Geophys. J. Int.
(2007)
169,
421–434 doi: 10.1111/j.1365246X.2007.03305.x
G J I G e o m a g n e t i s m , r o c k m a g n e t i s m a n d p a l a e o m a g n e t i s m
A study of spectral methods of estimating the depth to the bottom of magnetic sources from nearsurface magnetic anomaly data
D. Ravat,
1
A. Pignatelli,
2
I. Nicolosi
2
and M. Chiappini
2
1
Department of Geology, Southern Illinois University Carbondale, Carbondale,
IL 629014324,
USA. Email: ravat@geo.siu.edu
2
Istituto Nazionale di Geoﬁsica e Vulcanologia, Via di Vigna Murata
605, 00143
Roma, Italy
Accepted 2006 November 16. Received 2006 November 1; in srcinal form 2006 June 15
SUMMARY
Based on a critical evaluation of several different spectral magnetic depth determination techniques on areally large synthetic layered and random magnetization models, we recommend the following considerations in the usage of the methods as necessary prerequisites to successful bottom depth determinations: (1) using windows with sufﬁcient width to ascertain that theresponse of the deepest magnetic layer is captured and by verifying the spectra and computingthedepthestimateswiththelargestpossiblewindows(
>
300–500km);(2)avoidingﬁlteringtoremovearbitraryregionalﬁelds,accomplishedbycompilingmagneticanomaliesderivedfrommodern spherical harmonic degree 13 Earth’s main ﬁeld models [e.g. recent International Geomagnetic Reference Field models (IGRF) or Comprehensive models (CM)]; (3) ascertainingthe nearcircularity of the autocorrelation function to avoid analysing biased spectra containing strong anomaly trends; and (4) avoid determining the slopes from the exponential, lowwavenumber part of the spectra in the cases of layered magnetization. We also describe thedetails of the new spectral peak forward modelling method and discuss the conditions under which the method can lead to useful results. We found that, despite all these precautions, insomecases,theresultscanstillbeerroneousand,therefore,werecommendacriticalevaluationoftheresultsbymodellingheatﬂowandtakingintoaccountseismicinformationonthecrustaland lithospheric thicknesses and seismic velocities wherever possible. In the southcentral US,east of the Rockies, where the surface heat ﬂow ranges between 40 and 65 mW m
−
2
, we obtained the magnetic bottom depth of 40
±
10 km using the approach of the forward modellingof the spectral peak. This range is similar to the seismically derived crustal thickness of 45– 50km,suggesting,therefore,thattheentirecrustmaybemagneticinthisregion.Becauseoftheuncertaintiesinthevariousheatﬂowcontributingparameters,suchasthevariationsinthermalconductivity, radiogenic heat and hydraulic regime, we could not constrain the lithosphericthickness beyond an estimate ranging approximately from 100 to 200 km.
Keywords:
crust,geothermalevaluation,lithosphere,magneticanomalies,SouthcentralUS,spectral analysis.
INTRODUCTION
Deriving the depths to the bottom of magnetic sources in the lithosphere (in many cases identiﬁed with the Curie point of magneticminerals)isimportantforconstrainingtemperaturesinthecrustand thustherheologicalnatureoftheEarth’slithosphere.Inthelastfour decades, several methods and their variations have been proposed for estimating this depth from azimuthally averaged Fourier spectraof magnetic anomalies (e.g. Spector & Grant 1970; Bhattacharyya& Leu 1975, 1977; Shuey
et al.
1977; Connard
et al.
1983; Okubo
et al.
1985; Blakely 1988; Pilkington & Todoeschuck 1993; Maus& Dimri 1995; Tanaka
et al.
1999; Chiozzi
et al.
2005). In thisstudy, we evaluate the performance of several different methods using model studies on areally large, centrally upwarped, layered and random magnetic models. We also describe the details of the spectral peak of forward modelling method proposed simultaneously byRavat (2004), Finn & Ravat (2004) and Ross
et al.
