragments in the Body of a Vase from the Directivity of the Remanent Magnetization

If γ_earth is the inclination and ξ the declination of the local magnetic field of the Earth (B_Earth) during pottery firing (Figure 15), then its components with respect to the right-hand axial coordinate system, XYZ, of the vase, where the Z-axis of rotational symmetry of the vase deviates from the vertical at α deviation angle,α, of the vase on the kiln floor and points towards its base, are computed by the equations:

B_X^Earth = B_Earth. sinγ. cosξ, B_Υ^earth = B_earth.sinγ. sinξ, B_Z^Earth = B_earth. cosγ, γ =(90° − γ_Earth) ± α

(8)

The declination, ξ, of the geomagnetic field cannot be computed without knowing the vase orientation in the kiln. The inclination, γ_earth, of the geomagnetic field cannot be computed without knowing the deviation angle, α, of the kiln with respect to the horizontal plane.

The circular grooves in the vase body, which are parallel to its base, deviate from the horizontal plane according to the same angle, α, due to the rotational symmetry of the base. In any vase with rotational symmetry, and for any considered axial reference system where the z-axis coincides with the axis of rotation, the X,Y axes are defined on the plane of the same transverse groove on the vessel body.

The angle, γ, of inclination of the geomagnetic field from the vase axis of symmetry, is equal to the deviation angle of magnetization, M, calculated from measurements of the magnetic field of base fragments (Appendix B.6 and Appendix B.7) since the ceramic material magnetization obtains the orientation of the geomagnetic field during firing.

8.1. Formation of Remanent Magnetization in Vases with Cylindrical Symmetry

Measurements of the field (Figure 16a), B, are obtained in right-hand reference systems (xyz)*, where the x^*-axis has the direction of the transverse grooves, the y^*axis passes through the axis of rotational symmetry and the z-axis is tangential to the side walls, directed towards the vase base.

In the specific case of vases with cylindrical symmetry, because the z*-axis in the measurement system of the B field is parallel to the axis of symmetry, Z, (Figure 16b) the position of each longitudinal axial section is defined by the rotation angle, u, of the right-hand axial reference system, XYZ, of the cylindrical vase (Figure 16c), around the Z-axis of rotational symmetry.

Since magnetization, M, obtains the orientation of B_earth in the pottery kiln, then for every rotation angle, u, of the axial reference system, XYZ, the M-magnetization components at the fringe of each transverse groove are determined by the rotation transformation of the peripheral reference systems (xyz)* at an angle ω = ξ − u around the z*-axis:

M_x* = M·sinγ·cosω = M_xy*· cosω, M_y* = M· sinγ·sinω = M_xy*· sinω, M_z* = M·cosγ

(9)

where: ω = ξ − u, γ = (90° – γ_Earth) ± α

Although the angles u, ξ cannot be determined without knowing the orientation of the vase in the pottery kiln, the angle ω = ξ – u is determined by calculating the magnetization of the fragments of the vase body.

8.2. Formulation of Remanent Magnetization in Vases with Arbitrary Rotational Symmetry

In the general case of vases with arbitrary rotational symmetry (Figure 17a), the side walls diverge at different angles, κ, from the axis of symmetry in each longitudinal section, (α,β,γ,δ), of the vase and at the same angle κ in each transverse groove, (1,2,3).

The components M_i (i = x,y,z) in the reference systems (xyz) of the B_i field measurements (Figure 18) are derived from a rotational transformation (Figure 17b) of the reference systems (xyz)* of the cylindrical vase at an inclination angle κ around the x*-axes:

