A New Approach to Archaeological Dating Using Geomagnetic Field Modelling. 1 1 1 Alexandra Lodge , Richard Holme , Neil Suttie 2 1 1 , John Shaw , Mimi Hill , Paul Linford 1 2 University of Liverpool, Liverpool, U.K. English Heritage, Portsmouth, U.K. email: [email protected] CALS7K model (Korte & Constable, 2005). -A global geomagnetic field model from 7000 years of archaeomagnetic data. -Not designed or appropriate for archaeomagnetic dating. -Designed to be smooth in space and time at the core-mantle boundary (CMB). -Very useful to constrain magnetic field behaviour at the CMB. -Less useful for producing detailed models of the surface magnetic field. -Tends to under fit maxima in both space and time. -Such maxima are particularly useful for dating purposes. Summary -For a small number of single epoch data, relaxing the smoothness constraints, gives a model that fits the data better than CALS7K. -Build upon this successful result: (1) time dependent modelling, using the secular variation curves as data (2) using all the data available to produce a series of secular variation curves for any location in Western Europe. Results The results of decreasing the smoothness of the model to fit the data points are clearly shown in table 2. For each of the different observations, declination, inclination and intensity, the weighted residuals are lower for the new model than for CALS7K. Preliminary Method Development Using Secular Variation Curves as Data. Country/ region Relocation point Lat. (º) Long. (º) Bulgaria Sofia 42.7 23.3 Caucasus Gori 42.0 44.1 France Paris 48.9 2.3 Ukraine Kiev 50.4 30.5 UK Meriden 52.4 1.6 Table 1. Secular variation curve relocation (central) points (Daly & Le Goff, 1996). To demonstrate the viability of this modelling strategy: use secular variation curves as data (Daly & Le Goff, 1996). Considering single epoch: 1750 AD. See Table 1. Errors are:- D (2.5 (2.5 º º ), I ), F (1500 nT). (D and I from Korte & Constable, 2003, F from Korte & Constable, 2005). -Minimise data misfit of departure from CALS7K model. -Use CALS7K as an apriori model. -Vary damping parameter to produce trade-off curve (figure 3), which shows a "knee" in the curve at approx. an RMS misfit of 1s . CALS7K has 3s . -Figure 4 shows the Lowes power spectrum for CALS7K, our model and the difference between the two. -Increased power in the higher degree spherical harmonics, i.e. the non-dipole components. -increasing roughness in the model Û reduced misfit. References Bloxham, J., Jackson, A., 1992. Time-dependent mapping of the magnetic field at the core-mantle boundary. J.G.R. 97, 19537-19563. Clark, A. J., Tarling, D. H., Noël, M., 1988. Developments in archaeomagnetic dating in Britain. J. Arch. Sci. 15, 645-667. Daly, L., Le Goff, M., 1996. An updated and homogeneous world secular variation data base. 1. smoothing of the archaeomagnetic results. PEPI 93, 159-190. Korte, M., Constable, C., 2003. Continuous global geomagnetic field models for the past 3000 years. PEPI 140, 73-89 doi:10.1016/j.pepi.2003.07.013. Korte, M., Constable, C. G., 2005. Continuous geomagnetic field models for the past 7 millennia: 2. CALS7K. G3 6 (1), doi:10.1029/2004GC000801. Korte, M., Genevey, A., Constable, C. G., Frank, U., Schnepp, E., 2005. Continuous geomagnetic field models for the past 7 millennia: 1. A new global data compilation. G3 6 (2), doi:10.1029/2004GC000800. Noël, M., Batt, C. M., 1990. A method for correcting geographically separated remanence directions for the purpose of archaeomagnetic dating. GJI 102, 753-756. Zananiri, I., Batt, C. M., Lanos, P., Tarling, D. H., Linford, P., 2006. Archaeo- magnetic secular variation in the UK during the past 4000 years and its application to archaeomagnetic dating. PEPI, doi:10.1016/j/pepo.2006.08.006. Country/ D weighted residual I weighted residual F weighted residual region (CALS7K) (CALS7K) (CALS7K) Bulgaria -0.29 (-1.01) -0.73 (-0.90) 0.77 (3.80) Caucasus 1.47 (8.10) -0.45 (1.21) -0.47 (2.43) France -0.26 (-1.19) -0.26 (1.16) - Ukraine -0.32 (-0.75) 2.50 (3.83) 0.39 (1.33) UK -0.12 (-1.55) -0.64 (1.16) - -Cubic B-splines as temporal basis: -Regularisation, i.e. smoothing by constraining a property of the field, e.g. 2 minimising <B > at CMB, to find the simplest model explaining the data within the desired accuracy. Table 2. Weighted residuals between the data and our model (CALS7K in brackets). "Knee” 0 1 2 3 RMS Misfit 0 10000 20000 30000 40000 50000 RMS surface differences (nT) Damping parameter 0 2 4 6 8 10 Spherical harmonic degree 10 0 10 2 10 4 10 6 10 8 10 10 2 Power at surface (nT) CALS7K. new model difference Figure 3. Tradeoff curve Figure 4. Power Spectrum Global geomagnetic models -Linearised inverse method (e.g. Bloxham & Jackson, 1992). -Write magnetic field B as the gradient of a scalar potential, V: -Spherical harmonic basis functions, i.e. development into series of dipole, quadrupole, octupole, etc. -At any time t, the potential can be written as: where r is radius, a is the Earth's radius, { ; } are Gauss coefficients, and are Schmidt quasi-normalised associated Legendre functions. P m l (cos è ) g m l (t) h m l (t) Introduction Absolute dating is fundamental to the understanding and interpretation of archaeological material. Burnt material such as hearths, pottery, bricks and tiles contains a record of the Earth's magnetic field at the time which the material was last heated, and with a calibration curve the date of this heating can be obtained. The aims of this project are: - to produce a geomagnetic field model for the last 2000+ years specific to the UK, taking into account errors in field measurement and dating. - to use this model to produce high resolution archaeomagnetic calibration curves of declination, D, inclination, I and intensity, F for any location in Western Europe for the last 2000+ years. - to obtain high-quality, accurately dated archaeomagnetic intensities from well-dated British archaeological samples. -Publically available database of Korte et al. (2005) (see figure 1). -European data collated under the coordination of Cathy Batt (Bradford University) as part of the EU research training network AARCH (Archaeomagnetic Advances for the Rescue of Cultural Heritage). -Our own intensity data for the UK. Archaeomagnetic Data and Uncertainties Figure 1. Locations of the global compilation of archaeomagnetic and palaeomagnetic data. (a) lakes, (b) archaeomagnetic directional data, and (c) archaeomagnetic intensity data. Left: locations of lakes and archaeomagnetic regions. Right: concentration of data (from Korte et al. (2005)). Measurement uncertainties: -Exact orientation (directional data) -Alterations of initial magnetisation (thermal overprinting, etc.) Dating uncertainties: -Archaeological: -(i) varve counting (1 to a few yrs) -(ii) radiocarbon dating (decades to 1-2 centuries) -Sediments: -constant sedimentation rate assumed -means magnetisation might be younger than the sediment age (lock-in depth). Figure 2. UK secular variation curve reduced to Meriden, from Bayesian statistical modelling. Each black and grey segment of the curve represents a 100 years interval (from Zananiri et al. (2006)). Secular variation Curves -Standard dating procedure: compare archaeomagnetic observations (D, I & F) from samples of unknown age to well- dated reference curves for a particular geographic region (e.g. see figure 2). -Data used to construct reference curves, and data from samples to be dated, are reduced to a central location. -For D & I data by assuming an inclined axial dipole configuration of the magnetic field (reduction via a virtual geomagnetic pole (Noël & Batt, 1990)). -Same for F, unless no associated directional data exist, in which case a geocentric axial dipole is assumed. -But field is not simply dipolar: reduction introduces errors into both the reference curve and the particular date determined. -Furthermore, the construction of separate curves for each field element ignores an important constraint: the field elements (D, I & F) are not independent.
