Forward magnetic models: Creation and calculation

Forward magnetic models: Creation and calculation by: Bruce W. Bevan (Geosight)

date: 17 August 2016 DOI:10.13140/RG.2.1.2296.0242/1

Summary A magnetic model describes a magnetic object: Its shape, size, depth, and magnetic properties. A forward magnetic model is created in order to calculate the magnetic map of an actual or assumed object. This calculation may have several benefits: 1: Predict whether a feature might be detectable with a magnetic survey. 2: Verify if a feature found in an excavation caused the anomaly that was measured. 3: Discriminate the relative effects of different parts of a feature. 4: Create a good initial model for an automated inversion program. A forward magnetic model is different from an inverse magnetic model, which is an estimate of a magnetic object that has been interpreted from the measurements of a magnetic map. While the geometry of a magnetic object may be easy to estimate, the magnetic properties of that object may be more difficult to measure or guess. However, some of these magnetic properties may be measured in the field with simple techniques. A hand-held magnetic susceptibility meter can be essential. Remanent (permanent) magnetization of objects can be estimated with several techniques: Rotate a sample near a magnetometer, measure magnetic maps over the sample, or determine the magnetic anomaly of the sample in six perpendicular directions. After the parameters of the magnetic model have been found, the magnetic field of that model can be calculated with a computer program; many of these are available for no cost, such as: Pdyke, MagPrism, GaMField, Noddy, Potensoft, 3dFFT, grav mag prism, and Fatiando a Terra. Probably all large commercial programs for magnetic analysis will allow these forward calculations; examples include Potent, ModelVision, and Emigma. There are three sections to this report. The first part describes how a magnetic model may be constructed; complications are also noted. The second part illustrates how the magnetic field of models can be calculated. The third part has examples of about ten different magnetic models. Detailed notes and references to articles, books, and computer programs are included in an appendix. Hyperlinks to these details and to the figures (which are at the end of the report) are indicated with blue text. The captions for the figures have enough information that they can be understood separately, without the main text here. The captions also contain information that is not included in this text. Introduction A magnetic model is an approximation of a magnetic object; it may be a simplification of both the shape and the magnetic properties of that object. One can calculate the magnetic field of this model and the resulting patterns will suggest the measurements that could be made with a magnetometer over an object that is similar to the model. This magnetic model can be approximate and simple primarily because a magnetic map cannot reveal small details in a magnetic body. References to two excellent papers on magnetic models are in Appendix 1. Page 1

Construct a magnetic model A magnetic model may be generated from two "directions". It can be estimated from an analysis of some magnetic measurements (a map or a line of readings) that have been measured above a buried feature. This is called an inversion of the measurements, and the result is called an inverse model. A magnetic model may also be constructed by knowing (from excavations) or estimating (from other knowledge) the geometry and magnetism of a buried feature; the magnetic field from that model may then be calculated. This is called the calculation of a forward model. In other words: A forward magnetic model starts with measurements of dimensions and magnetic properties of a real or assumed feature and ends with a calculated magnetic map. An inverse magnetic model starts with a measured magnetic map and ends with an interpreted model that suggests the shape and magnetic properties of a buried feature. Examples of forward magnetic models are illustrated in Figure 1; these have simple shapes, and they do not reveal the exact magnetic properties of the bodies. The magnetic field of these models may be calculated, and then one can: Compare the calculated anomaly of an excavated feature with the anomaly that was measured above the feature. This can indicate if the model explains the full anomaly; if not, there may be additional parts of the feature that are undiscovered, perhaps because they are deeper underground. Find how the amplitude of anomalies changes with the depth of the feature (this is difficult to do otherwise); one can also estimate how the feature would appear on sloping ground and also with the disturbance of noise and unwanted magnetic objects. Determine if small parts of a feature may be resolved or visible in a magnetic map. Start an inversion with a closer approximation to the final model; this may prevent the inversion from finding an unreasonable solution. Estimate the relative effect of different parts of a complex feature, revealing what is most detectable and what is least detectable; Figure 2 has an example of this. This report was prepared because there are many articles about inverse models, but rather few about forward models. While this report has archaeological examples, the ideas here can also be applied to engineering and environmental geophysics. The magnetic measurements here emphasize the total magnetic field, although the ideas apply to surveys with a gradiometer also. Construct a magnetic model The first step in creating a magnetic model requires the estimation of the shape of the body. The actual shape may be known from excavations of a similar feature; sometimes this shape can be estimated from other knowledge. The shape of the object in the model will almost always be much simplified from the exact shape of the body. This simplification is possible because a magnetic survey can never reveal small details about features; the magnetic images are quite blurred, and they become increasingly blurred where objects are deeper underground. At first, it may be necessary to make the model more detailed than necessary; compare the calculations of this detailed model to those of a simpler model and see if that Page 2

Measure magnetic properties simple model is good enough. As a general rule, parts of a model that are deeper (below the magnetometer) than their separation cannot be distinguished from each other. Objects that are deeper than their diameters may possibly be modeled as spheres, even though the objects are much more complicated than that. Spheres are an important type of model. It was discovered long ago that all spheres cause magnetic anomalies with the same shape; if the center of a sphere is at a fixed depth, the shape of its magnetic anomaly does not change with the radius of the sphere, as long as it is less than the depth underground. Therefore, it is not possible to estimate the diameter of objects that are deep and compact (with all dimensions roughly equal) unless the magnetic properties of the object are known. However, if the susceptibility of a sphere is known, the radius of that sphere may be determined. Uniform magnetic layers or strata are another important type of magnetic model. They have the valuable property that a constant value of magnetic susceptibility may be added or subtracted from any layer without changing the magnetic calculations. In fact, the susceptibility of any planar layer that is infinitely-wide can be set to zero, for this layer is not detectable by a magnetic survey. Fortunately, a continuous magnetic layer is detectable with an electromagnetic (EM) induction meter operating in its susceptibility mode, and this can be a good reason to apply this instrument. Figure 1 shows a pair of examples of two-dimensional magnetic models; the rest are three-dimensional. Two-dimensional models are most suitable for features that extend for at least a short distance along a straight line; furthermore, these 2-D models allow initial approximations of short or compact 3-D bodies. These 2-D models have the advantage of simplicity; it is easy to see and change the shape of the body. Three-dimensional models allow better approximations of most bodies. Those in Figure 1 are rather simple, but perhaps adequate. The magnetic model in Figure 3 is more complicated, mostly because of the sloping surface of the ground. As 3-D models become more complex, it is more difficult for the viewer to understand their shape; illustrations from several directions may help, or a dynamic display that the user controls may allow the details to be understood. It may be difficult for even the creator to change the shape of some threedimensional models; see Appendix 16 for polyhedral models, which can be very complex. Since the magnetic field of a forward model is calculated only once or a few times, it can be quite detailed, if that is necessary; however, much of the fine detail may have little effect on the calculations. By comparison, when an inversion is done, the magnetic model is calculated hundreds or thousands of times; a simpler model can speed this analysis. Measure magnetic properties Once the geometry of the model has been determined, its magnetic properties must also be measured or estimated. If an example of the body is visible in an excavation, an examination can start with simple magnetic tests. Collect samples of soil from outside the body and perhaps inside it, and let them dry if necessary. Pour the loose soil slowly over a strong permanent magnet (perhaps encased in a plastic bag), and see what fraction of the soil sticks to the magnet. Make a pendulum magnet to see if stones or other materials are strongly magnetic. Page 3

Measure magnetic properties This can be a cylindrical bar that is about 2 mm in diameter and 30 mm long; suspend the bar with a light string (perhaps 30 cm long) at its middle so that it balances. Hold the end of the string and touch the magnet to an object; then pull the string away gently and see if the magnetic sticks slightly to the object. The simple tests above are not very sensitive, and better measurements are necessary for quantitative models. The most important measurement is that of magnetic susceptibility. This magnetic parameter describes the ease with which a material may “conduct” a magnetic field. This parameter is the ratio of two magnetic fields, the source field and the resultant field in the object; it is therefore a pure number and has no unit name. However, one may state that the susceptibility was measured in the convention of the Système International and to call it an SI unit. This quantity is usually quite small and a typical value for soil will be about 0.001; therefore, these susceptibility values will usually be multiplied by 1000 here, and called a susceptibility in parts per thousand (abbreviated ppt). Further information about susceptibility is in Appendix 2. Several manufacturers have portable susceptibility meters for sale, and they cost about US$ 2000; Appendix 3 has descriptions and comparisons of some that are easily held in one’s hand. These small instruments are very portable; they allow a measurement to a depth of about 1 cm; therefore samples should be rather flat for the best accuracy. A rough sample means that the susceptibility of air, which is zero, is included in the measurement’s average. Larger instruments, such as those made by Bartington Instruments and Geonics, can allow readings to be made to a depth of 0.1 to 0.5 m. In addition to being magnetic “conductors”, many materials are also weak permanent magnets. This is called remanent magnetization, while magnetic susceptibility creates what is called induced magnetization. It is not possible to measure remanent magnetization with a magnetic susceptibility meter; however, it can be important to determine this parameter, for its effect may be larger than induced magnetization, at least for burned earth and objects made of iron. Remanent magnetization can be discovered easily with a magnetometer. Get a fistsized sample of soil or another object that is compact (roughly similar to a cube or sphere). Place it near the sensor of the magnetometer and rotate the sample 360 degrees. If the readings change significantly and consistently with orientation, then the sample is probably a permanent magnet. A more accurate test may be done by fixing the direction and distance of the sample as illustrated in Figure 4. The resulting measurements, plotted in Figure 5, allow a calculation of the remanent and induced magnetization of the sample. Further details about these measurements and calculations are in Appendix 4. The ratio of remanent to induced magnetization is called the Q ratio (or Konigsberger ratio). My measurements of this ratio for many samples of ceramic, brick, and fired earth have suggested that a typical value is about ten; for soils, perhaps a value of one is typical. Several publications summarize the magnetic parameters of materials; these tables can assist when it is not possible for one to measure samples of stone, soil, or artifacts. These lists are primarily for rocks, but with some soils; only a few parameters are available for archaeological materials. See Appendix 5.

Page 4

Complications Complications There are several complications to the creation of magnetic models. It appears that none of these cause serious difficulties, but all of these effects must be considered. Magnetic susceptibility readings can change with direction. A material (even a spherical object) may be more magnetic in one direction than another. This effect is called the anisotropy of magnetic susceptibility, and it may be abbreviated AMS. Measurements of susceptibility in a number of directions allow the determination of this anisotropy; some good references on this topic are in Appendix 6. I have never tried to measure AMS, and it may not be practical to do this in the field. This directionality of susceptibility appears to be caused primarily by a preferred orientation of magnetic grains, which may be flattened or elongated; as a result of a shearing motion such as the flow of water, these grains may tend to be aligned in one direction. Figure 6 shows how AMS might cause the magnetic anomaly of a rotated sample to change with direction; note that the same change could be created by a sample with constant susceptibility added to remanent magnetization. Very different magnetic models can result in the same calculated field. This is called equivalence or ambiguity in magnetic surveys. It has already been mentioned how spheres of different size can appear identical to a magnetic survey. Many other examples have been described in publications (Appendix 7). A common example of equivalence is illustrated in Figure 7: A broad and shallow feature can cause almost the same anomaly as a small and deep feature. While measurements with an EM meter may help to distinguish depth, this distinction may be impossible with a magnetometer. Figure 8 and Figure 9 show some examples of different magnetic models whose calculated fields are identical. It may not be possible to measure the magnetic susceptibility of mixtures. A typical example would be soil with many stones in it. While one may measure the soil and stones separately, what is the combined susceptibility? It appears to be a good approximation that one can calculate the volume-weighted average susceptibility of a mixture. As an example, soil (with a susceptibility of 2 ppt) is composed of 20% of stones by volume (with the stones having a susceptibility of 1 ppt). The average susceptibility would then be (0.8 * 2) + (0.2 * 1) = 1.8 ppt. For a more accurate analysis, see the reference in Appendix 8. Parts of a magnetic model may be invisible to a magnetic survey. This effect is sometimes called annihilation. The annihilator that is found most often is the infinite magnetic slab. A uniform topsoil could be an example, and this layer can be invisible even if it is quite magnetic. For this to be true, the layer must be quite uniform; any change in thickness or susceptibility would cause that contrast to be detectable to a magnetometer. Annihilation can allow a simplification of a magnetic model. Consider an air-filled void (nonmagnetic) in soil that is magnetic. While one could create a magnetic model with soil extending to infinity, it is easier to just subtract the susceptibility of that soil from all features and layers. The new “soil” then has a susceptibility of zero, while the void has a negative apparent susceptibility. This modified model yields the same calculation, but is much simpler. Further discussions about other unusual annihilators are given in the references of Appendix 9. Strongly-magnetic features may require corrections. Steel artifacts and fired features, such as kilns, provide examples of this. For these features, the magnetic anomaly may be Page 5