$$\begin{gathered} \mathrm{M}{\mathrm{x}}_{ }= {\mathrm{M}}_{\mathrm{x}}{*}^{ }= \mathrm{M}·\sin \mathsf{\gamma }·\cos \mathsf{\omega } =\mathrm{M}·\sin \mathsf{\theta }·\cos \mathsf{\phi } = \mathrm{M}·\cos {\mathsf{\phi }}_{\mathrm{x}}, \\ \mathrm{M}{\mathrm{y}}_{ }= {\mathrm{M}}_{\mathrm{y}}*. \cos \mathsf{\kappa }-{\mathrm{M}}_{\mathrm{z}}*·\sin \mathsf{\kappa } = \mathrm{M}. \sin \mathsf{\gamma }. \sin \mathsf{\omega }·\cos \mathsf{\kappa } - \mathrm{M}. \cos \mathsf{\gamma }·\sin \mathsf{\kappa } = \mathrm{M}.\cos \mathsf{\theta } \\ \mathrm{M}{\mathrm{z}}_{ }= {\mathrm{M}}_{\mathrm{y}}*·\sin \mathsf{\kappa } + {\mathrm{M}}_{\mathrm{z}}*.·\cos \mathsf{\kappa } = \mathrm{M} \sin \mathsf{\gamma }·\sin \mathsf{\omega }. \sin \mathsf{\kappa } + \mathrm{M}·\cos \mathsf{\gamma }·\cos \mathsf{\kappa } = \mathrm{M}. \sin \mathsf{\theta }·\cos \mathsf{\phi }, \\ \mathsf{\omega } = \mathsf{\xi }-\mathrm{u} \mathsf{\kappa }\mathsf{\alpha }\mathsf{\iota } \mathsf{\gamma } = (90° - {\mathsf{\gamma }}_{\mathrm{earth}}) \pm \mathsf{\alpha } \end{gathered}$$

(10)

❖ Based on the above transformation, the circumferential position of the fragments in each cross section of the vase is determined by the angle ω,which is computed by the formula:

$${\mathsf{\omega }=\cos }^{-1}\frac{\sin \mathsf{\theta }.\cos \mathsf{\phi }}{\sin \mathsf{\gamma }}$$

(11)

The angles $\tan \mathsf{\phi }=\frac{{\mathrm{M}}_{\mathrm{z}}}{{\mathrm{M}}_{\mathrm{x}}}=\frac{{\mathrm{B}}_{\mathrm{z}}}{{\mathrm{B}}_{\mathrm{x}}}$, $\cos \mathsf{\theta }=\frac{{\mathrm{M}}_y}{\mathrm{M}}$ are calculated from the magnetic field measurements of the body fragments, while the angle γ is calculated from the magnetic field measurements in co-belonging fragments of the vase base.

❖ The height-position of the fragments in each longitudinal section of the vase is determined by the inclination, κ, which is computed by the equation:

$${\mathsf{\kappa }=\tan }^{-1}\frac{{\mathrm{M}}_{\mathrm{z}}.\sin \mathsf{\gamma }{.\sin \mathsf{\omega }-\mathrm{M}}_y.\cos \mathsf{\gamma }}{{\mathrm{M}}_y.\sin \mathsf{\gamma }{.\sin \mathsf{\omega }+\mathrm{M}}_{\mathrm{z}}.\cos \mathsf{\gamma }}$$

(12)

Determining the change in directionality of remanent magnetization, M, in the vase body in a manner dictated by its rotational symmetry, is performed from the calculated angles ω, κ at each measurement position, using the inverse rotational transform of the calculated magnetization components at an angle -κ around the x-axis (Figure 19a), from the reference system (xyz), where the magnetic field measurements are taken, to the (xyz)* reference system (Figure 19b) of the cylindrical vase:

$$\begin{gathered} {\mathrm{M}}_{\mathrm{x}}*={\mathrm{M}}_{\mathrm{x}}=\mathrm{M}. \sin \mathsf{\gamma }.\cos \mathsf{\omega }, {\mathrm{M}}_{\mathrm{y}}{*}_{ }= {\mathrm{M}}_{\mathrm{y}}.\cos \mathsf{\kappa } + \mathrm{M}{\mathrm{z}}_{ }\sin \mathsf{\kappa } = \mathrm{M}.\sin \mathsf{\gamma }.\sin \mathsf{\omega }, \\ {\mathrm{M}}_{\mathrm{z}}{*}_{ }= {\mathrm{M}}_{\mathrm{z}}.\cos \mathsf{\kappa } – {\mathrm{M}}_{\mathrm{y}}.\sin \mathsf{\kappa } = \mathrm{M}.\cos \mathsf{\gamma } \end{gathered}$$

(13)

In the reverse transformation (Figure 19b), the components M_x*_,M_y* and the angle ω should display similar values in each longitudinal section of the body, based on the above equations, while M_z*, which depends only on angle γ, should display similar values in each measurement position. Indicative experimental results with the reverse transformation are given from measurements of body fragments (

Table 3. Indicative values of the transformed components, Mi* (i = xyz), of magnetization in the reference systems (xyz)* of the cylindrical vase, in fragments of the body (a) and the base (b) from vase 6. The values inside the colored frames computed by the insufficient length (5) relationships.