(2004) and discuss conditions under which the method can lead to useful results.Furthermore,weinvestigatethefeasibilityofforwardmodellingthe powerlaw expressions (e.g. Maus & Dimri 1995) for the depths of deepest layers from azimuthally averaged magnetic spectra. Eventhoughsomeoftheroutinelyusedmethodsappearedtoworkwithina tolerable error (a few km), no single technique worked on our model studies with complete reliability and at times produced com pletely erroneous depths to top and bottom.More exhaustive model studies showing additional and morecomplex examples, especially using fractal source models, are possible, but they would not change the conclusion that all depth
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D. Ravat
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Table1.
Thefactorsforconvertingthespectralslopestodepthsusingthe relationship, slope
=
factor
∗
Z
. Observe the two different deﬁnitions of wavenumber. The usage of the term ‘wavenumber’ withoutspecifying the units has led to confusion in the literature in termsof determining the depths. In this paper, we use the terms frequency(1 km
−
1
) and wavenumber (2
π
km
−
1
).Ordinate of the spectral plot Factor for amplitude Factor for power spectrum spectrumFrequency (
f
) (1 km
−
1
) 2
π
4
π
Wavenumber (1 km
−
1
)Wavenumber (
k
) (2
π
km
−
1
) 1 2
estimates must be carefully performed and assessed using tectonicframework, geological and shallow as well as deep geophysical evidence, and heat ﬂow modelling, especially in the cases where highvalues of heat ﬂow are inferred based on magnetic bottoms lyingwithin the upper and middle crust. In addition, we found that thedepth determination methods have been incorrectly used in several published studies and labelled in a way that has caused confusionregarding the usage of the method. Therefore, we also summarizethe caveats and usage in this paper.
THE BACKGROUND OF THEMETHODS, THE RATIONALE, ANDTHE KEY OBSERVATIONS
Two types of methods have been commonly used in the spectral estimation of the depth to the bottom of the magnetic layer:
00.10.20.30.40.50.60.70.80.9642024681012
Fourier spectrumFrequencyscaledFourier spectrumk
3
corrected spectrumWavenumber (2
π
/ km)
l n o f P o w e r
Figure 1.
An example of forward modelling of the spectral peak showing the Fourier spectrum (continuous line) and the modelled spectrum (dashed line), and also the frequencyscaled spectrum (heavy dash and dot line) and
k
3
corrected spectrum (dotted line) discussed later in the paper. The straight line segments inthe
k
0.1–0.2 range are linear slope segments for computing depths based on Spector & Grant (1970) and Bhattacharyya & Leu (1977) and Okubo
et al.
(1985)methods. See the text for details. This example is based on the results from ‘thin’ random source model used later in deriving the depths to deepest layer and will be brought up again in that context. Actual depth to the top is 7.29
±
1.81 km and bottom is 8.30
±
2.00 km and depth from Spector and Grant slope is4.24 km and frequencyscaled spectrum is 6.88 km.
k
3
correction overcorrects the spectrum and thus is inappropriate. The continuous dashed line is a resultfrom forward modelling; its depth to the top is
∼
6 km and bottom is
∼
9 km.
the spectral peak method srcinally given in a landmark paper bySpector & Grant (1970) and used by Shuey
et al.
(1977), Connard
et al.
(1983) and Blakely (1988) among others, and the centroid method srcinally presented by Bhattacharyya & Leu (1977) and used with certain caveats and variations by Okubo
et al.
(1985) and Tanaka
et al.
(1999). Both methods need
a priori
estimation of thedepth to the top of the same layer. During this study, we found thatthe spectral peaks in the azimuthally averaged spectra are observed only when sources are randomly magnetized as prescribed by Spector & Grant (1970); with uniform magnetization layers, the spectrahave powerlaw form and no spectral peaks are observed.Spector & Grant (1970) showed that the slopes of logarithms of azimuthally averaged Fourier spectra of magnetic anomalies fromensemble of simple sources are related to the depth to the top of theensemble and also the spectra have peak positions on the frequencyor wavenumber axis that are related to the thickness of the magnetic source layers (see also Table 1). The Spector & Grant (1970)equation, in the notation after Blakely (1995), is