Calculate the field of a magnetic model lower than expected, and the direction of magnetization within the features may be different than expected. These effects need to be considered only when the most accurate analyses are needed, and Appendix 10 has my thoughts on this topic. Magnetic anomalies and properties may change with time. These changes may happen in minutes, months, or decades; Appendix 11 has further details. Calculate the field of a magnetic model A simple calculation of the anomaly of a magnetic model may be all that is needed, at least at first. Perhaps the feature is sufficiently deep that it can be approximated by a spherical body, and perhaps the inclination of the Earth’s field is rather steep. Then one can apply a little arithmetic to the estimated values of the volume of the body, along with its magnetic susceptibility and depth. The calculation below distinguishes four different parameters that may quantify the magnetization of an object: k, Magnetic susceptibility: A measurement with a simple field instrument I, Intensity of magnetization: Adjusts for the magnitude of the Earth's field M, Magnetic moment: Further adjusts for the volume of the object Ba, Magnetic anomaly: Further adjusts for the distance to the object The following is an illustration of how to calculate the peak magnetic anomaly of a compact object: measurement calculation example k = Magnetic susceptibility of the object; no unit name 0.002 Be = Magnitude of the Earth's magnetic field 50,000 nT I = Intensity of magnetization of the object = k * Be / (400 * pi) 0.08 A/m V = Volume of the object 0.1 m3 M = Magnetic moment of the object = V * I 0.008 Am2 r = Distance: center of the object to the center of the magnetic sensor 0.8 m 3 3 nT Ba = Peak magnetic anomaly = 200 * M / r Further information about this calculation: Magnetic anomaly: The calculation is for a total-field magnetic sensor. It will be good for a vertical component magnetometer if the inclination of the Earth's field is steep. For a magnetic gradiometer, the difference between the readings from the two sensors will be smaller than this calculation. If you wish, calculate the magnetic anomaly at the heights of the two sensors and subtract them (lower - upper). Compact object: The calculation is probably okay if the object's maximum length is less than twice its minimum width. It can still be a valuable estimate for objects that are more elongated. Susceptibility: To convert from old values to new ones: k(SI) = 4 * pi * K(cgs); k(SI) is the new System International value, k(cgs) is the old centimeter-gram-second value. Magnetic remanence: As an approximation, just add it to the induced magnetic moment above. An accurate calculation requires a vectorial addition, but this is more difficult. Alignment: This peak anomaly will be found where the line between the object and the magnetic sensor is along the line of the Earth's magnetic field (defined by its inclination and declination). For other orientations, the anomaly will be lower. Page 6

2-D calculations Equatorial location: The anomaly will be about a quarter of this calculation, and the negative of the value above. The anomaly is halved because the magnitude of the Earth’s magnetic field near the equator is halved; it is halved again because the direction of the magnetic field is horizontal. Earth's magnetic field: Determine this from your measurements, from published maps of the geomagnetic field, applying IGRF (International Geomagnetic Reference Field) calculations, or on the web at: https://www.ngdc.noaa.gov/geomag/magfield.shtml. If the Earth’s magnetic field is determined from your map of magnetic measurements, one can select the reading in an area where spatial changes are small, or one can calculate the median measurement in the total-field magnetic map (not measured with a gradiometer). 2-D calculations When a greater accuracy is needed for the magnetic anomaly, consider doing a twodimensional calculation; this will be simpler than a three-dimensional analysis. A 2-D analysis assumes that a body is prismatic, that is, that it extends for a long distance in one direction. However, the error in a 2-D calculation of a three-dimensional body is not always serious. Figure 10 is a plot of the magnetic anomaly of a cube, and compares that to the anomaly that would be found if opposite faces of the cube were extended to form a prism. In doing these calculations of two-dimensional (or 3-D) models, it can be valuable to remember that all dimensions can be multiplied by any constant and the calculations will be the same. These dimensions are both the size of the model and the distance from the model to the calculation points (sensor height plus model depth). The magnetic susceptibility of the model must remain the same between the original and modified models; if there is remanent magnetization in the model, its intensity of magnetization must remain the same. This identity of the calculations between models can have several advantages, including one that applies to models that have dimensions in feet; some computer programs assume that dimensions are in meters. Related to the above: If the dimensions of the model remain fixed, but its susceptibility is multiplied by a constant, then the anomaly amplitudes are multiplied by the same constant. It can also be important to remember that changes in magnetic anomalies may not appear reasonable or obvious at first; Figure 11 has an example that shows how the amplitude of an anomaly may rise and then fall with increasing size of an object. An unfortunate fact: A few points are more difficult to calculate than others. Calculations on the surface of a body or inside it may have errors or may not be possible. Some computer programs will create errors where calculations are made at points that are outside bodies but are along the extended line of a polygonal or polyhedral edge of a body. These errors will often be visible as abrupt and impossible changes at one or a few separate points in a magnetic map. One can ignore these errors, correct them with the average of nearby calculations, or shift the bodies or calculations by a slight amount. Figure 12 is an example of the calculation of a 2-D model; the magnetic body is shown by the red rectangle; the parameters of the body and calculations are listed in the figure. When any of these parameters is changed (with numerical entries or with the up-down arrows Page 7

3-D calculations to the right of the numbers) the calculated curve will immediately change. The calculation is for a total-field magnetic anomaly (TMI, Total Magnetic Intensity); only a single body is allowed, and it has the cross-section of a rectangle or parallelogram. Further details about this Pdyke program are in Appendix 12. The calculated anomaly from another free computer program is shown in Figure 13. Again, the parameters of the model and calculation are listed in the program’s window. Like the Pdyke program above, the calculation shows the magnetic values along a single line. While the calculations of the Pdyke program cross the middle of the magnetic body, the MagPrism program of Figure 13 allows the line of the calculations to be anywhere. The model for this program can be three-dimensional. In addition to these two programs, there more than five other free programs that allow the forward calculation of magnetic models; further information on these is in Appendix 13. Most of these programs work directly with Windows; however, some apply an auxiliary program called MatLab that supplies and controls the display and input of data. While the MatLab software is moderately expensive, some of these programs can operate with a similar, but free, program called Octave. A few of the older programs described in Appendix 13 apply DOS; while there may be equivalent programs for Windows, these DOS programs will still run in some versions of Windows. 3-D calculations There are a smaller number of free programs that do the forward calculation of threedimensional models. The GaMField program allows a great flexibility in the shape of its models because they are a collection of rectangular boxes. While this program was written in the MatLab language, it has been compiled so that it may run in Windows, without having to buy the MatLab program. Figure 14 shows the first two windows that set the parameters for the magnetic model. This magnetic model is distributed on the rectangular plates that are shown in the lower figure; see the two views in Figure 15 for an example. Colors in Figure 16 reveal the amplitudes of the calculated anomalies; these calculations can also be saved to files for further analysis or for plotting with a different program. While the GaMField program will calculate the magnetic field of simple forward models, it is ideal for models with rather complex shapes. The description of the software that was written by the authors is good, but there are additional options at some steps that are discovered by right-clicking on windows and images. Some further notes about this program are included in Appendix 14. Another free program for the calculation of 3-D models is called Noddy; screen images that illustrate a magnetic model and its calculated field are in Figure 17 and Figure 18. This program was written specifically for modeling geological stratigraphy; some archaeological features can also be approximated with this software. The operational manual for this program is very thorough, and there are several publications that describe the modeling that has been done with the program. There is also a related and more recent (but perhaps preliminary) version of this program that has been written in the Python language; this version is called pynoddy. Page 8

Examples of magnetic models Two other suites of free programs that allow the calculation of 3-D magnetic models are summarized in Appendix 13. The suite called “Fatiando di Terra” is written in Python and it allows many additional types of analysis of magnetic or gravity data. A much older suite of programs that was prepared for DOS by the US Geological Survey is still valuable; like Fatiando di Terra, the source code is available and this allows the software to be modified. All commercial (not free) computer programs will probably allow the forward calculation of magnetic models also; Appendix 15 has a short description and comparison of some of these programs. Examples of magnetic models Compare model calculations with measurements The shape of a simple magnetic model of a stone wall is shown at the top of Figure 19; the remaining part of this wall is now about 0.3 m underground. While the model is simple, it does have a complication: The soil extends for an infinite distance to the left and right, and computer programs usually do not allow this infinity. There are several ways around this difficulty. First, one can write a simple program that will allow this calculation of a magnetic layer that goes to infinity in one direction (this is called a half-infinite slab); the references in Appendix 16 have the equations that are needed. Second, one can just extend the magnetic layers for a long, but finite, distance, perhaps 10 to 50 times the width of the central model. This can work well, but it must be done with caution so that those distant ends of the model do not affect the calculations in the central part of the model. It does not help to taper those distant layers to end at triangular points. Instead, it is better to make the magnetic layers as thin as possible; for example, end the bottom magnetic layer at the bottom of the stone wall, and not deeper. While it might seem reasonable to extend this bottom layer to a good depth, that just increases the magnetic anomaly from the distant end of the layer; it is important to keep the angular size of that distant edge (as seen from the central part of the model) as small as possible. It may be easier to modify the magnetic model as shown in Figure 19; the goal of this simplification is that of changing the material outside the center of the model so that it has a susceptibility of zero, and therefore has no effect on the calculation of magnetic field This is done by subtracting the susceptibility of an infinite layer from all of the susceptibility readings that are found in the depth range of that layer. This has been done twice in Figure 19. While the resulting and equivalent model (D) now has six magnetic layers instead of the five in the original model (B), magnetic materials are no longer found at infinity. The calculated magnetic anomaly of this model is plotted in Figure 20 for two orientations of the wall; magnetic lows predominate in both calculations because the stone facing of the fortification wall is less magnetic than the surrounding soil. Since the actual magnetic measurements that were made over this stone wall were greater than this prediction, the magnetic model is not correct. Forward magnetic models can also help one to analyze magnetic measurements. One estimates what is underground, then calculates the magnetic field of that model. After comparing those calculations with the measurements, the model is modified and calculated again. This cycle can be repeated until the calculations are similar to the measurements, and Page 9

Examples of magnetic models the model might now be a good estimate of what is underground. Figure 21 has an illustration of this; the process might be called “inversion by repeated forward modeling”. An automated inversion can yield a model whose calculated curve is much closer to the measurements; however, the shape and magnetic property of the resulting model may be unreasonable. An excavation was made to find the source of the red “bulls-eye” anomaly near the top of Figure 22. This exposed a shallow lens of dark soil. A magnetic model of this feature was found to have a calculated anomaly that was too faint; the actual source was found to be deeper underground. After the field of a magnetic model has been calculated, it can be educational to subtract those calculations from the measurements. Figure 23 has an example of this, and it shows that the simple magnetic model was good, although it lacks some small details. Predict that anomalies will be faint Important features are too frequently found in excavations that were not revealed by an earlier magnetic survey. Also, it may be valuable to determine if a feature that has been discovered in an excavation could have been detected by a magnetic survey. Magnetic models may help this understanding. Refilled fortification trenches were found to be almost invisible to a magnetic survey. The cause of this failure was discovered only later when excavations exposed the features underground, allowing their dimensions and magnetic properties to be determined. Figure 12 has the calculation of the magnetic model, shown in red there. While an anomaly with an amplitude of 0.2 nT can be detected with a magnetic survey, a few small iron artifacts in the soil can easily hide this anomaly. Another example of a prediction of a weak magnetic anomaly is in Figure 13. Historic graves are typically difficult to detect with magnetic surveys because modern iron debris in the area creates too many unwanted anomalies. The analysis with Figure 24 shows that a prehistoric post hole that was found in an excavation could not have caused the distinct magnetic anomaly that was measured. Something else created that anomaly. Granite is usually quite magnetic, and therefore easy to locate with a magnetic survey. Figure 25 reveals the magnetic model of a now-buried aqueduct that was constructed of granitic stone. This granite was unusual, for it was less magnetic than the surrounding soil. Interesting or unusual models In the northern hemisphere, magnetic lows are most commonly found on the north side of the predominant magnetic highs that are measured over magnetic features. However, where mapping holes in magnetic layers, and also below overhead objects, those lows will be toward the south. Figure 26 illustrates these effects. Archaeological surveys frequently encounter objects with strong remanent magnetization that is oriented rather randomly; this is typical of modern steel debris, and also walkways and pavements constructed of brick or igneous stone. Figure 27 shows the magnetic patterns of a cluster of objects; these maps become increasingly complex as the Page 10