Table 3a. (Body fragments measurements)
Vase 6		A	B	Γ	Δ	E	Z	H	Θ	Ι	Κ
1	My*(mA/m)	−10 ± 6			45 ± 4	35 ± 5	11 ± 7	−16 ± 5			−38 ± 3
	Mx*(mA/m)	−34 ± 2			−8 ± 3	8 ± 3	38 ± 2	37 ± 2			2 ± 2
	Mz*(mA/m)	37 ± 3			49 ± 4	39 ± 4	42 ± 3	43 ± 3			40 ± 3
	ω°	197 ± 8			100 ± 10	77 ± 12	16 ± 9	336 ± 5			272 ± 2
	κ°	−40 ± 8			−31 ± 3	−29 ± 4	−22 ± 9	−33 ± 5			−30 ± 3
2	My*(mA/m)	−10 ± 8	10 ± 6	33 ± 4	43 ± 5	42 ± 4	−7 ± 9	−22 ± 6	−31 ± 6	−34 ± 5	−44 ± 4
	Mx*(mA/m)	−41 ± 2	−35 ± 2	−28 ± 2	−9 ± 2	9 ± 2	40 ± 2	38 ± 2	24 ± 2	10 ± 2	−2 ± 2
	Mz*(mA/m)	45 ± 2	39 ± 2	46 ± 2	46 ± 3	46 ± 3	43 ± 2	47 ± 3	42 ± 3	37 ± 3	46 ± 2
	ω°	194 ± 7	163 ± 8	131 ± 4	101 ± 3	78 ± 3	350 ± 10	331 ± 3	308 ± 3	286 ± 3	268 ± 2
	κ°	−16 ± 8	−19 ± 8	−12 ± 3	−14 ± 3	−6 ± 2	−18 ± 11	−18 ± 5	−11 ± 5	−16 ± 5	−6 ± 2
3	My*(mA/m)	3 ± 18	15 ± 6	26 ± 5	40 ± 5	41 ± 5	−6 ± 11	−12 ± 6	−26 ± 4	−43 ± 4	−41 ± 3
	Mx*(mA/m)	−43 ± 2	−32 ± 2	−21 ± 2	−5 ± 2	3 ± 2	41 ± 2	40 ± 2	25 ± 2	9 ± 2	−9 ± 2
	Mz*(mA/m)	46 ± 2	37 ± 3	35 ± 3	43 ± 3	43 ± 3	45 ± 2	44 ± 2	38 ± 2	46 ± 2	44 ± 2
	ω°	176 ± 2	155 ± 5	129 ± 3	97 ± 2	85 ± 2	352 ± 12	344 ± 7	313 ± 5	281 ± 3	258 ± 4
	κ°	24 ± 23	13 ± 7	19 ± 5	17 ± 4	21 ± 4	9 ± 13	14 ± 7	16 ± 4	14 ± 2	18 ± 2
4	My*(mA/m)	9 ± 7	12 ± 6	32 ± 5	42 ± 5	41 ± 5	−10 ± 9	−14 ± 7	−24 ± 5	−43 ± 3	−40 ± 3
	Mx*(mA/m)	−40 ± 2	−41 ± 2	−18 ± 2	−2 ± 2	8 ± 2	37 ± 2	35 ± 2	33 ± 2	2 ± 2	−4 ± 2
	Mz*(mA/m)	44 ± 3	46 ± 3	38 ± 4	45 ± 4	44 ± 4	41 ± 3	40 ± 3	43 ± 3	45 ± 3	43 ± 3
	ω°	168 ± 9	164 ± 6	119 ± 3	92 ± 2	79 ± 2	345 ± 11	338 ± 10	324 ± 8	273 ± 20	265 ± 14
	κ°	42 ± 9	34 ± 6	32 ± 5	31 ± 4	40 ± 4	19 ± 11	37 ± 9	32 ± 5	41 ± 2	38 ± 2
5	My*(mA/m)	−11 ± 9	15 ± 6	30 ± 3	42 ± 4	36 ± 4	6 ± 13	−19 ± 6		−37 ± 5
	Mx*(mA/m)	−38 ± 2	−31 ± 2	−18 ± 2	−13 ± 2	11 ± 2	39 ± 2	32 ± 2		7 ± 2
	Mz*(mA/m)	42 ± 3	36 ± 4	37 ± 3	46 ± 4	40 ± 5	42 ± 3	40 ± 4		40 ± 5
	ω°	197 ± 13	154 ± 8	120 ± 3	107 ± 2	73 ± 3	9 ± 20	330 ± 11		280 ± 58
	κ°	41 ± 11	57 ± 8	59 ± 4	48 ± 4	52 ± 5	59 ± 18	41 ± 8		49 ± 6
	Table 3b (Vase fragments measurements)
	Mz(mA/m)	42,5±0,2
	Mxy(mA/m)	39,8±0,2
	M(mA/m)	58,3±0,2
	γο	43,3±0,3