F
(
k
)

2
=
4
π
2
C
2
m

θ
m

2

θ
f

2
M
2
o
e
−
2

k

z
t
1
−
e
−
k

(
Z
b
−
Z
t
)
2
S
2
(
a
,
b
)
,
(1)where
F
is the Fourier power spectrum,
k
is wavenumber in cycles km
−
1
or 2
π
km
−
1
,
C
m
is a constant related to units,
θ
m
isa factor related to magnetization direction,
θ
f
is a factor related to magnetic ﬁeld direction,
M
o
is magnetization,
Z
t
and
Z
b
arethe depths to the top and the bottom of the ensemble of magneticsources,and
S
2
(
a
,
b
)isthefactorrelatedtohorizontaldimensionsof sources.
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The spectral peak method
The observed spectral peak position (
k
peak
) is a function of
Z
t
and
Z
b
and is given by the following transcendental equation (Connard
et al.
1983; Blakely 1995) from which
Z
b
can be obtained by trialand error:
k
peak
=
ln
z
b
−
ln
z
t
z
b
−
z
t
.
(2)Thelimitationofthemethodisthatthespectralpeakisnotalwaysobserved (e.g. spectra of uniformly magnetized layer keeps risingat low wavenumbers with a powerlaw form (see Blakely 1995), or,trivially,forinsufﬁcientlylargedatawindows).Also,manytimesthe peak is represented by a single point and may be uncertain in termsof its position on the abscissa because of either small size windowsor the difﬁculty of deriving precise estimates of power spectrumat the low wavenumbers. To avoid the latter difﬁculty, careful windowing or multitapering must be performed. Unfortunately, manyof the published studies have not been sufﬁciently careful in takingthis precaution and, therefore, the low wavenumber portions of thespectra can have either false peaks or incorrect spectral estimateswhich could lead to inaccurate depths.
Forward modelling of the spectral peak
Recently, Ravat (2004), Finn & Ravat (2004) and Ross
et al.
(2004)simultaneously proposed forward modelling (iterative matching)
Figure 2.
(a) The top of the ‘thin’ layered source model used in the study and (b) the bottom of the ‘thin’ layered source model used in the study.
of the spectral peak to better estimate the bottom depth by using the part of the eq. (1) that depends on the top and the bottomdepths,

F
(
k
)

2
=
C
(
e
−
k

Z
t
−
e
−
k

Z
b
)
2
.
(3)Here, the constant
C
, which consists of nondepthdependent termsin eq. (1), can be adjusted to move the modelled curve up or downto ﬁt the observed peak. The depth to the top of the layer is adjusted also to match the slope adjacent to the spectral peak. Fig. 1 showsan example of the forward modelling procedure that is based on oneof the model study results we use later. At this juncture, we are onlyconcerned with the continuous solid line (labelled as the Fourier spectrum) and the dashed line (which is the modelled spectrumgenerated using an assumed
Z
t
and
Z
b
in eq. 3). The assumed
Z
b
controls the location of the spectral peak (near
k
∼
0.1 in Fig. 1)and
Z
t
controls the slope in the high wavenumber range (
k
>
0.5).The slope immediately adjacent to the peak is controlled by thecombination of both
Z
t
and
Z
b
. For example, deepening the
Z
b
leadstoraisingthepeakrelativetotherestofthemodelledcurve,and vice versa, and consequently affects the low wavenumber slope ontherightsideofthepeak.Deepeningthe
Z
b
alsoshiftsthemodelled peak to the left relative to the rest of the curve and also broadensthe peak, whereas shallowing the depth to bottom moves the peak to the right and makes them sharper.The advantage of forward modelling is that it allows one to ﬁtiteratively the position and the width of the peak and match theadjacentpartoftheslopemorepreciselyandexplorethemodelspace
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D. Ravat
et al.
(ratherthanrelyingonasingleestimateoftheabscissapositionofthe peakwhichcanbeunreliableduetohavingtoofewspectralsamplescovering the peak from smallerthan adequate sizewindows). Based on the ﬁt of modelled spectra with the observed, one can accept or reject the results more conﬁdently in this overall subjective processof ﬁtting speciﬁc parts of the spectra.
The centroid method
In 1977, Bhattacharyya and Leu published a method for determinationofthecentroidofrectangularparallelepipedsources,whichtheyhad used earlier (Bhattacharyya & Leu 1975) in their study of Curiedepths of Yellowstone caldera. In this method, used also by Okubo
et al.
(1985), the estimate of the depth to the centroid (
Z
o
) is obtained from the slope of an azimuthally averaged frequencyscaled Fourier spectra in the low wavenumber region [
G
(
k
)
=
1
/
fF
(
k
),where
f
is frequency (in 1 km
−
1
)] and the estimate of the depthto the top of the source is obtained from the slope of azimuthallyaveraged Fourier spectrum. Fig. 1 shows the examples of selectingthe slopes from the low wavenumber range parts of the Fourier and the frequencyscaled spectra. The depth to the magnetic bottom isthen obtained from
Z
b
=
2
Z
o
−
Z
t
.Okubo
et al.
(1985), elaborating on the application of theBhattacharyya & Leu (1977) method, suggested that one could obtain centroid estimates from the highpass ﬁltered (as low as
Figure 2.
(
Continued.
)
40–50 km highpass) magnetic data; however, this leads to elimination of meaningful part of the spectra related to the depths tothe deepest layers of interest. There appears to be a recent consensus among the researchers in the ﬁeld (Hansen, private communication, 2004; Chiozzi
et al.
2005; and the results of thisstudy) that the dimension of the window analysed may need to be, in some cases, up to 10 times the depth to the bottom. Manytimes, it is not practical to analyse windows greater than about200–300 km because disparate tectonic regimes can be ‘averaged’ within a single window, diluting the meaningfulness of theresults.In obtaining the depth to the top of the source using this method,Tanaka
etal.
(1999)advocatedﬁttingtheslopetoahigherwavenum ber part of the spectrum than the lowest wavenumber straight slopesegment(asshowninFig.1),arguingthatthelinearizedequationfor the depth to the top is valid for wavelengths greater than the thickness of the layer. This leads to deeper magnetic bottom estimatesthat, at times, appear to be desirable. However, in applications withreal data, different slopes of the Fourier spectra imply existence of multiplylayered magneticstructures.Therefore, themajorpracticallimitation in this usage is that the slope from the highwavenumber part of the spectrum can give the depth to the top of a shallower layer—not the same layer as in the determination of the centroid— and, consequently, give incorrectly a deeper estimate of the bottomof the deeper layer.
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Powerlaw corrections
Recently,importantcaveatsregardingthedepthdeterminationmethods have been recognized but have not been considered by manyrecent users interested in the problem of determination of the magnetic bottom depth, perhaps because not enough time has passed to digest the results. Fedi
et al.
(1997) noted that when the randomsource variation is large, the Spector & Grant (1970) equation hasan inherent powerlaw form,