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A New Approach to Archaeological Dating Using Geomagnetic Field Modelling. 1 1 1
Alexandra Lodge , Richard Holme , Neil Suttie21 1
, John Shaw , Mimi Hill , Paul Linford1 2University of Liverpool, Liverpool, U.K. English Heritage, Portsmouth, U.K. email: [email protected]
CALS7K model (Korte & Constable, 2005).
-A global geomagnetic field model from 7000 years of archaeomagnetic data.
-Not designed or appropriate for archaeomagnetic dating.
-Designed to be smooth in space and time at the core-mantle boundary (CMB).
-Very useful to constrain magnetic field behaviour at the CMB.
-Less useful for producing detailed models of the surface magnetic field.
-Tends to under fit maxima in both space and time.
-Such maxima are particularly useful for dating purposes.
Summary
-For a small number of single epoch data, relaxing the smoothness constraints, gives a model that fits the data better than CALS7K.
-Build upon this successful result:(1) time dependent modelling, using the secular variation curves as data(2) using all the data available to produce a series of secular variation curves for any location in Western Europe.
Results
The results of decreasing the smoothness of the model to fit the data points are clearly shown in table 2.
For each of the different observations, declination, inclination and intensity, the weighted residuals are lower for the new model than for CALS7K.
Preliminary Method Development Using Secular Variation Curves as Data.
Country/ region Relocation point Lat. (º)
Long. (º)
Bulgaria Sofia 42.7 23.3
Caucasus Gori 42.0 44.1
France Paris 48.9 2.3
Ukraine Kiev 50.4 30.5
UK Meriden 52.4 1.6
Table 1. Secular variation curve relocation (central) points (Daly & Le Goff, 1996).
To demonstrate the viability of this modelling strategy: use secular variation curves as data (Daly & Le Goff, 1996). Considering single epoch: 1750 AD. See Table 1.
Errors are:- D (2.5 (2.5º º), I ), F (1500 nT). (D and I from Korte & Constable, 2003, F from Korte & Constable, 2005).
-Minimise data misfit of departure from CALS7K model.
-Use CALS7K as an apriori model.
-Vary damping parameter to produce trade-off curve (figure 3), which shows a "knee" in the curve at approx. an RMS misfit of 1s. CALS7K has 3s.
-Figure 4 shows the Lowes power spectrum for CALS7K, our model and the difference between the two.
-Increased power in the higher degree spherical harmonics, i.e. the non-dipole components.
-increasing roughness in the model Û reduced misfit.
ReferencesBloxham, J., Jackson, A., 1992. Time-dependent mapping of the magnetic field at the core-mantle boundary. J.G.R. 97, 19537-19563.Clark, A. J., Tarling, D. H., Noël, M., 1988. Developments in archaeomagnetic dating in Britain. J. Arch. Sci. 15, 645-667.Daly, L., Le Goff, M., 1996. An updated and homogeneous world secular variation data base. 1. smoothing of the archaeomagnetic results. PEPI 93, 159-190.Korte, M., Constable, C., 2003. Continuous global geomagnetic field models for the past 3000 years. PEPI 140, 73-89 doi:10.1016/j.pepi.2003.07.013.Korte, M., Constable, C. G., 2005. Continuous geomagnetic field models for the past 7 millennia: 2. CALS7K. G3 6 (1), doi:10.1029/2004GC000801.Korte, M., Genevey, A., Constable, C. G., Frank, U., Schnepp, E., 2005. Continuous geomagnetic field models for the past 7 millennia: 1. A new global data compilation. G3 6 (2), doi:10.1029/2004GC000800. Noël, M., Batt, C. M., 1990. A method for correcting geographically separated remanence directions for the purpose of archaeomagnetic dating. GJI 102, 753-756.Zananiri, I., Batt, C. M., Lanos, P., Tarling, D. H., Linford, P., 2006. Archaeo- magnetic secular variation in the UK during the past 4000 years and its application to
archaeomagnetic dating. PEPI, doi:10.1016/j/pepo.2006.08.006.