) from vase 6.

The components M_x*_,M_y* and the angle ω display similar values (Table 3a) in each longitudinal section of the body. Inclination, κ, displays similar values in each transverse groove of the vase.

M_z* displays similar values (Table 3b) in each measurement position and with that which computed for the base fragments of the vase.

8.3. Experimental Results: Screening of Co-Belonging Fragments and Determination of Their Position in the Body of a Vase from Their Magnetization

The computed values for the inclination, κ, and the angle ω are displayed (Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30 and Figure 31) on the bodies of the six vases. The magnetization magnitude M (mA/m) in body fragments displays similar values (

Table 4. The magnetization magnitude in body and vase fragments of vase 1 display similar values. The values inside the colored frames computed by the insufficient length relationships (5).

Vase 1	M(Body)
	A	B	Γ	Δ	E	Z	H	Θ	Ι
1	120 ± 4	138 ± 6	130 ± 7	135 ± 7	129 ± 7	133 ± 5	141 ± 5	128 ± 3
2	135 ± 3	126 ± 5	136 ± 6	132 ± 6	133 ± 5	132 ± 5	132 ± 3	127 ± 3
3	138 ± 3	139 ± 5	139 ± 5	138 ± 6	127 ± 5	137 ± 4	128 ± 3	124 ± 2
4	130 ± 3	125 ± 4	130 ± 4	131 ± 5	135 ± 5	121 ± 4	130 ± 3	131 ± 3	123 ± 3
5	119 ± 3	137 ± 3	140 ± 5	132 ± 5	138 ± 4	128 ± 3	129 ± 3	138 ± 3	131 ± 2
6	125 ± 3	128 ± 3	131 ± 4	131 ± 4	129 ± 4	123 ± 3
7	138 ± 3	132 ± 3	121 ± 4	121 ± 4	124 ± 4	130 ± 3
8	133 ± 3	141 ± 3	123 ± 3	132 ± 4	135 ± 3	134 ± 3
9		125 ± 3	131 ± 3	131 ± 4	133 ± 3	133 ± 4
	M(Base)		131.3 ± 0.2(mA/m)

,

Table 5. The magnetization magnitude, M, in body and vase fragments of vase 2 display similar values. The values inside the colored frames computed by the insufficient length relationships (5).

M(Body)