F
(
k
)

2
∝
k
−
2
.
9
.
(4)A similar
k
−
β
dependence has also been observed for fractalsource distributions (e.g. Pilkington & Todoeschuck 1993; Maus &Dimri 1995). For such cases, the spectra can be premultiplied withthe factor
k
β
prior to computation of depths (Pilkington
et al.
1994;Fedi
etal.
1997).Weexaminetheeffectofthepowerlawcorrectionfor the layered and uniform random sources, but not fractal sourcedistributions. The power law correction straightens and ﬂattens ﬁctitiousslopeschangesintheusualFourierpowerspectraofmagneticanomalies and leads to shallower depth estimates (see examples inthe above studies). Because both the depth to the bottom estimatorsmentioned earlier rely on the
a priori
estimates of the depth to thetop, the powerlaw correction is important for the determination of the depth to the bottom.In this study, we found that when the correction is grossly inappropriate, it overcorrects the spectra such that low wavenumber
Figure 3.
(a) The magnetic ﬁeld from the ‘thin’ layered model shown in Fig. 2 and (b) the magnetic ﬁeld from the ‘thin’ random source model contained withinthe source layer shown in Fig. 2. See text for details.
portion of the spectrum turns over (or dips down toward the lowwavenumber end of the axis) (compare the
k
3
corrected and Fourier spectra in Fig. 1). An example of appropriate amount of correctionis given later. In employing the correction factor, it is possible toexplore a strategy of reducing the
β
factor to yield an appropriatecorrection for the given situation, but this is cumbersome and wehave not attempted it.It is interesting to note that the approach of Tanaka
et al.
(1999),in which the depth to top is picked from the slope of the adjacentless steep portion of the spectrum compared to the centroid depth, partly achieves the effect of the above powerlaw corrected ﬂattening. Thus, for single layered model studies, the Tanaka
et al.
approach might be perfectly ﬁne, but as discussed earlier, with their approach in multiplelayer cases, one would most certainly end up pickingthedepthtothetopoftheshallowerlayerwhichwouldresultin a deeper bottom estimate based on
Z
b
=
2
Z
o
−
Z
t
.Sincethespectraoflayeredsourcesdonotproducespectralpeaks but have a power law increase in the low wavenumber region, itshould be possible to implement forward modelling to estimate the parametersofthedeepestlayerinthiscase;aninversionthroughlinear programming for the powerlaw situation for the top of the magnetic basement has been accomplished by Maus & Dimri (1995).However, the inversion requires that the range of wavenumbersover which the spectra are modelled be preselected and be correct;this range, for the deepest layer, is different for different windowsand, therefore, difﬁcult to employ in a moving window fashion.
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