Conclusion proportion of remanent magnetization increases. Water wells may be vertical steel pipes or they may be cylinders that have been dug into the soil and lined with brick. Figure 28 reveals the magnetic map that was measured over a well, and also the calculated map of a simple magnetic model. A line from the magnetic high through the magnetic low goes straight toward magnetic north. The calculated graph of Figure 29 shows that the bottom depth of a well may not be determined accurately as the well becomes deeper. It is also impossible to estimate the diameter of a well. Magnetic surveys have difficulties near standing buildings, for there may be massive steel objects inside. It is possible to create a magnetic model of this interference; the calculated field of that model may then be subtracted from the measurements in order to decrease the effect of the magnetic objects. Figure 30 has an illustration of this. Conclusion Forward magnetic models can aid one’s understanding of a magnetic survey. While the magnetic models can be quite simple, the calculations of these models can be very accurate. Magnetic models also allow a comparison of a magnetic map with the findings of a later excavation. If the calculated model (which is based on the excavation) has much weaker anomalies than the original magnetic measurements, then the complete source of the magnetic anomaly has not been discovered. Perhaps it is deeper underground, possibly it is nearby but in a different location, perhaps it was removed without noticing it during the excavation, or possibly the feature has unexpected remanent magnetization. This calculated model may also show that the feature could never have been detected by the magnetic survey. Additional detailed information and references follow; the final appendix here (Appendix 17) has further notes on the figures that are included here. Appendix: Further information for those who wish it Appendix 1: Magnetic models: Excellent descriptions of magnetic modeling are given in the following papers: "Quantitative methods for interpreting aeromagnetic data: a subjective review", by P. J. Gunn, 1997, Journal of Australian Geology and Geophysics, volume 17 number 2, pages 105-113. "Three-dimensional geological modelling of potential-field data", Mark Jessell, 2001, Computers and Geosciences, volume 27, pages 455-465. Appendix 2: General information on magnetic susceptibility and its measurement Workshop on soil magnetism, edited by Jacqueline A. Hannam, Remke L. Van Dam, and Russell S. Harmon, 2009; this is the proceedings of a conference that was devoted to magnetic soil and how it applies to metal detectors in the search for buried mines: http://www.gichd.org/ fileadmin/GICHD-resources/rec-documents/AbstractSoilMagn_Proceedings2008.pdf Page 11

Appendix 3: Comparison of magnetic susceptibility meters The slides for some of the talks are available at: https://www.msu.edu/~rvd/soilmag08/presentations/Monday-Talks.pdf "Magnetic susceptibility", by Rinita A. Dalan, 2006, chapter 8 in: Remote Sensing in Archaeology: An Explicitly North American Perspective, edited by Jay K. Johnson, University of Alabama Press, pages 161-203. Environmental magnetism, by Michael E. Evans and Friedrich Heller, Academic Press, 2003. "A review of the role of magnetic susceptibility in archaeogeophysical studies in the USA: Recent developments and prospects", by Rinita A. Dalan, 2008, Archaeological Prospection, volume 15, pages 1-31. Environmental Magnetism, by Roy Thompson and Frank Oldfield, 1986, Allen and Unwin. Still an excellent book, and now available at the web site of one of its authors: http://www.geos.ed.ac.uk/homes/thompson/envmag/ J. Duncan McNeill, recently retired as President of Geonics, prepared several excellent reports in 2012 - 2013 that describe the interpretation of magnetic susceptibility surveys that are done with the model EM38 instrument from Geonics; these reports are available at www.geonics.com. Technical Note TN-34, "Application of Geophysical Surveys Measuring Soil Magnetic Susceptibility to Locate the Site of the Eighteenth-Century Parish Church of Saint-Charles-des-Mines at Grand Pré National Historic Site"; Technical Note TN-35, "Archaeological Mapping Using the Geonics EM38B to Map Terrain Magnetic Susceptibility (With Selected Case Histories)", November 2012; Technical Note TN-36, The magnetic susceptibility of soils is definitely complex", April 2013; Technical Note TN-37, "Time-domain response of a magnetically susceptible soil". Appendix 3: Comparison of magnetic susceptibility meters "A comparative study of hand-held magnetic susceptibility instruments", by Deng Ngang Deng, 2015, MSc thesis, Laurentian University, Sudbury, Ontario, Canada, 184 pages; available at: https://zone.biblio.laurentian.ca/ dspace/bitstream/10219/2313/1/MSC%20Thesis%20Submission_Deng_2.pdf "Comparison of magnetic-susceptiblity meters using rock samples from the Wopmay Oregen, Northwest Territories", by M. D. Lee and W. A. Morris, 2013, Geological Survey of Canada, Technical Note 5, 7 pages; available at: http://publications.gc.ca/collections/collection_2013/rncan-nrcan/M41-10-5-2013-eng.pdf Manufacturers or sellers of portable magnetic susceptibility meters: ZH Instruments, model SM-30: http://www.zhinstruments.com/sm-30.html Bartington Instruments, models MS2 and MS3 (USB): http://www.bartington.com/ms3.html Geonics Ltd., model EM38: http://www.geonics.com/html/em38.html GDD Instrumentation, model MPP probe: http://www.gddinstrumentation.com/index.php/mpp-probe Terraplus Inc., model KT-20, www.terraplus.ca Heritage Geophysics, models SM-30 and SM-100, www.heritagegeophysics.com Fugro Instruments, model RT-1: The company appears to be closed and out-of-business now. Page 12

Appendix 4: Measure the magnetization of soils and small samples Appendix 4: Measure the magnetization of soils and small samples: Field procedures are described here; nothing about laboratory methods is included. "Applications manual for portable magnetometers", by Sheldon Breiner, Geometrics, 1973, available at: www.geometrics.com; see the chapter starting at page 33. "Application of Magnetic and Geotechnical Methods for Archaeological Site Investigations", by Bradley G. Fritz, Douglas McFarland, William Hertz, and Jeffrey Gamey, 2011, SERDP Project RC-1697, Department of Defense Strategic Environmental Research and Development Program (SERDP), about 60 pages. The Qmeter, from the Magnetic Earth company, is an instrument for measuring the induced and remanent magnetization of samples and artifacts, http://www.magneticearth.com.au/ My own experiments with different techniques are listed next. These reports are available at either ResearchGate or the Digital Archaeological Record. General summary: Project "20050209 GeoExcav", file GeoExcav.pdf, page 16 of "Geophysics in Excavations"; this report is available at ResearchGate. Test soils with a magnet: Project "19780104 VaForge", file VaForge.pdf, page 8 of "Patterns of the Past: Geophysical and Aerial Reconnaissance at Valley Forge". Project "20050209 GeoExcav", file GeoExcav.pdf, page 16 of "Geophysics in Excavations", available at ResearchGate. Rotate a sample near a magnetic sensor: Project "19910123 Geophys", file GeophysC.pdf, Figure C40 of "Geophysical Exploration for Archaeology, Volume C: Detailed survey procedures", available at ResearchGate. Project "20120911 Braendes", file Braendes.pdf, Figure 43 of "A magnetic exploration of Brændesgård in 2012", available at the Digital Archaeological Record, document #381225. Measure a magnetic map over the sample: Project "20050209 GeoExcav", file GeoExcav.pdf, page 19 of "Geophysics in Excavations", available at ResearchGate. "20020626 Hopeton 2", Bevan02a.pdf, page 15 of "Geophysical Tests in the Hopeton Excavations", available at the Digital Archaeological Record, document #381057. Measure the anomaly on six faces of a sample: Project "20020626 Hopeton 2", file Bevan02a.pdf, page 9 of "Geophysical Tests in the Hopeton Excavations". Project "20060622 Hopeton 6", file Bevan06a.pdf, page 23 of "Geophysical Tests at Hopewell and Hopeton", available at the Digital Archaeological Record, document #381064. Appendix 5: Collected values of the magnetic susceptibility of soil and rock: "Magnetic effects and properties of typical topsoils", by John C. Cook and S. L. Carts, Jr., 1962, Journal of Geophysical Research, volume 62 number 2, pages 815-828; has a valuable listing of many soils in the US. "Magnetic properties of rocks and minerals", by Christopher P. Hunt, Bruce M. Moskowitz, and Subir K. Banerjee, 1995, pages 189-204 , in Rock Physics and Phase Relations: A handbook of physical constants, edited by Thomas J. Ahrens, American Geophysical Union. "Magnetic susceptibility of soil: an evaluation of conflicting theories using a national data set", by John A. Dearing and 4 others, 1996, Geophysical Journal International, volume Page 13

Appendix 6: Anisotropy of magnetic susceptibilty 127, pages 728-734. "The influence of soil type on the magnetic susceptibility measured through soil profiles", by M. Hanesch and R. Scholger, 2005, Geophysical Journal International, volume 161, pages 50-56. "Magnetic susceptibility of forest topsoils in mountain regions of southern Poland based on field measurements", by Tadeusz Magiera. Zygmunt Strzyszcz, and Marzena Rachwa», 2006, Polish Journal of Soil Science, volume 39 number 2, pages 101-108. Handbook of the physical properties of rocks, volume 2, edited by R. S. Carmichael, 1982, CRC Press. This reference, like the one below, is still good even though it is old; there are newer books of tables, including one by Carmichael, but I do not know the details of these books. Handbook of Physical Constants, revised edition, by Sydney P. Clark Jr., 1966, Geological Society of America Memoir 97, 587 pages. Many of the measurements that I have made are included in a report: Project "19990819 MagParam", file MagParam.pdf, available at the Digital Archaeological Record, document #381108. Appendix 6: Anisotropy of magnetic susceptibility: These are some of the clearest papers that I have found on this subject. "Anisotropy of Magnetic Susceptibility", chapter 2 in: Understanding an Orogenic Belt Structural Evolution of the Himalaya, by Ashok Kumar Dubey, 2014, Springer, pages 17-34. "Sixty years of anisotropy of magnetic susceptibility in deformed sedimentary rocks", by Josep M. Parés, 2015, Frontiers in Earth Science, volume 3, article 4, pages 1-13. "On the reliability of the AMS ellipsoid by statistical methods", by Sara Guerrero Suárez and Fátima Martín-Hernández, Tectonophysics, 2014, volume 629, pages 75-86. "Determination of fundamental magnetic anisotropy parameters in rock-forming minerals and their contributions to the magnetic fabric of rocks", by Fátima Martín Hernández, 2002, doctoral dissertation, Swiss Federal Institute of Technology, Zurich. Equations are garbled in printing. "The magnetic fabric of some loess/palaeosol deposits", by J. J. Hus, 2003, Physics and Chemistry of the Earth, volume 28, pages 689-699. "Paramagnetic and ferromagnetic anisotropy of magnetic susceptibility in migmatites: measurements in high and low fields and kinematic implications", by Eric C. Ferré, Fàtima Martín-Hernàndez, Christian Teyssier, and Mike Jackson, 2004, Geophysical Journal International, volume 157, pages 1119-1129. "The magnetic anisotropy of rocks: Principles, techniques and geodynamic applications in the Italian peninsula", by Aldo Winkler and 4 others, 1997, Annali di Geofisica, volume 40 number 3, pages 729-740. "Weak-field magnetic susceptibility anisotropy and its dynamic measurement", by William F. Hanna, 1977, US Geological Survey, Bulletin 1418. Appendix 7: Magnetic equivalence, ambiguity, and non-uniqueness: Many different shapes and distributions of magnetic material can cause the same or similar Page 14