Country/ D weighted residual I weighted residual F weighted residual
region (CALS7K) (CALS7K) (CALS7K)
Bulgaria -0.29 (-1.01) -0.73 (-0.90) 0.77 (3.80)
Caucasus 1.47 (8.10) -0.45 (1.21) -0.47 (2.43)
France -0.26 (-1.19) -0.26 (1.16) -
Ukraine -0.32 (-0.75) 2.50 (3.83) 0.39 (1.33)
UK -0.12 (-1.55) -0.64 (1.16) -
-Cubic B-splines as temporal basis:
-Regularisation, i.e. smoothing by constraining a property of the field, e.g. 2
minimising <B > at CMB, to find the simplest model explaining the data within the desired accuracy.
Table 2. Weighted residuals between the data and our model (CALS7K in brackets).
"Knee”
0 1 2 3
RMS Misfit
0
10000
20000
30000
40000
50000
RM
S su
rface
dif
fere
nce
s (n
T)
Damping parameter
0 2 4 6 8 10
Spherical harmonic degree
100
102
104
106
108
1010
2Pow
er
at
surf
ace
(nT)
CALS7K.new modeldifference
Figure 3. Tradeoff curve Figure 4. Power Spectrum
Global geomagnetic models
-Linearised inverse method (e.g. Bloxham & Jackson, 1992).-Write magnetic field B as the gradient of a scalar potential, V:
-Spherical harmonic basis functions, i.e. development into series of dipole, quadrupole, octupole, etc. -At any time t, the potential can be written as:
where r is radius, a is the Earth's radius, { ; } are Gauss coefficients, and are Schmidt quasi-normalised associated Legendre functions. Pm
l (cos è ) gm
l (t) hml (t)
Introduction
Absolute dating is fundamental to the understanding and interpretation of archaeological material. Burnt material such as hearths, pottery, bricks and tiles contains a record of the Earth's magnetic field at the time which the material was last heated, and with a calibration curve the date of this heating can be obtained.
The aims of this project are:
- to produce a geomagnetic field model for the last 2000+ years specific to the UK, taking into account errors in field measurement and dating.
- to use this model to produce high resolution archaeomagnetic calibration curves of declination, D, inclination, I and intensity, F for any location in Western Europe for the last 2000+ years.
- to obtain high-quality, accurately dated archaeomagnetic intensities from well-dated British archaeological samples.
-Publically available database of Korte et al. (2005) (see figure 1).
-European data collated under the coordination of Cathy Batt (Bradford University) as part of the EU research training network AARCH (Archaeomagnetic Advances for the Rescue of Cultural Heritage).
-Our own intensity data for the UK.
Archaeomagnetic Data and Uncertainties
Figure 1. Locations of the global compilation of archaeomagnetic and palaeomagnetic data. (a) lakes, (b) archaeomagnetic directional data, and (c) archaeomagnetic intensity data. Left: locations of lakes and archaeomagnetic regions. Right: concentration of data (from Korte et al. (2005)).
Measurement uncertainties: -Exact orientation (directional data) -Alterations of initial magnetisation (thermal overprinting, etc.)
Dating uncertainties: -Archaeological: -(i) varve counting (1 to a few yrs) -(ii) radiocarbon dating (decades to 1-2 centuries) -Sediments: -constant sedimentation rate assumed -means magnetisation might be younger than the sediment age (lock-in depth).
Figure 2. UK secular variation curve reduced to Meriden, from Bayesian statistical modelling. Each black and grey segment of the curve represents a 100 years interval (from Zananiri et al. (2006)).
Secular variation Curves
-Standard dating procedure: compare archaeomagnetic observations (D, I & F) from samples of unknown age to well-dated reference curves for a particular geographic region (e.g. see figure 2).
-Data used to construct reference curves, and data from samples to be dated, are reduced to a central location.
-For D & I data by assuming an inclined axial dipole configuration of the magnetic field (reduction via a virtual geomagnetic pole (Noël & Batt, 1990)).
-Same for F, unless no associated directional data exist, in which case a geocentric axial dipole is assumed.
-But field is not simply dipolar: reduction introduces errors into both the reference curve and the particular date determined.
-Furthermore, the construction of separate curves for each field element ignores an important constraint: the field elements (D, I & F) are not independent.
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