Download DETERMINATION OF THE EFFECTIVE ELASTIC THICKNESS OF

DETERMINATION OF THE
EFFECTIVE ELASTIC THICKNESS
OF THE CRUST BY SPECTRAL ANALYSIS
by
John Moores
[990161846]
Submitted:
April 24, 2003
Toronto Ontario
Instructor:
R. Bailey
PHY 495S
JOHN MOORES
DETERMINATION OF THE ELASTIC THICKNESS OF
THE CRUST USING SPECTRAL ANALYSIS ( 2003 )
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TABLE OF CONTENTS
1.
2.
3.
4.
5.
6.
7.
8.
Introduction and Motivation……………………………………….
Notation………………………………………………………………………
Theory and Approach………………………………………………….
Implementation…………………………………………………………..
Error Considerations…………………………………………………..
Results and Discussion……………………………………………….
Conclusions…………………………………………………………………
References and Bibliography……………………………………..
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19
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Appendix A – Graphs and Figures
Appendix B – MATLAB Codes
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INTRODUCTION AND MOTIVATION
While it is important to understand the stratified division of the Earth into inner and outer
core, mantle and lithosphere, one must keep in mind that none of these layers is
homogenous. For the surface of the lithosphere this inhomogeneity is as obvious as the
differences between plains, mountains and oceans. However, these features are not
exclusively the result of internal processes but also interact with the layers below in a
feedback loop of sorts. For instance, while tectonic processes deep in the mantle drive the
formation of high mountain chains, given enough time the additional bulk will deflect the
lithosphere and the entire mountain chain will sink until an hydrostatic equilibrium is reached.
This is referred to as isostatic compensation.
As such, we are interested not only in the composition and distribution of these layers, but
also how they respond to loading. In particular, this paper is concerned with the elastic
deformation of the lithosphere in response to the periodic loading of the terrain. Since the
layers of the earth near the surface may be approximated as plates, this suggests the
definition of an 'effective elastic thickness' as the depth of lithosphere over which the
response can be likened to the deformation of a plate under load.
However, some caution is warranted. By no means should this be interpreted to be the actual
thickness of the lithosphere itself or that there is a discontinuity demarcating a boundary
between two layers at this depth (even though we will model the lithosphere in this way). In
fact, it is well known from seismic and thermodynamic arguments that the boundary between
lithosphere and mantle is somewhat 'blurred' and occurs over several kilometers1.
Instead, it is best to conceptualize this effective elastic thickness as the thickness of a
perfectly elastic infinite plate composed of lithospheric material located on top of an
essentially fluid (at least on geological time scales) mantle. Thus, using the principles of
structural/continuum mechanics we can determine this thickness by observing the
deformation in the plate due to a known loading pattern. While this may not give us the
lithospheric thickness directly, it does allow for the comparison of different regions since it is
logical to suppose that if a particular region has a higher effective elastic thickness, it also has
a thicker lithosphere.
We can learn a great deal about the lithosphere from this thickness variation. For instance,
we know from thermodynamics that as the temperature of rock increases its elasticity
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decreases thus we can argue that areas with greater elastic thicknesses are also colder and
thus presumably older. It is also reasonable to postulate that mountain building would be
easier in a region where the lithosphere is thin and thus more prone to buckling from tectonic
stresses. The aligned rows of alternating mountains and valleys of the basin and range region
of the Rocky Mountains, is highly suggestive of such a buckled plate – a thin elastic thickness
here would tend to reinforce this supposition.
Thus, we can gain a number of insights into the nature of the upper mantle and lithosphere
by determining this elastic thickness for different regions. In particular, we shall compare
several regions in North America including the basin and range region of the Rocky Mountains
- thought to be a relatively young region - in addition to the Appalachian Mountains and
Canadian Shield - both thought to be very old surface features.