Appendix 8: Magnetic susceptibility of mixtures magnetic anomalies. Some very practical examples of this have been given in the following articles: "Non-uniqueness and the next generation magnetics" by Phil Schmidt and Dave Clark, Preview (a magazine of the Australian Society of Exploration Geophysicists), August 1994, pages 29-31; available at: https://aseg.org.au/preview "Some equivalent bodies and ambiguity in magnetic and gravity interpretation", by B. David Johnson and Gerry van Klinken, Bulletin of the Australian Society of Exploration Geophysicists, March 1979, volume 10 number 1, pages 109-110. "Unique geologic insights from 'non-unique' gravity and magnetic interpretation", by Richard W. Saltus and Richard J. Blakely, GSA Today (Geological Society of America), December 2011, volume 21 number 12, pages 4-11; available at: http://www.geosociety.org/gsatoday/archive/21/12/article/i1052-5173-21-12-4.htm "Using archaeological models for the inversion of magnetometer data", by Armin Schmidt and Martijn van Leusen, presented as a poster at the 11th International Conference on Archaeological Prospection, held 15 - 19 September 2015 in Warsaw, Poland. More detailed studies of ambiguity are described in the following articles: "Some studies relating to nonuniquenes in gravity and magnetic inverse problems", by Mahboub Al-Chalabi, 1971, Geophysics, October 1971, volume 35 number 5, pages 835-855. "Ambiguity analysis and the constrained inversion of potential field data", by Fabio Boschetti, Frankiln G. Horowitz, and Peter Hornby, published at: http://www.per.marine.csiro.au/staff/Fabio.Boschetti/papers/ambiguity.pdf "Practical applications of uniqueness theorems in gravimetry: Part I—Constructing sound interpretation methods", by João B. C. Silva, Walter E. Medeiros, and Valèria C. F. Barbosa, 2002, Geophysics, volume 67 number 3, pages 788-794. Appendix 8: Magnetic susceptibility of mixtures: see the following reference: "Effective permeability of mixtures", by A. H. Sihvola and I. V. Lindell, 1992, PIER 6, Progress in Electromagnetics Research, Dielectric Properties of Heterogeneous Materials, edited by A. Priou, pages 153-180. Available at: http://www.jpier.org/PIER/pier06/04.900106.pdf Appendix 9: Annihilators: Some features can be very magnetic, but can still be invisible to a magnetic survey; these are called annihilators, but perhaps the word "invisible" is clearer. A good description of these features is found in the book: Potential theory in gravity and magnetic applications by Richard J. Blakely, Cambridge University Press, 1995; page 95 describes the effect of an infinite slab, while pages 97 and 296 give a general discussion of annihilators. Early studies of the infinite slab were done by James Affleck, "Interrelationships between magnetic anomaly components" by James Affleck, Geophysics, volume 23, number 4, October 1958, pages 738-748. The annihilator of an infinite slab is important for all magnetic studies; this annihilator applies to both magnetic susceptibility and remanence. Two other annihilators apply only to remanence; they are unusual and interesting, but not generally important. The first is found Page 15

Appendix 10: Self-demagnetization with magnetization on a sphere, such as the Earth. If a spherical shell is magnetized from a source that is inside, and then that source is removed, the shell causes no external magnetic field. This effect was first revealed in a publication by S. K. Runcorn, "On the interpretation of lunar magnetism", Physics of the Earth and Planetary Interiors, 1975, volume 10, pages 327 -335. Other authors have described how this effect complicates the interpretation of the magnetic field of the Earth: "Magnetic field annihilators: Invisible magnetization at the magnetic equator", by S. Mauss and V. Haak, Geophysical Journal International, 2003, volume 155, pages 509-513. "Exact solutions for internally induced magnetization in a shell", 2000, by Vincent Lesur and Andrew Jackson, Geophysical Journal International, volume 140, pages 453-459. It is possible that these types of spherical effects may have some importance to the magnetization of a kiln or furnace as it cools. A third type of magnetic annihilator results from the magnetization of the soil by a lightning strike. The horizontal flow of the electrical current underground creates remanent magnetization in a circular cylinder around the line of the current. Where this cylinder is complete, the magnetization remains invisible at the surface. However, parts of the magnetized cylinder will be interrupted or missing because of the surface of the earth; this part of the magnetization is readily detected by a magnetic survey. Examples of these anomalies are included in a poster: "Lightning Strikes in Archaeological Magnetometry Data: A Case Study from the High Bank Works Site, Ohio, USA", by Jarrod Burks, Andreas Viberg, Bruce Bevan, that was presented at the 11th International Conference on Archaeological Prospection, held 15 - 19 September 2015 in Warsaw, Poland; article at ResearchGate. This lightning annihilator is explained in a newsletter article of mine: "The magnetic anomaly of a lightning strike", FastTimes (available at the web site of the Environmental and Engineering Geophysical Society, www.eegs.org), issue March 2009. Project folder "20080428 Lightning". Further information on the analysis of the magnetic anomalies of lightning strikes is in a report (LightDepth.pdf) of mine titled "The depth of lightning magnetization" in folder "20141107 lightning2". This and some other recent reports that are mentioned here might not have been published to an on-line repository by the time you read this. Appendix 10: Self-demagnetization: Most writers have introduced this by considering magnetic poles on surfaces; I find it easier to think of this as an interaction of cells or dipoles in a volume, and some of my writing is noted below: Project "20090323 CAAconf", file Direct.pdf, page 4 of "Directions of magnetization". Project "19950905 Snorup", file Snorup.pdf, page 20 of "Magnetization Directions of Iron Slag at Snorup", available at ResearchGate. Project "20081124 ArchDate", file ArchDate.pdf, page 138 of "Archaeological dating from magnetic maps: Some failures"; published in the Journal of Environmental and Engineering Geophysics, September 2009, volume 14, issue 3, pages 129 - 144. Appendix 11: Temporal changes in magnetic properties: A good reference is: "Advanced UXO Discrimination using Magnetometry: Page 16

Appendix 12: Notes on the Pdyke program Understanding Remanent Magnetization", by Stephen Billings, Yaoguo Li, and Whitney Goodrich, 2009, final report, SERDP Project MM-1380 (Strategic Environmental Research and Development Program), 73 pages. During two geophysical surveys where I have worked, magnetic measurements appear to have detected the magnetic viscosity of nearby iron objects; magnetic measurements changed significantly when there was no suggestion that the Earth's field had changed. A survey at Hopeton Mounds was done near an iron frame of a shelter that covered an excavation. The frame was moved somewhat and then magnetic readings were made nearby, while the nearby iron was stationary; the magnetic readings changed by over 50 nT; see page 10 (note for Figure 18) of the report: "Conductivity and Magnetic Surveys in Trench 8 at Hopeton", project "20050514 Hopeton 5"; report available at ResearchGate. During another survey, a magnetic shift of 10 nT appears to have been caused by a stationary car at a distance of 30 m. See page 3 of the report "A magnetic survey at Store Krusegård", project "20110911 Krusegaard", available at the Digital Archaeological Record, document #381299. During two separate surveys, I have looked for seasonal changes in the magnetic anomalies of archaeological features. These changes appear to have been reliably detected over a buried brick foundation; see Figures C14 and C15 in the report "Geophysical Exploration for Archaeology, Volume C: Detailed survey procedures" in Project "19910123 Geophys", GeophysC.pdf, available at ResearchGate. However, a test over a prehistoric earthen ridge at Hopeton Mounds found no detectable change between the two dates of my tests: See page 9 in a report titled "Conductivity and Magnetic Surveys in Trench 8 at Hopeton", in Bevan05a.pdf, project "20050514 Hopeton 5", available at the Digital Archaeological Record, document #381062. If a magnetic model includes magnetic susceptiblity, one must remember that the strength of the Earth's magnetic field can change enough to affect future anomalies. While these long-term changes in the magnitude of the Earth's magnetic field are included with the parameters of the International Geomagnetic Field, I was impressed to find that the field had dropped by over 1000 nT in a period of about a decade at a site that I have revisited. See Figure C19 of the report "Geophysical Exploration for Archaeology, Volume C: Detailed survey procedures" in Project "19910123 Geophys", GeophysC.pdf, available at Research Gate. Appendix 12: Notes on the Pdyke program: The program Pdyke calculates the magnetic field (or its vertical gradient) along a line over a four-sided prism; this prism has the cross-section of a parallelogram. The parameters of the prism (its size, depth, tilt, and susceptibility), along with the parameters of the Earth's magnetic field are easily changed. After each change, the program immediately calculates the new magnetic anomaly. The program can also calculate the gravity anomaly (and its gradient) of the prism. Pdyke is a public domain program written by two Australian companies: Geophysical Software Solutions Pty. Ltd., and TGT Consulting. The program is available on the web site: www.geoss.com.au This web site also has a similar program called Pblock, which is almost the same as Pdyke Page 17

Appendix 12: Notes on the Pdyke program except that the sides of the prism can be tilted in Pdyke. The compressed file (PdykeSetup.exe) can be run in order to extract the original Windows program, Pdyke.exe (file size = 304 kB, date = 4 December 2005, 2:54 pm). The program Pdyke can then be run, and this will display two windows: One for the plot of the calculations and one for the parameters of the model. Information about the operation of the program is included at its Help menu; further information is below. The parameters of the magnetic model can be adjusted to a wide range of values. Parameters may be stepped up or down by clicking the up and down arrows to the right of each numerical value; the new calculated curve will be displayed immediately. After a parameter is selected, the up and down arrow keys on the keyboard will also step the values. New numbers may also be entered from the keyboard; to see the new calculations, click the Apply button or press the Enter key. To move between parameters, one can use the mouse, or the Tab key. While dimensions are listed in meters, they can be assumed to be either feet or cm, and the calculation will still be correct as long as all units of length are the same. The line of calculation is over the midpoint between the ends of the prism. Enter the parameters of the prism as follows: Width, of the top or bottom of the prism, Height, the vertical dimension of the prism, Dip, the tilt angle of the left and right sides of the prism, Strike length, the total length perpendicular to the display, Depth, below the sensor, not the distance underground, Susc, the magnetic susceptibility in SI units, and Density, if gravity calculations are wished. If the dip angle is not 90 degrees, then the strike length is assumed to be infinite. The strike length can be set to a small value for compact bodies, but this length can be adjusted only if the Dip is set at 90 degrees. The 4 * pi factor for converting from cgs to SI units of susceptibility is listed; a negative value may be entered for susceptibility. Enter the magnitude of the Earth's magnetic field at H. Inclination is positive in most of the northern hemisphere. Declination is the angle from grid north or true north (the vertical direction on the display) to magnetic north; it is positive if magnetic north is east of true north. Place a check mark at "Plan view" to see a map that is viewed from above; the blue symbol locates the direction of magnetic north. Other parameters: Traverse bearing is the angle of the line of calculation, measured clockwise from true or grid north; turn on the "Plan view" option to see the rotation of the prism and the line of calculation. The line of calculation is always perpendicular to the prism. Traverse length is the span of the calculation; the prism is located at the middle of this span. Any one of four calculated components can be plotted. TMI is the anomaly (Total-field Magnetic Intensity), in nT. Vert. Grad. TMI is the vertical gradient, in nT/m. Gravity plots the anomaly in milligal. The gradient calculations assume that the vertical spacing between the measurements is almost zero, that is, it is infinitesimal. The plot window lists the maximum and minimum anomalies, and shows a zero-anomaly line. The plot window can be enlarged to full-screen if wished. The display can Page 18