A

B

Γ

Δ

E

Z

H

Θ

Ι

K

Λ

M

Ν

Ξ

O

Π

P

Σ

1

182 ± 12

153 ± 12

179 ± 14

169 ± 3

171 ± 3

168 ± 4

2

185 ± 10

167 ± 5

172 ± 2

163 ± 4

173 ± 3

178 ± 4

168 ± 5

170 ± 6

3

181 ± 7

184 ± 7

181 ± 8

167 ± 3

167 ± 3

174 ± 4

164 ± 3

159 ± 4

170 ± 6

170 ± 6

163 ± 5

4

185 ± 5

169 ± 4

164 ± 4

173 ± 4

162 ± 3

170 ± 3

169 ± 4

174 ± 4

166 ± 4

174 ± 6

173 ± 5

170 ± 5

5

186 ± 4

175 ± 5

168 ± 4

163 ± 3

164 ± 3

161 ± 2

168 ± 2

169 ± 3

169 ± 4

173 ± 5

178 ± 7

174 ± 7

167 ± 6

6

179 ± 3

171 ± 2

174 ± 3

172 ± 4

158 ± 4

164 ± 4

169 ± 3

169 ± 3

168 ± 3

168 ± 2

164 ± 3

171 ± 4

174 ± 5

170 ± 7

173 ± 7

182 ± 7

7

156 ± 6

175 ± 3

170 ± 2

162 ± 3

169 ± 3

168 ± 4

161 ± 4

167 ± 4

170 ± 3

168 ± 3

173 ± 3

164 ± 3

176 ± 4

173 ± 5

177 ± 6

185 ± 8

170 ± 7

178 ± 8

8

175 ± 7

163 ± 4

167 ± 2

164 ± 3

170 ± 4

176 ± 4

183 ± 5

180 ± 4

161 ± 3

165 ± 3

159 ± 2

165 ± 2

162 ± 3

171 ± 5

167 ± 6

179 ± 9

9

175 ± 7

163 ± 4

167 ± 2

164 ± 3

170 ± 4

176 ± 4

183 ± 5

180 ± 4

161 ± 3

159 ± 3

169 ± 2

171 ± 3

166 ± 3

173 ± 7

178 ± 10

M(Base)

169.3 ± 0.6 (mA/m)

,

Table 6. The magnetization magnitude in body and base fragments of vase 3 display similar values. The values inside the colored frames are computed by the insufficient length relationships(5).

M(Base)	M(Body)	A	B	Γ	Δ	E	Z	H	Θ	Ι	Κ
164.1 ± 0.4 (mA/m)	1	167 ± 8	178 ± 8		163 ± 4	159 ± 3	165 ± 4	175 ± 3	173 ± 3		194 ± 6
	2	166 ± 5	154 ± 4		153 ± 3			177 ± 3		169 ± 3	169 ± 5
	3	160 ± 3		158 ± 3	161 ± 4	171 ± 6	176 ± 6	160 ± 4	164 ± 3	157 ± 3	173 ± 3
	4	168 ± 3	161 ± 3	162 ± 3	162 ± 5	161 ± 5	181 ± 7	162 ± 5	166 ± 5	167 ± 4	155 ± 3
	5	156 ± 4	175 ± 3	159 ± 2	154 ± 4	172 ± 6	160 ± 7	164 ± 5	155 ± 2	173 ± 3	167 ± 3

,

Table 7. The magnetization magnitude in body and vase fragments of vase 4 display similar values. The values inside the colored frames are computed by the insufficient length relationships (5).

M(Base)	M(Body)	A	B	Γ	Δ	E	Z	H	Θ	Ι
84.1 ± 0.2 (mA/m)	1			81 ± 3	78 ± 3		93 ± 3		89 ± 2
	2		97 ± 3	86 ± 3	85 ± 3	92 ± 3	72 ± 3	76 ± 3	76 ± 3	77 ± 2
	3	82 ± 3	78 ± 3	78 ± 4	77 ± 3	84 ± 3	85 ± 3	88 ± 3	89 ± 3	76 ± 2
	4			87 ± 4	76 ± 4	90 ± 4	80 ± 4	78 ± 4	78 ± 3	82 ± 3
	5			84 ± 4	85 ± 4	82 ± 4	77 ± 4	85 ± 4	85 ± 3	89 ± 3

,

Table 8. The magnetization magnitude in body and vase fragments of vase 5 display similar values. The values inside the colored frames are computed by the insufficient length relationships (5).

M(Body)	A	B	Γ	Δ	E	Z	H	Θ
1	110 ± 3	114 ± 5	110 ± 3	115 ± 4	111 ± 3	111 ± 3	113 ± 3	116 ± 3
2	114 ± 3	114 ± 3	111 ± 3	113 ± 3	112 ± 3	110 ± 3	112 ± 3	110 ± 3
3	108 ± 3	111 ± 3	112 ± 3	112 ± 3	113 ± 3	112 ± 3	111 ± 2
4	113 ± 4	112 ± 3	111 ± 3	113 ± 3	111 ± 3	115 ± 3	118 ± 3
5		116 ± 4	116 ± 3	113 ± 3	114 ± 3	110 ± 3	113 ± 3
6			114 ± 4	116 ± 3	110 ± 3	115 ± 3	111 ± 3
7			113 ± 4	117 ± 4	114 ± 3	108 ± 4	111 ± 5
8			119 ± 4	113 ± 3	112 ± 3	113 ± 4	117 ± 4
9			111 ± 3	114 ± 3	113 ± 3	116 ± 3	116 ± 4
10			119 ± 3	116 ± 3	115 ± 3
M(Base)		112.7 ± 0.3 (mA/m)

and

Table 9. The magnetization magnitude in body and base fragments of vase 6 display similar values. The values inside the colored frames are computed by the insufficient length relationships (5).

M(Base)	M(Body)	A	B	Γ	Δ	E	Z	H	Θ	Ι	Κ
58.3 ± 0.2 (mA/m)	1	51 ± 2			67 ± 4	53 ± 4	58 ± 3	58 ± 2			56 ± 3
	2	62 ± 2	53 ± 2	64 ± 3	63 ± 4	63 ± 3	59 ± 2	64 ± 3	57 ± 4	51 ± 4	64 ± 3
	3	63 ± 2	51 ± 3	48 ± 4	59 ± 4	60 ± 4	61 ± 2	61 ± 2	52 ± 3	63 ± 3	61 ± 3
	4	60 ± 3	63 ± 2	53 ± 4	62 ± 4	61 ± 4	56 ± 3	55 ± 3	59 ± 3	62 ± 3	59 ± 3
	5	57 ± 3	50 ± 4	51 ± 3	64 ± 4	55 ± 4	58 ± 3	54 ± 4		55 ± 5

), and with those calculated from the base fragments of the vases.

The slope, κ (Figure 20, Figure 22, Figure 24, Figure 26, Figure 28 and Figure 30) which displays similar values in each cross-section and determines the height-position of the fragments, is compared with the measured inclination of the walls with respect to the axis of symmetry in each vase. The computed values for the inclination, κ, from the sensor readings are similar to the measured ones.

The angle ω (Figure 21, Figure 23, Figure 25, Figure 27, Figure 29 and Figure 31) displays a similar value in each longitudinal section and varies at the transverse measurement positions, defining the peripheral position of the fragments on the body of each vase.

From the orientation (ω) of the transformed M_xψ* in the vessel body and of M_xψ in the fragments of the base (Figure 21, Figure 23, Figure 25, Figure 27, Figure 29 and Figure 31), the jointing position of the base and the body of each vase is determined. The shape of the original vase can be reconstructed when only a few non adjacent fragments are available.

9. Conclusions and Proposals for Future Research

From the experimental results it can be seen that despite the small range of the magnetization magnitude in the six vases, the screening of excavation fragments can be done on the basis of their similar magnetization values. The position of co-belonging body fragments of the vases is determined with satisfactory precision by calculating the angles ω, κ.

The laboratory environment in which the proposed device is intended to be used is a large space, in which the vase fragments of each excavation layer are placed on platforms and separated into the excavated squares in which they were found. From each excavation block the characteristic fragments belonging to the bases, the orifices or handles of the vases and the fragments with written or embossed decoration are separated. The characteristic fragments determine the number of surviving vases and the search for co-belonging shells, which, in the simplest case, are located in adjacent excavated squares. In the common cases of perturbed excavation layers due to natural phenomena (such as floods) or illegal excavations, the procedure is more difficult, because the co-belonging fragments must be sought in the excavation layers above or below, in each excavation square, or in remote excavation squares. The positioning of the fragments in adjacent excavation layers and squares is of critical importance for the finding of co-belonging fragments. The investigation and screening of co-belonging fragments is performed by conservators.

The apparatus is not recommended to be used with fragments of characteristic portions of the vases (nozzles, handles), with fragments with written or embossed decoration, because it is easier for them to be located in an empirical manner.

Small vases of daily use (e.g., plates, cups), without characteristic portions on their surface may have been manufactured from the same clay and fired in the same kiln under the same conditions. In that case the co-belonging vase fragments cannot be distinguished from the magnitude of thermoremanent magnetization, because it displays similar values. This is a very useful information on their manufacturing technique and on the origin of the raw material. The fragments from each vase can be distinguished from their different magnetization directivity at the assembly stage. The method is ineffective in the rather rare case of vases of the same size and shape from the same clay, placed in the same orientation in the kiln, relative to the Earth’s magnetic field.

It should be noted that the position of small vase fragments with high curvature is hampered, at least with the ~3 cm fluxgate sensors used in this research. The error in calculating the angles ω, κ increases as the sensor contact with the fragment surface becomes smaller. Magnetic field measurements in larger curvature fragments can be performed with shorter-length sensors.

However, in any case, it is a small number of fragments with characteristic curvature, which are easier to categorize and assemble empirically.

The apparatus is recommented to be used for the location and assembly of unidentified fragments from medium-size vases (basins, hydries) or larger-size (usually pithos vases), which are the most common excavation findings.