Appendix 13: Other forward-modeling software be printed. Quit the program by clicking on the X at the upper right corner of either window. Appendix 13: Other forward-modeling software: Inversion programs are not listed. The best programs are toward the top of this list: Fatiando a terra: "An open source toolkit for geophysical modeling and inversion". A wide collection of potential field programs that are written in Python, at the web site: http://www.fatiando.org/ Grav Mag Prism: "Grav_Mag_Prism: A MatLab / Octave program to generate gravity and magnetic anomalies due to rectangular prismatic bodies", by Alessandra de Barros e Silva Bongiolo, Jeferson de Souza, Francisco José Fonseca Ferreira and Luís Gustavo de Castro, 2013, Revista Brasileira de Geofísica, Volume 31 number 3, pages 347-363. Article available at: http://sys2.sbgf.org.br/revista/index.php/rbgf/article/view/310 FFT3D, a MatLab program: "Rapid interactive modeling of 3D magnetic anomalies", by Fabio Caratori Tontini, 2012, Computers and Geosciences, volume 48, pages 308-315. This program has many similarities to GaMField. Potensoft: "Potensoft: MatLab-based software for potential field data processing, modeling and mapping", by M. Özgü Arisoy and Ünal Dikman, 2011, Computers and Geosciences, volume 37, pages 935-942. USGS potential field software: version 2.2, DOS software, programs MagPoly and PfMag3D are particularly good for forward models, available at: http://pubs.usgs.gov/of/1997/ofr-97-0725/pfofr.htm MagMod, a MatLab program: "Multisensor methods for buried unexploded ordnance detection, discrimination, and identification", by Dwain K. Butler, Ernesto R. Cespedes, Cary B. Cox, and Paul J. Wolfe, 1998, Technical Report SERDP-98-10, US Army Corps of Engineers, Waterways Experiment Station, 183 pages. The publication includes a listing of the MatLab program for forward modeling. "Forward modeling of applied geophysics methods using Comsol and comparison with analytical and laboratory analog models", by S. L. Butler and G. Sinha, 2012, Computers and Geosciences, volume 42, pages 168-176. Requires Comsol, a high level language like MatLab. Grama: "GRAMA, an interactive 2.5-dimensional gravity and magnetics modeling program, version 1.3", 1990, by Pacific Geoscience Centre, Geological Survey of Canada, Open File 2290. This is a DOS program. Model2d: "Interactive 2D gravity and magnetic modeling for IBM PC compatible computers", version 3, 1992, by Michael Roach (University of Tasmania). Another DOS program. Appendix 14: Notes on the GaMField program: A published article has good additional information about the program: "Graphical interactive generation of gravity and magnetic fields" by A. Pignatelli, I. Nicolosi R. Carluccio, M. Chiappini. and R. von Frese, Computers and Geosciences, volume 37 (2011), pages 567–572. Page 19

Appendix 14: Notes on the GaMField program After installation of the program, the only file that is needed in the working folder is GaMField.exe. One can move the program GaMField and its files to any other location on the computer where it was installed. One must be careful if this program and also the Microsoft Access program are installed on the same computer, for both use the file extension "mat"; it is the MatLab parts of GaMField that apply this extension for matrix files. It appears that if the Access program is installed after GaMField, the Access program attempts to load the MatLab files. This causes Windows Explorer to repeatedly halt and reload. Information about this problem is available at: http://www.mathworks.com/matlabcentral/newsreader/view_thread/269365 One should generally have the granularity (spacing) of the calculation grid smaller than the grid spacing for the rectangular boxes; otherwise the calculated maps are not very smooth. There is a numerical entry on the right between "Show 3-D Model" and "Multiple Layer Copy". If the value here is changed, the program allows entry of magnetic bodies on a vertical plane that goes north-south; the value here is the Easterly distance in the 3-D matrix. To set prisms onto a new layer, the depth of that layer must be selected from the pull-down list (click on the down arrow). A new window will not be created when the depth already shown is clicked. The file of saved "Survey Parameters" is loaded to the "Observation Grid"; the file of saved "Model Parameters" is loaded to the "Source Grid"; the file of "Total Parameters" is saved to both the Observation and Source grids. The file from "Save Model Parameters to GaMField" can be loaded at "File / Model / Load Model". The visibility of the Built Model may sometimes be poor, with all bodies rather black. This can be improved by right-clicking the image and then going to Rotate Options / Stretch-to-Fill Axes is selected. On a slower computer, the rotation of the image is improved by selecting "Plot Box Rotate" at Rotate Options. Note that each of these settings will return to their original values when the view is closed. It appears that ModelBuilder fails if the XY dimensions and spacing of the survey parameters are the same as those for the model parameters. If there is only one prism in ModelBuilder, then Show 3-D Model will not display it; there must be more than one prism for this to work. A circular icon (snake chasing its tail) for a cursor indicates that the current map can be rotated for a different view; this rotation can be a bit slow, so do not try to do it quickly. After the first box is marked in the magnetic model, the checkerboard disappears. When using the polygon to select boxes, the checkerboard remains; do not mark the last point that closes the polygon, for the program will continue the line from the last point to the first. The color scale on the right side of the source volume changes as the maxmum value of the plotted parameter changes. Some warnings will be hidden by the ModelBuilder window, but these will appear on the Taskbar, and can be viewed by clicking on them there. With a calculation matrix of 100 by 100 points, there can be a period of about 30 seconds with no display of what is going on except for hard disk activity. With this number of Page 20

Appendix 15: Major commercial programs for magnetic analysis calculations and roughly 30 prisms, the program fails with "out of memory" on a computer with 1 GB of RAM. The text file created by "Save Model Parameters" has nine parameters for each prism, in this order: minimum X of prism, minimum Y of prism, Z depth to top of prism, magnetization (A/m), X width of prism, Y width of prism, thickness of prism, inclination angle of magnetization, declination angle of magnetization. The program allows saving magnetic maps in many different graphics formats. There is an expiration date for the program; it is likely that a newer version will be available by then. I thank Alessandro Pignatelli for assisting me with his program; while he is no longer with INGV he remains very interested in this software. Appendix 15: Major commercial programs for magnetic analysis: Probably all of these programs will also do forward modeling. It takes a large amount of time to learn the good and bad points of these complex programs and I know only a few of them well enough for an evaluation. Potent: Least expensive; allows the calculation of true (as-measured) gradients and not just theoretical gradients. Emigma: Excellent visualization of models; it is complex in operation because of its generality (includes EM and resistivity). ModelVision: Includes a detailed description of the program and the principles of geophysical analysis. This program was recently marketed by Pitney-Bowes, but it is now sold by Tensor Research, the primary author of the software. Pitney-Bowes now sells another program, called Discover PA. Other companies with excellent potential field software include Geosoft, Intrepid Geophysics, and Zond, but I do not know very much about these programs. Appendix 16: Magnetic calculations of slabs and polyhedra: These references have the equations that are needed for the calculation of the magnetic anomaly of a half-infinite slab: "Automatic inversion of magnetic anomalies of faults", by I.V. Radhakrishna Murthy, K.V. Swamy, and S. Jagannadha Rao, 2001, Computers and Geosciences, volume 27, pages 315-325. "LIMAT: a computer program for least-squares inversion of magnetic anomalies over long tabular bodies", 2003, Computers and Geosciences, volume 29, pages 91-98. "Magnetic modelling of two and three-dimensional bodies", by C. Tarlowski, 1989, Bureau of Mineral Resources, Geology and Geophysics, Australia, report 291, 23 pages. General magnetic models may be constructed from polyhedra, although the shapes may be difficult to create or understand. The Emigma program (Appendix 15) allows these calculations, and the following references have the equations: “Optimum expression for computation of the magnetic field of a homogeneous polyhedral body”, by M. Ivan, Geophysical Prospecting, 1996, volume 44, number 2, pages 279-288. Page 21

Appendix 17: Notes on the figures “New scheme for computing the magnetic field resulting from a uniformly magnetized arbitrary polyhedron”, by D. Guptasarma and B. Singh, Geophysics, 1999, volume 64, number 1, pages 70-74. “New method for fast computation of gravity and magnetic anomalies from arbitrary polyhedra”, by Bijendra Singh and D. Guptasarma, Geophysics, 2001, volume 66, number 2, pages 521-526. Appendix 17: Notes on the figures: These notes will be of little value to most readers. However, if you have a question about one of the figures in this report, the information here might help you. Almost all of the reports that are mentioned here are to be found on the internet, in particular at www.ResearchGate.net or at the Digital Archaeological Record: core.tdar.org/document/#, where # is replaced by the tDAR document number here. If you would wish a report, it may be easiest to do a web search for the report title that is given here; this will have the advantage of finding similar reports that may also be of interest to you. Figure 1: File Model3C.gcd in folder "example models". Figure 2: File pc395gp3.grf in folder "03 separate model components". This is Figure 8 in a report (LinearMound.pdf) titled "Appendix: Geophysical tests on the linear mound northwest of Kosh-Oba" in project "20120928 LinearMound"; the report is available at ResearchGate. This analysis was for the study of a magnetic map that was measured by T. N. Smekalova (St.-Petersburg State University). No measurements of magnetic susceptibility have yet been made at this mound, and the values in the figure are typical of the area; however, the susceptibility of topsoil was estimated from the magnetic anomaly caused by a linear rut (10 cm deep) in a nearby dirt road. Figure 3: File Fig14m.srf in folder "06 model below topo". This has been modified from Figure 7 in the publication (ArchDate.pdf) titled "Archaeological dating from magnetic maps: Some failures" in project "20081124 ArchDate" and also from Figure 2, file (23)CKelM.grd, of the report "A Magnetic Analysis of Kiln Alpha at Choban-Kule" in project "19980401 Choban". This model was created from a magnetic map that was measured by Tatiana Smekalova; the height of the magnetic sensor was 0.1 m above the surface for this survey. The topography was determined by the measurements of elevation that were made by Natalia and Alexander Mitsyn (St.-Petersburg State University). Figure 4: File "mag test.psp" in folder "17 meas remnance". This photo was Figure 43 in a report (Braendes.pdf) titled "A magnetic exploration of Brændesgård" in project "20120911 Braendes"; this report is available at the Digital Archaeological Record, document 381225. This survey was done for Olfert Voss and the geophysical work was directed by Tatiana Smekalova. The photo shows a Geometrics model G-858 cesium magnetic gradiometer with both of its sensors stationary. The wooden table and the other items near the sensors were non-magnetic. This test was done in Virginia, USA. Figure 5: File m5r.grf in folder "17 meas remnance". This is Figure 44 from the report (Braendes.pdf) titled "A magnetic exploration of Brændesgård" from project "20120911 Braendes"; this report is available at the Digital Archaeological Record, document 381225. Page 22

Appendix 17: Notes on the figures The stone that was tested was from the archaeological site of Brændesgård on Bornholm Island in Denmark; this stone was approximately centered in the clear plastic sphere that is shown in the prior figure. The stone had dimensions of 8.0 x 5.3 x 3.2 cm and its volume was 48.3 cm3; errors result if the magnetic center of the stone is not placed at the middle of the sphere. This work was done as part of a project for Olfert Voss. Figure 6: File rotation.grf in folder "object rotation". The arbitrary equations for the red and blue lines were as follows: red = 5 * sin(angle + 30) and blue = 6 + {3 * sin(angle + 70)}. Figure 7: File 3ne735n.grf in folder "01 del feat same anom" and in folder "20130728 xyz pnt". This is Figure 5 of a report (OrtliField.pdf) titled "An analysis of the magnetic anomalies at the corners of the Ortli field" from project "20130725 OrtliField", folder "20130728 xyz pnt". The magnetic survey was done by T. N. Smekalova and her colleagues at the ancient Greek farm called Ortli, in Crimea. The calculations of the 2D models were made using the algorithm of Won and Bevis. The magnetic parameters of the Earth were assumed to be: Magnitude = 49,050 nT (a typical measurement at the site), Inclination = 62.6/ (from the IGRF11 values at the time and location of this survey), Declination = 11/ (from a magnetic compass angle). The axis on the left also shows depth underground. Figure 8: File mm4.grf in folder "15 same anom". This is also Figure 21 in a report (Huseby.pdf, available at ResearchGate) called "A Geophysical Survey at Huseby" for project "20090603 Huseby", folder "S mag models". This project was done for Frans-Arne Stylegar and the geophysical work was directed by T. N. Smekalova. Figure 9: File Band3.grf in folder "13 same anom del k". This is Figure 29 from a report (MagLin.pdf) titled "Analysis of Linear Magnetic Anomalies" in project "20050923 MagLin"; the report is available at ResearchGate. Figure 10: File cubex.grf in folder "cube extended". The peak anomaly from the cube is 6.74 nT, while the peak anomaly from the infinitely long prism is 7.40 nT. With a vertical-component, vertical-gradient magnetometer (with a sensor spacing of 1 m), the peak anomalies are as follows: prism = 5.40 nT, cube = 5.75 nT. Therefore, for this instrument, the anomaly of the cube is 7% larger than that of the prism. Figure 11: File WidthP.grf in folder "04 prism length". This is Figure 31 from a report (MagLin.pdf) titled "Analysis of Linear Magnetic Anomalies" from project "20050923 MagLin"; the report is available at ResearchGate. The 2-D calculations were made with the aid of the algorithm of Won and Bevis. The magnetic north direction is the same as the direction of the length of the prisms. Figure 12: File "trench model.psp" in folder "09 trench anomaly". The parameters for this model are from Figure 9 in a report (PetExcav.pdf) titled "Geophysical Tests in the Fort Morton Excavations" in project "199806019 PetExcav"; data are in project "19910123 Geophys", disk (280b). The report is available at the Digital Archaeological Record, document 381089. The figure is a screen capture of the Pdyke program. While this magnetic anomaly is weak, John Weymouth (University of Nebraska) discovered that the refilled fortification ditch was actually visible in one area where the anomaly was stronger. Figure 13: File "model grave.psp" in folder "07 model grave". This is a screen capture Page 23

Appendix 17: Notes on the figures from the program MagPrism. The parameters of the model are listed on page E3 of the report (ChristCh.pdf) titled "Geophysical Tests in the Christ Church Excavations" with project "19850403 ChristCh"; the report is available at the Digital Archaeological Record, document 381146. The grave shafts would be detected primarily as magnetic lows. Modifications of the model show that if the rather non-magnetic fill was assumed to extend between a depth of 0.5 ft and 3 ft, the anomaly would increase to -1.1 nT; if the magnetic sensor was lowered to the surface of the earth, then the anomaly would increase to about -2 nT. Figure 14: File GFinput.psp in folder "GaMField / test 2". These are a pair of screen captures from the program GaMField. Figure 15: File GFmodel.psp in folder "GaMField / test 2". Another pair of screen captures from the program GaMField. Figure 16: File GFcalc.psp in folder "GaMField / test 2". These are the final pair of screen captures from the program GaMField. Figure 17: File pita.psp in folder "Noddy tests". This is a screen capture from the Noddy program. Figure 18: File plow.psp in Folder "Noddy tests". This is another screen capture from the Noddy program. Figure 19: File Wall4.gcd in folder "Ak-kaja model". This is derived from Figure 13 in a report (Ak-Kaja.pdf) titled "A magnetic survey at Ak-Kaja" in project "20150609 Ak-kaja". This magnetic survey was directed by Tatiana Smekalova and the work was done for archaeological excavations which were directed by Juriy Zaitcev. Figure 20: File Walls.grf in folder "Ak-kaja model". This is Figure 14 from the report (Ak-Kaja.pdf) titled "A magnetic survey at Ak-Kaja" in project "20150609 Ak-kaja". The magnetic survey was directed by Tatiana Smekalova; the archaeological excavation was directed by Juriy Zaitcev. Figure 21: File gm180c2.grf in folder "02 calc compare change"; see also folder "Guaj lecture". This has been derived from Figure 8 in the publication (Guajara.pdf) titled "Geophysical exploration of Guajara, a prehistoric earth mound in Brazil" and project "20030305 Guajara" and also from Figure 171 in the report (Amazon87.pdf) titled "Geophysical Surveys at Three Sites Along the Lower Amazon River" and project "19871004 Amazon87"; the report is available at the Digital Archaeological Record, document 381140. The archaeological project was directed by Anna Roosevelt and the magnetic measurements were made by her field crew. It is possible that the deep magnetic strata continue across the width of the mound; it is reasonable that they would not be very detectable where they are deeper and covered by shallower features. Figure 22: File 4nec.srf in folder "18 Ortli NE pit". Modified from Figure 1 in the report (OrtliField.pdf) titled "An analysis of the magnetic anomalies at the corners of the Ortli field" in project "20130725 OrtliField". The magnetic survey at this site was done by Andrei Chudin; T. N. Smekalova also did the magnetic survey and directed the project. A summary of the project is included with the article (Ortli.pdf) titled "The Discovery of an Ancient Greek Vineyard" in project "20140912 Ortli" that was published in the journal Archaeological Prospection in 2016 (volume 23, pages 15-26). The sensor height for the total field magnetometer was about 0.3 m. The spacing between parallel lines of traverse was 0.5 m, Page 24

Appendix 17: Notes on the figures and the spacing between measurements on each traverse was 0.25 - 0.5 m. The dipolar anomaly of a shallow iron object near E299 N296 hides the weak magnetic low that would be expected from the feature that is to the south. Figure 23: File MsSe22.srf in folder "14 residual map". This is a modification of Figures 8 - 10 in a report (Brick.pdf) titled "Geophysical Detection of Brick Structures" in project "20011011 Brick"; this report is available at ResearchGate. The inclination of the Earth's magnetic field at this test site was about 66 degrees. This brick was from the Greek and Roman site of Aleria in Corsica. The 1971 geophysical project was directed by Froelich Rainey and Elizabeth Ralph, and the work was for Jean Jehasse. Note that the arithmetic subtraction here results in some errors; a vectorial subtraction would have been better. Figure 24: File tu8c.gcd in folder "08 posthole". This figure has been modified from Figure 37 in a report (Bevan02a.pdf) titled "Geophysical Tests in the Hopeton Excavations" in folder "20020626 Hopeton 2"; the report is available at the Digital Archaeological Record, document 381057. This work was done as part of a project that was directed by Mark Lynott; the magnetic survey at this site was directed by John Weymouth. The susceptibility contrast of the pit was 0.8 ppt, the estimate volume of the pit was 0.0035 m3, and this yields a magnetic moment of 0.121 mAm2. The Earth's field had a magnitude of 54,000 nT at an inclination of 68.2 degrees. Figure 25: File Xsection.gcd in folder "16 aqueduct model". This figure has been modified from Figure 5 in a report (AqueMag.pdf) titled "Magnetic properties of the granite aqueduct at Ammaia" from folder "20110704 AqueMag". This work was done with Tatiana Smekalova as a demonstration for a course on geophysical exploration that was held at the Roman site of Ammaia in Portugal; that course was coordinated by Cristina Corsi and Frank Vermeulen. The Earth's field has the parameters of: Magnitude = 44,100 nT, Inclination = 54 degrees. Figure 26: File hover.srf in folder "11 special models". This is a modification of Figure B31 in the report (GeophysB.pdf) titled "Geophysical Exploration for Archaeology; Volume B: Introduction to geophysical exploration" in project "19910123 Geophys"; this report is available at ResearchGate. These models are calculated for a magnetic inclination of 45 degrees. The slab is 6 m on a side and it has a 1 m hole in it. Its magnetic susceptibility is 0.01 and its thickness is 1/6 m; the top of the slab is 1 m below the sensor. For the overhead object, a dipole with a moment of 3.13 Am2 is assumed (about 10 kg iron); this dipole is 3 m above the sensor and is located at the star in the figure. Figure 27: File Do.srf in folder "05 random remnance". This has been modified from Figure 30 in a report (MagMaps.pdf) titled "Understand Magnetic Maps" in folder "20060501 MagMaps"; this report is available at ResearchGate. The Earth's field was assumed to be: Be = 57,000 nT; Ie = 70/; De = 0. The calculations were made 0.5 m above the model; the algebraic sum of the magnetic moments for each dipole is 0.01 Am2. The directions of remanence are randomly chosen over a sphere; these directions remain the same in panels B - D. The anomaly range in the four panels is: A = -43.9 to 115.9 nT; B = -23.9 to 69.7 nT; C = -19.2 to 45.2 nT; D = -58.1 to 46.6 nT. An example of random remanence is in my article: “The magnetic anomaly of a brick foundation”, in Archaeological Prospection, 1994 volume 1, number 2, pages 93 - 104. Brick foundations may also have a simple magnetization if they Page 25

Appendix 17: Notes on the figures have been refired in place; see Figure B34 of my publication: Geophysical Exploration for Archaeology: An introduction to geophysical exploration, 1998, Midwest Archaeological Center (Lincoln, Nebraska), special report number 1. Figure 28: File "iron well.srf" in folder "10 well monopole". This is a modification of Figures C54 and C55 in the report (GeophysC.pdf, available at ResearchGate) titled "Geophysical Exploration for Archaeology; Volume C: Detailed survey procedures" in folder "19910123 Geophys" where the gridded files are MMiron.grd and MMironM.grd. An archaeological excavation to the top of this well was done by David Orr; he discovered a dug well that was filled with iron debris. Some notes about that excavation are included with a report (PetExcav.pdf) titled "Geophysical Tests in the Fort Morton Excavations" in folder "1998019 PetExav", and also with a presentation titled "Tracing leveled earthworks at Petersburg" at the CAA2009 (Computer Applications in Archaeology) conference which is in folder "20090323 CAAconf" and which is available among the online proceedings at http://archive.caaconference.org/2009/articles/Bevan_Contribution157_a.pdf. Figure 29: File MonoLin.grf in folder "10 well monopole". This is a modification of Figure C57 from a report (GeophysC.pdf, available at ResearchGate) titled "Geophysical Exploration for Archaeology; Volume C: Detailed survey procedures" in folder "19910123 Geophys". Figure 30: File MsNo23L3c.srf in folder "12 bldg iron". This is a modification of Figure 9 in a report (Watt.pdf) titled "A Geophysical Survey at the Watt House, Richmond National Battlefield" in folder "19991012 Watt". This geophysical survey was done for Allen Cooper; the report is available at the Digital Archaeological Record, document 381211.

Page 26

Figure 1: A wide variety of magnetic models. The pair of models at the upper left are two-dimensional and are shown as cross-sections; the other models are three-dimensional. The color red marks magnetic features, while blue shows where the features are less magnetic than normal. These models are greatly simplified from the true geometry of the buried features; since magnetic surveys provide such a low resolution of the features, these simplifications are almost always acceptable. In fact, most magnetic anomalies can be approximated by one or more of the following three basic magnetic models: Spheres, rectangular boxes, or flat slabs.

117 Linear mound NW of Kosh-Oba Magnetic features and topography Vertical exaggeration = 5

Elevation, m

116

Surface elevation

115 k = 0.007 k=0

114

k=0

surrounding k = 0.001 113 0

10

20 30 North distance, m

40

50

Magnetic anomaly, nT

40 Linear mound NW of Kosh-Oba Anomalies of individual features Calculation 0.25 m above surface

30 20

Earth's magnetic field: Be = 49,176 nT Ie = 63.1o De = 46o CW from N

10 0 -10 0

10

20 30 North distance, m

40

50

Figure 2: The separate parts of a magnetic model. One of the applications of a magnetic model is an estimation of the effects of its separate components. The calculation of this twodimensional model indicates that the magnetic feature that is outlined in red has a greater effect than the mound itself (green) or the non-magnetic walls of stone that are inside (blue). The resultant anomaly of all three magnetic components is a vector sum of these three parts; this is similar to an algebraic sum of the anomalies of the three components. This feature is an earthen mound that is about 2 m tall; it extends for a distance of over 50 m. Magnetic anomalies caused by features within the mound are also linear and this allows a two-dimensional analysis. This model was created as part of a study of the magnetic map of this mound. While the magnetic soil (red) between the walls was revealed by a magnetic high, this feature may not be a triangular distribution of uniform soil; instead it could be caused by an increase in the magnetic susceptibility or remanence of a flat stratum of soil near the middle of the mound.

Magnetic model below surface, contours at 0.5 m 15 M=8 I = -3 D = 11

2-3m

3

1 .6 -

2m

2m

1.6 - 2 .8 m

10

M = 107 Am2 I = 78o o

4.5

4

D = 105

3.5

North distance, m

1 .6 -

5 5

10

15

East distance, m

Figure 3: The magnetic model of a buried ceramic kiln. The calculated field from this model was similar to the magnetic map that was measured over the kiln. The red E-shaped outline locates the model; black text on the arms of the E lists the elevation of the upper and lower sides of the separate parts of the model. A separate red box at the top of the figure locates another magnetic object. The two dashed blue rectangles list the magnetic parameters of the models, showing their total magnetic moment (M) and the direction of this magnetization (Inclination and Declination). The elevation of the ground surface is contoured with green lines. While this magnetic model has a simple shape, it is difficult to detect which part of the model is closest to the surface; a separate figure would help to clarify the geometry. More complex models may require that the viewing angle be rotatable by the reader.

Figure 4: Measuring the magnetic properties of a small sample. The sample is supported by plastic foam inside a transparent sphere. A triangular frame on the wooden table (which is about 1 m square) allows the sample to remain centered while it is rotated below the sensor of a cesium magnetometer. The angle between the active midpoint of the sensor and the middle of the sample is that of the Earth's magnetic field. A second magnetic sensor measures natural changes in the magnetic field. The control unit here records the measurements from both sensors and the difference is plotted in the next figure. Since the two magnetic sensors must be distant from each other, this measurement is difficult to do with a gradiometer having fixed sensors. This type of measurement allows an estimation of the remanent magnetization of an object; the induced magnetization may also be measured, but this can be easier to determine with a magnetic susceptibility meter. Fired archaeological materials often have remanent magnetization that is ten times larger than induced magnetization, so it is important to measure this parameter.