In medium-size vases, which display a variety of shapes due to their different uses, the risk of deformation during drying is minimized with the admixture of aggregates, depending on their shape and size. The possibility of finding medium sized pots of the same clay fired in the same kiln in the same excavation layer is small. In the event that the measured magnitude shows similar values, the above applies.

Regarding large size vases for storage use, these were fired in one-use kilns built around the vase, from antiquity to today. Therefore the possibility of two large vases from the same clay to have been fired in the same kiln is very small.

The methodology can be used to screen fragments from archaeological excavations under the following conditions:

-

Finding co-belonging fragments of the vase base is essential for calculating the magnetization deviation angle, γ, and the parameters α^˪, α^//, λ from measurements of their magnetic field. This is not a problem, because in the usual process of screening co-belonging fragments, the base fragments, which are easily identified, are initially collected and the neighboring fragments of the vase body are then searched.

-

In the fragment reference system, where measurements are taken, the x-axis is oriented in the direction of the vase grooves while the y-axis is perpendicular to the surface and directed towards the inside of the fragments. To take measurements on all fragments in relation to the same reference system, the z-axis must be oriented towards the base (or the aperture) of the vase. The orientation of each fragment with respect to the base or the aperture of the vase can be done easily, on the basis of its shape and the curvature of the grooves on its surface.

Proposals for future research, based on additional preliminary investigations, are given below.

1) Experimental application of the method in fragments from archaeological excavations.

2) Construction of ceramic specimens from different raw materials and with different granulometry, fired in the same pottery kiln under different slopes of the kiln floor.

-

Construction of ceramic specimens from the same raw material, fired in the same kiln, with the same slope, but under different oxidative conditions and at different temperatures.

-

Analysis of the composition of the specimens and their content in magnetic oxides.

The above are necessary in order to study the dependence of remanent magnetization on the clay composition, the temperature/ventilation conditions in the pottery kiln, the orientation of the geomagnetic field, and the degree of magnetic anisotropy of the ceramic material. For the above research a traditional wood kiln is required. In ceramic specimens made from different commercially available clays fired in an electric kiln, the measured magnetic field was less than 10nT and without clear orientation, as a result of magnetic fields from the electric currents of the kiln.

3) Investigation of the sensitivity radii dependence of the vertical (α^˪) and parallel (α^//) sensors on the magnetization magnitude of the ceramic material.

-

-Investigation of the mode, the sensors’ degree of excitation, and of the dependence of the normalization constant, λ, of the vertical and parallel sensor readings as well as of α^˪, α^//, on measurements of the same specimens at different distances from their surface. A prerequisite for these are magnetic field measurements in ceramic materials with higher thermoremanent magnetization, so that their magnetic field is attenuated at longer distances from their surface.

-

Investigation of the dependence of the normalization constant, λ, of the vertical and parallel sensor readings and their correlation with the dimensions and technical characteristics of the sensors from measurements of the same specimens with different types of fluxgate sensors.

4) Investigation of the feasibility of constructing a handy three-sensor fluxgate magnetometer for local measurements of weak surface fields (1–1000 nT) in ceramic materials with a magnetization of 1–1000 mA/m for archaeological use.

5) Investigation of induced magnetization of ceramic materials in alternating magnetic fields as an additional criterion for the screening of co-belonging excavation fragments.

Experiments were carried out by the successive placement of cylindrical ceramic specimens from co-belonging vase fragments at specific locations inside a solenoid (L~3 mH) in an RLC circuit in a field (~50 μΤ), and the induced magnetization was calculated by increasing the inductance of the coil at the resonance frequency [34] (pp. 21–34) of the circuit (Null Detection technique). In sequential addition of co-belonging tiles at different resonance frequencies (f = 40–95 KHz), similar increase in their induced magnetization was observed, which is a criterion for distinguishing samples from different vases.

-

Investigation of the possibility of constructing a handy instrument for local surface measurements of induced magnetization of ceramic materials from an alternating magnetic field for archaeological use for the purpose of screening the co-belonging excavation fragments.

6) Investigation of the application of the method in rocks with remanent magnetization, for the localization of changes in local soil morphology, by comparing the orientation of the magnetic field of the parent rock and its detached parts. Experiments were conducted in fragments from serpentinite and were found to be oriented in a joining manner, from their thermoremanent magnetization.

These studies will be presented in follow up publications in the near future.