40 Changing anomaly of basalt stone (#3) when it was rotated randomly at a center-to-center distance of 16 cm from a total-field magnetic sensor Earth's field = 50,935 nT; stone mass = 123 g high = 21.492 nT

Magnetic anomaly, nT

20

0 background = -3.940 nT

-20

low = -30.731 nT

-40 0

50

100 150 Measurement number

200

250

Figure 5: Magnetic recordings of a rotated sample. The sphere in Figure 4 was rotated rather randomly, causing these oscillations. The highest and lowest readings are noted with red and blue. A series of readings were also made after the sample was removed from near the magnetic sensor; these are shown as a horizontal line at the right end of the curve. Three values from this curve (high and low anomaly, along with the background value) allow a calculation of the remanent magnetic moment of the stone; when divided by the mass of the sample, this value was 4.3 mAm2/kg. Since the remanent moment was so much larger than the induced moment (the ratio was about 150), the induced moment was inaccurate from these measurements. Instead, the magnetic susceptibility was measured to be 1.9 ppt, and the induced moment was calculated from this value. The calculation of the remanent moment from the plot above is as follows: Mr = [B(high) - B(low)] * r3 / 400 = (21.5 + 30.7) * 0.163 / 400 = 0.53 (10-3) Am2

The effect of rotation on the anomaly of an object 10

8

susceptibility is anisotropic

6 constant susceptibility

Magnetic anomaly, nT

4

2 non-magnetic or no object 0

-2 remanent magnetization

-4

-6

-8

-10 0

90

180 Rotation angle, degrees

270

360

Figure 6: Rotating a sample near a magnetic sensor. If there is no sample, or if the sample is not magnetic, the anomaly is always zero, the black line. If the sample has induced magnetization alone, and it is constant, the readings shift to a higher value and remain constant with rotation; this is the green curve. If the sample has remanent magnetization alone, the anomaly is high in one direction, and negative in the opposite direction. The magnitude of the high and low are equal; see the red curve. If the sample has only magnetic susceptibility, but its value changes with direction, the readings also oscillate, but the anomaly is always positive, as shown in the blue curve. Many samples will have constant susceptibility and also magnetic remanence; the readings may then oscillate and may always be positive. The resulting curve will look just like the blue curve above that shows the effect of anisotropic susceptibility alone. This indicates that one cannot detect this anisotropy with these measurements.

12

10

8

2-dimensional calculations: shallow and broad intermediate deep and narrow

Magnetic anomaly Ortli field NE corner Line N73.5 Total magnetic field Sensor height = 0.3 m

Magnetic anomaly, nT

6

4

2

magnetic measurements

0

-2

-4

-6 90

95 East distance, m

100

105

Figure 7: Similar anomalies from very different features. The cross-sections of three features are drawn here with color: A broad and shallow triangle (blue), a deeper rectangle (green), and the deepest rectangle (red). The calculated anomalies of these three features are plotted with the same color as the feature, and all calculations are similar. This illustrates the inescapable geophysical problem called equivalence. While the problem is serious, it may not be as bad as suggested here. First, one may still get reliable indications of the maximum depth of a feature. Second, the approximate breadth of the feature may be known, and this will narrow the range of possible models. The measurements of the magnetic field are plotted with a black line. While the positive parts of all curves are almost identical, the negative values differ, and the calculations all show more negative values than the measurements. This is caused by the application of two-dimensional calculations to a compact (somewhat circular) anomaly; see Figure 10 for more information. Even with this 2D - 3D difference, two-dimensional models allow a quick and easy analysis of many magnetic anomalies.

Figure 8: Two different bodies that cause identical anomalies. On the left, there is a rectangular hole in magnetic soil or rock; the resulting anomaly is predominantly a magnetic low. On the right, there is a rectangular object that has remanent magnetization whose direction is the opposite of the Earth's field (the blue arrow). Magnetic holes are most commonly found with the air-filled chambers of tombs or underground cellars. Reversed magnetization may be found with iron artifacts; only a small fraction of iron objects will be oriented for this reversal. In this figure, k indicates magnetic susceptibility (in parts per thousand), M indicates magnetic moment, and I indicates the inclination angle of the magnetic field (zero inclination is horizontal to the north).

Calculated magnetic anomalies Rectangular prisms 1 x 4 m Be = 53,500 nT; De = 0 Sensor height = 1 m above top

12

0

sensor line

0

-6

1 k = 2 ppt

magnetic prisms

k = -2 ppt 2

vertical exaggeration = 2

3

-12

4 0

5

10

15 North coordinate, m

20

25

Depth below sensor, m

Magnetic anomaly, nT

6

30

Figure 9: Another example of identical anomalies from different features. This example is not as obvious as that in the prior figure, and it is also less important. The illustration on the left in red shows that a magnetic feature such as a filled ditch that was measured in the northern hemisphere would be detected as a magnetic high. The blue part of the illustration shows that the same anomaly could be found for a non-magnetic wall that was measured in the southern hemisphere, where the inclination of the Earth's field is upwards. The magnetizing directions for the two illustrations are 90 degrees apart; the arrows shows these directions.

The magnetic anomaly of a prism and cube both with sides 1 m wide Calculation: total magnetic field Earth's field: Be = 50,000 nT; Ie = 90o prism (infinitely long)

8

Magnetic anomaly, nT, or elevation, m

6

4

2

cube

calculation surface

0

cube or prism, k = 1 ppt -2 -5

-4

-3

-2

-1

0 1 Distance, m

2

3

4

5

Figure 10: The effect of length on the anomaly of a feature. It is less than one would probably guess. The black curve indicates the magnetic anomaly of a cube. If two opposite faces of this cube are extended to infinity, creating a prism, the red curve shows a rather small change in the anomaly; the peak has increased by only 10%. There is a larger change to the magnetic lows. This is due to the fact that distant magnetic objects (at about the same elevation) cause magnetic lows, and an infinitely-long prism has a lot of magnetic material that is distant. Calculations like these can provide a basis for approximating the anomalies of other features. The dimensions of depth and cube size can both be multiplied by any constant without changing the anomalies. If the parameters of the magnitude of the Earth's field or the susceptibility of the cube are multiplied by a constant, the anomaly will be multiplied by the same constant. These approximations can be applied where the inclination angle of the Earth's field is steep; near the equator, the anomalies will be negative.

15

Prism goes north-south Calculated magnetic anomalies Rectangular prism, k = 2 ppt Be = 53,500 nT; Ie = 68o; De = 90o Sensor height = 1 m above top

10

Magnetic anomaly, nT

5

0

sensor line

0 1

-10 2 Increasing width of prism

vertical exaggeration = 2

3

-15

4 -5

0

5

10 East coordinate, m

15

20

Depth below sensor, m

-5

25

Figure 11: The effect of width on the anomaly of a magnetic slab. The width of these infinitely long prisms increases from black, green, blue, to red. Starting with a very thin prism, the amplitude of the anomaly increases as it thickens. A peak anomaly is reached quickly, and then the peak anomaly slowly decreases as the prism keeps getting wider. This decrease may seem unreasonable, since the amount of magnetic material is increasing, however remember that distant magnetic materials cause magnetic lows. The anomalies from the two ends of the slab are mirror images. The droop or sag between these two anomalies approaches zero as the slab becomes wider, and the amplitude of the magnetic lows increases as the slab becomes broader.

Figure 12: Calculating the 2-D magnetic anomaly of a filled trench. This model was created for an estimate of the anomaly that would be measured over a refilled trench from the US Civil War (1864) battlefield at Petersburg, Virginia. The weak anomaly, less than 0.2 nT, indicates that a trench like this would not be detected in the area of interest, for the spatial magnetic variability in that area is larger than this. The magnetic parameters of the model have been entered into the table on the right. The feature has a contrast in magnetic susceptibility of 0.1 ppt, and the magnetic object (shown in red) is 3.5 ft (1.1 m) below the magnetic sensor and has a cross-section of 1.5 by 5 ft (0.4 m by 1.5 m). This two-dimensional model was calculated with the program Pdyke, which is freely-available from the company Geophysical Software Solutions at www.geoss.com.au; some notes on the operation of the Pdyke program are included in Appendix12. The program assumes that distances are in meters, but the actual values, in feet, have been entered into the program for this calculation. This causes no error.

Figure 13: The magnetic model of a grave shaft and its calculation. The magnitude of the anomaly would be about 0.5 nT, and this feature would therefore be difficult to detect with a magnetic survey. The model here was derived from measurements that were made in an archaeological excavation that exposed the top of a grave shaft. This excavation also allowed a determination of the dimensions of the grave shaft. The surrounding soil had a susceptibility of 0.5 ppt, while the soil within the grave shaft had a susceptibility of 0.3 ppt. Therefore, the magnetic contrast of the soil within the grave shaft is -0.2 ppt. This calculation was made with the computer program called MagPrism, written by Markku Pirttijärvi (University of Oulu, Finland), and it is available at no cost from: https://wiki.oulu.fi/display/~mpi/Magnetic+field+of+a+prism+model Magprism allows the calculation of the magnetic field of a single finite-length prism that may be rotated; the box has a flat top and its sides may be sloping. The model can have remanent magnetization and the effect of demagnetization is determined. Calculations are made along a horizontal line that may be offset from the prism; these calculations can be total field or one directional component. The program does not allow the entry of susceptibilities that are negative, and so the polarities shown in the figure should be inverted, but the values are otherwise correct. The parameters of the model, for the calculations, and for the Earth's field are listed on the left.

Figure 14: Initial steps with a forward-modeling program. The program is called GaMField (Gravity and Magnetic Field) and it allows the calculation of the magnetic field of a large collection of rectangular boxes. This figure illustrates the information that can be entered to the first two pages of the program. The upper image lists the overall parameters of the calculation and specifies the volume that includes the model. On the left side are listed the span that will be calculated, the spacing and elevation of those calculations, and also the direction of the Earth's field. On the right side, the dimensions of a volume that encompasses all of the models is listed; this volume will usually have a smaller surface area than that to be calculated. The lower image reveals the magnetic model on a series of layers that are colored dark blue; the two separated models are difficult to see in this view, but they are colored light blue (near the top) and red (near the bottom).

Figure 15: Two views of the magnetic model for GaMField. These may clarify the magnetic model whose field is to be calculated. The upper image is an oblique view of the shallow (comma-shaped) and deep (linear) features. The lower image is a vertical view; while this does not show depth, it does reveal the patterns of the boxes on the two layers. The color scale on the right indicates the intensity of magnetization of the bodies, in A/m. Information about the rectangular boxes that are next to be entered to the model are listed on the upper side of the lower image in Figure 14; these parameters are the intensity of magnetization and its direction for the body, and also its depth. The GaMField program is available at no cost at the web site: http://geosoftware.sci.ingv.it/ A signup is required before downloading the program.

Figure 16: Two views that show the calculations of the GaMField program. The color scales on the right indicate the magnitude of the values, with blues showing lows and reds marking highs. The upper image is a vertical view, with calculations at a height of 0.25 m. The lower image is an oblique view that shows calculations at two heights (0.25 and 1.25 m). Since vertical-component calculations may be made and saved as data files; one may calculate the readings of a fluxgate gradiometer. The GaMField program was written by Alessandro Pignatelli and others at the Instituto Nazionale di Geofisica e Vulcanologia, in Italy.

Figure 17: The magnetic model of a refilled pit. The model at the lower left was created in the program Noddy, and its calculated field is shown at the upper right. The pit (colored red) is surrounded by subsoil (green) and covered by another layer of soil (brown). This model, in the lower left side of the screen, has a section cut away in order to reveal the internal shape. At the upper left, the steps in creating the model are listed: 1 - the horizontal strata are entered (the green layer has a susceptibility of 1 ppt); 2 - the pit is defined as an intrusive plug (an ellipsoid) and it has a susceptibility of 3 ppt; 3 - the upper part of the model is truncated and the soil there is replaced (this is called an unconformity U/C) and the brown soil has a susceptibility of 2 ppt. The calculated map at the upper right has red colors to mark highs and violet for low values. While there is no scale that indicates the amplitudes of the anomalies, these numbers are listed where the cursor is located; the curve at the lower right shows amplitudes along a cross-section through the anomaly. It indicates a peak anomaly of about 7 nT for this model. The Noddy program is made for geological applications, and small archaeological dimensions do not appear to work well. It is best to enter dimensions in mm, so that a distance of 5000 means 5 m, rather than 5 km.

Figure 18: A magnetic model of plow scars in topsoil. This model was also created and studied with the Noddy program. The base stratum (green) is entered first; this has a magnetic layer on top of it (black, and originally flat). The strata are then folded (corrugated) so that the top layer extends into the lower layer to simulate furrows, which are slightly oblique from the sides of the model. The calculated map on the right illustrates the bands of high and low magnetic field. The Noddy program is available after registration at url: http://www.tectonique.net/tectask/. The original version of this program was written by Mark Jessell; for a period, a commercial version was sold by Encom Technology, but the program is now free.

Figure 19: The simplification of a magnetic model. The upper panel (A) shows the cross-section of a fortification wall whose remnants are now underground. There are a pair of solid stone walls (diagonal red hatching) that provide a strong facing for the defensive work. A mixture of stone and soil fills the space between the solid walls. The magnetic susceptibilities of the materials are listed in the second panel (B); the stone and soil were measured in an excavation. The susceptibility of the stone rubble was estimated by assuming that it was half stone and half soil; the soil that was mixed with the stone rubble was estimated to have a susceptibility of 0.35 ppt (a value between subsoil and topsoil). Therefore, the susceptibility of the stone rubble is estimated to be (0.017 + 0.35) / 2 or 0.18 ppt. In the third panel (C), a susceptibility of 0.21 ppt (subsoil) has been subtracted from all susceptibilities. Finally, in the fourth and bottom panel (D), a susceptibility of 0.3 ppt has been subtracted from all strata that are shallower than 0.5 m. The result is a model that is surrounded by a susceptibility of zero. This simplifies the calculation of the anomaly, even though it now consists of six separate rectangles.

1.5

1

Calculated magnetic anomaly of a stone wall, 4 m wide Faced with stone blocks and filled with stone rubble Wall height = 1.5 m, depth to top = 0.3 m Susceptibilities: Stone 0.017 ppt, topsoil 0.51 ppt, subsoil 0.21 ppt Earth's magnetic field: Be = 49200 nT, Ie = 63o Total magnetic field, sensor height = 0.3 m

wall E-W Magnetic anomaly, nT

0.5

0 wall N-S

-0.5

-1

-1.5 -10

-5

0 Distance from middle of wall, m

5

10

Figure 20: The calculated anomaly of the model shown in Figure 19. The calculation was made with the routine of Won and Bevis for two orientations of the length of the wall. These calculations indicate that the magnitude of the peak anomaly should be only about 1 nT. Magnetic mapping of unexcavated sections of the wall found that the anomalies over the buried stone of the wall facings typically had anomalies of about -7 nT, much larger than the estimate from the model. The model is therefore wrong. It is unlikely that the depth of the wall is much less than assumed by the model, or that its vertical extent is much greater. It is almost certain that the susceptibility values that were measured for stone are correct. The only parameter that remains is the magnetic property of the soil; there are two possibilities. This soil may have significant remanent magnetization (not measured by the susceptibility meter) or its susceptibility may be higher in other areas outside where it was exposed in the excavation. While both effects may contribute, perhaps remanent magnetization is most likely.

Magnetic profile, Guajara Mound Line E180 (average E173 - 187)

20

Magnetic anomaly, nT

10

0

-10

-20 measured calculated

-30

-40 70

80

90

100

110

120

130

Magnetic model, Guajara Mound Cross-section of line E180 Vertical exaggeration = 4X

12

Elevation, m

10 ground surface

8 measurement plane k = 0.035

6 k = 0.005

k = 0.01

4 70

80

90 100 110 North coordinate, m

120

130

Figure 21: The magnetic stratigraphy of an earthen mound. This mound is elongated, and a two-dimensional approximation of its shape is given in the bottom half of the figure. The red polygons there locate magnetic soil and list the susceptibility in SI units. The upper part of the figure is a plot of the calculated (total) magnetic field; since the site is in Brazil, the inclination of the Earth's field is nearly horizontal, and magnetic lows are found over magnetic features. The measurements of magnetic field are shown as the blue curve in the upper section. The magnetic model was manually iterated until the calculations were similar to those measurements. While automated inversions are common, a simple cycle of: Calculate-Compare-Change can also result in a good magnetic model. This procedure has the advantage of easily constraining some parameters to values that are reasonable.

N (magnetic) magnetic high magnetic low

Ortli vineyard, NE corner, contour interval = 1 nT 300

North distance, m

295

290

285

280

290

295

300

305

East distance, m Figure 22: The importance of magnetic models. The interpretation of the magnetic high (with red contours) near E299 N293 in this map suggested that the source was deeper than 1 m underground; since limestone bedrock is known to start at a depth of 1 m, the interpreted depth appeared to be wrong. An excavation at this location revealed a shallow lens of dark soil near this location. The magnetic susceptibility of this soil was 0.5 ppt greater than the surrounding soil. Since the volume and depth of the soil lens were known, its magnetic anomaly could be calculated. This anomaly was only 2 nT, much less than the actual anomaly of 12 nT. The excavation then continued deeper, and a pit was discovered that went to a depth of about 3 m, well into bedrock. The soil that filled this pit had a susceptibility contrast of 4 ppt; since the dimensions of the pit were known, the anomaly of this new model could also be calculated. It was found to be about 6 nT, and therefore still too small. Since the soil that fills the pit was not stratified, anisotropy seems unlikely, and it must be concluded that the soil in the pit has significant remanent magnetization.

Measurements Total magnetic field Sensor height = 2.6 cm Measurement spacing = 1 cm Contour interval = 20 nT Map area = 40 by 30 cm

magnetic high magnetic low

Difference: Measurements - Calculations

Calculations

Roman brick (black square) Dimensions: 19.2 x 20.7 cm, and 4.3 cm thick Mass: 2750 g, Volume: 1735 ml, Density: 1585 kg/m3 Color: moderate reddish brown, 10R6/6 Susceptibility: 6.0 ppt N (magnetic) Calculations: The brick has uniform magnetization magnetization of the model = 2.16 mAm2 inclination of total magnetization = 19.7 degrees declination = -107.5 degrees.

B(Earth) = 52,560 nT

Figure 23: The residual anomaly of the model of a brick. A black rectangle locates this brick, which is described at the lower right. The parameters for the model of the brick are listed at the lower left in this figure, and the calculated field of that model is plotted above. The map on the right is a subtraction of the Calculations from the Measurements. The residual or difference map locates faults in the model; the brick was not actually magnetized uniformly. Instead, there must be at least one area where the mineral content or firing of the brick was different from usual. The magnetic measurements were made by moving a small tri-axial fluxgate sensor along lines on a flat plate over the brick. The magnetic model was created by assuming that the whole volume of the brick was uniformly-magnetized.

Figure 24: The invisibility of a posthole. A magnetic survey is frequently followed by test excavation of a sample of anomalies. If a feature is found, was it the source of the anomaly? The excavation shown in cross-section here stopped when the dark soil of a posthole was encountered. The magnetic susceptibility of the soil was measured, both within the posthole and outside it. The diameter of the posthole was known, and its further depth could be estimated as about 23 cm (from nearby excavations). This magnetic model yields a calculated anomaly (total field) of 0.04 nT at the height of 0.3 m above the original ground surface. This pit could not have caused the magnetic anomaly. Even if there was a large remanent component to the magnetization of the fill of the pit, it still could not have caused the magnetic anomaly that was measured here. Since this location was farmed, perhaps a bit of wire or a bolt caused the magnetic anomaly, but this iron debris was not noted during the excavation.

Figure 25: The magnetic model of a stone aqueduct. A short length of this feature had been exposed in an excavation. The goal of this field test was an estimation of the magnetic anomaly that could be caused by the aqueduct in those areas where it is buried and invisible. The magnetic parameters of the stone and soil are included in this cross-section. Note that the granite is almost non-magnetic; it appears to be composed entirely of quartz and feldspar, and no dark mineral grains (which could be magnetite) were visible. This magnetic model allowed a two-dimensional calculation of the anticipated anomaly and it was only -1.8 nT; the anomaly is negative because most of the aqueduct is less magnetic than the surrounding soil. Spatial variations in magnetic maps made in the area suggest that it will be difficult to detect this aqueduct where it is buried. While the model here is moderately complex, the same anomaly is calculated when the model has the simple cross-section of a square with sides that are 0.6 m long. Models like these are valuable for indicating the polarity of the expected magnetic anomaly, and for revealing the offset between the anomaly and the buried feature. This information will allow a small excavation to expose a feature in a magnetic map.

Figure 26: Some interesting magnetic models. The top model is a square magnetic slab with a square hole in it. The calculated contour lines (red = high, blue = low) show a magnetic low on the upper side of the slab, but on the bottom side of the hole. The bottom model has the calculation of an overhead object, at the star. While magnetic north is toward the top of the page, this magnetic low is toward the south, and not to the north, as it will be for objects that are underground.

Induced alone

2

2

B North coordinate, m

North coordinate, m

A 1 0 -1 -2 -2

2

-1

0

1

1 0 -1 -2 -2

2

-1

0

1

East coordinate, m

East coordinate, m

Remanent = 3 * induced

Remanent alone

2

2

D North coordinate, m

C North coordinate, m

Remanent = induced

1 0 -1 -2 -2

-1

0

1

East coordinate, m

2

1 0 -1 -2 -2

-1

0

1

2

East coordinate, m

Figure 27: The effect of random directions of remanent magnetization. This magnetic model is composed of a matrix of 11 by 11 dipoles on a single level. The proportion of remanent magnetization (compared to induced magnetization) increases from panel A (upper left) to panel D (lower right). These types of anomalies can be found with pavements of brick or igneous stone. If the pavement was burned in place so that it was remagnetized, the anomalies could look like panel A; otherwise the pattern could be like panel D.

measured anomaly

calculated anomaly

Figure 28: The magnetic model of a well. The model was simply a magnetic monopole, and it is located with a red star in the calculation shown on the right. The magnetic measurements are on the left side. The interval between contour lines in these maps is 5 nT (colored) and 100 nT (black); the peak anomaly is over 2700 nT. The vertical length of both maps is 61 m. The upper part of this well was underground, and not visible. However, the only magnetic model that fits the measurements is a monopole. A magnetic monopole provides an excellent approximation of the magnetic anomaly of a very long magnetic feature that is vertical; while the object here could be a well pipe or a rod for grounding lightning, the intense anomaly indicates a much larger magnetic mass. The model that is calculated on the right is a monopole with a strength of -193 Am and a depth of 1.9 m. The magnetic map does not allow an interpretation of the diameter of the feature or its maximum depth. An archaeological excavation at this location uncovered a well that was dug into the soil; it was dense with iron debris, and it was neither necessary nor practical to find its bottom.

80 The magnetic model of a well or vertical pipe The top of the well is 1.8 m underground The sensor height is 0.8 m

Magnetic anomaly, nT

60

40

20

0 0

2

4 Depth extent of well, m

6

8

Figure 29: The effect of the vertical extent of a magnetic object. Starting with a very thin object, the magnetic anomaly increases rapidly with increasing length (or depth). When the depth reaches about 4 m, the anomaly no longer increases as fast, and then it becomes rather constant for depths greater than about 8 m. This calculation applies a pair of monopoles in order to model the upper and lower ends of a vertical magnetic object. The upper monopole has a strength of -3.92 Am and is at a depth of 1.8 m and the lower monopole has a strength of +3.92 Am. The elevation of the calculation is 0.8 m. Note that this result is different from that in Figure 11, which shows a lengthening horizontal object. In Figure 11, the anomaly reaches a peak value and then the anomaly decreases as it gets longer.

Magnetic model of nearby buildings, calculated contour lines at 25 nT interval 1000

950

North distance, ft

900

850

800

750

800

850

900

950

1000

1050

1100

East distance, ft

Figure 30: A magnetic model of two standing buildings. They cause magnetic lows that become stronger closer to the buildings; the anomalies can be several hundred nanotesla. While this is a total-field magnetic map, the anomalies were still very strong in a map of the vertical magnetic gradient. The anomalies are caused by iron inside a house (near the middle of the map) and also iron in a garage and other machinery (on the left). The five red stars in the map locate magnetic dipoles whose field approximates that of the interfering iron. This magnetic map can then be subtracted from the measurements, and this allows fainter anomalies near the buildings to be identified. However, the improvement in the magnetic map is far from perfection.