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NOTATION
φ
North latitude of a particular site from the Equator (Degrees)
ψ
ωx ,ωy
ωn
ρm , ρc
E
g
λ
λchar
ωchar
t
h
p
q
Q
East Longitude of a particular site (Degrees)
Frequency of oscillation in eastward and northward directions (Rads/km)
Nyquist Frequency (Rads/km)
Density of the mantle and crust respectively (kg/m3)
Elastic modulus of the lithosphere (N/m2)
Gravitational Acceleration (9.81 N/kg)
Wavelength (m)
Characteristic Wavelength of the lithosphere (m)
Characteristic Angular Frequency of the lithosphere (Rads/km)
Effective Elastic Thickness of the lithosphere (m)
Loading Amplitude of the Topography (m)
Horizontal Loading of the lithosphere (N/m2)
Vertical Loading of the lithosphere at the surface - i.e. the datum (N/m2)
Hydrostatic restoring pressure exerted by the mantle on deflected
lithosphere (N/m2)
Response Amplitude of the Lithosphere (m)
Coherence (dimensionless)
Flexural Rigidity of the lithosphere (N·m)
Degree of Compensation (dimensionless)
Poisson’s Ratio (dimensionless)
Cartesian coordinates in spatial domain (km)
The number of samples in a spatial or frequency domain
The spatial or temporal separation between successive samples
The period of the domain, given by N ∆t
Matrix indices (dimensionless)
Fast Fourier Transform function
Fast Fourier Transform function in 2 dimensions
Topography matrix in space domain
Gravity anomaly matrix in space domain
Topography matrix in frequency domain
Gravity anomaly matrix in frequency domain
Power Spectrum of the topographic map
Power Spectrum of the topographic map
Cross-Power Spectrum of the topographic and gravity maps
The error in the Power Spectrum of the topographic map
The error in the Power Spectrum of the topographic map
The error in the Cross-Power Spectrum
The error in the Coherence
z
Coh
D
C
ν
x,y
N
∆t
τ
i,j,k
fft( )
fft2( )
topo
gamap
T
G
PT
PG
STG
δPT
δPG
δSTG
δCoh
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THEORY AND APPROACH
The most obvious question is how does one determine this elastic thickness? While we can
estimate the mass of mountains, it is not obvious as to how to measure the deflection of the
lithosphere. The key however lies in the concept of isostacy. As mentioned in the introduction,
this is a form of hydrostatic equilibrium - that is large surface features will sink over time until
the weight of adjacent columns is equivalent [Figure 1a]. As a result, large surface features
have 'roots.'
This mechanism, however, is
fundamentally different from
elastic support [Figure 1b]. In this
case, we do not have an
equivalence between adjacent
columns of material, in fact, there
is an overabundance of mass in
the vicinity of features which are
elastically supported since the
deflection of the lithosphere is
less than it would be given
Figure 1: Isostatically compensated terrain (a)
Elastically supported Terrain (b)
isostatic compensation. As such,
it is possible to determine the
elastic thickness of the
lithosphere by determining the characteristic wavelength at which the topography shifts from
one type of behavior to the other.
We may relate this crossover wavelength by considering the differential equation for a plate
(Please note that the derivation that follows parallels that found in Turcotte and Schubert2,
chapter 3):
D
d 4z
d 2z
+
p
= q ( x) − Q ( z )
dx 4
dx 2
[1]
Where p is the horizontal stress in the plate, q is the loading at the theoretical surface of the
lithosphere (i.e. at the surface of a hypothetical equipotential surface), Q is the hydrostatic
restoring force of the liquid mantle and D is the flexural rigidity given by:
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D=
Et 3
12(1 − ν 2 )
[2]
If we assume that the deflection of the lithospheric plate is ‘filled in’ [ Figure 2 ] by crustal
rocks, the buoyancy force exerted by the mantle may be expressed as the weight of mantle
displaced subtracting the crustal fill:
Q( z ) = ( ρ m − ρ c ) gz
[3]
If we further assume that the
horizontal stress in the plate is
negligible (p≈0 – may not necessarily
be true of a tectonically active region)
we derive:
D
d 4z
+ ( ρ m − ρ c ) gz = q( x)
dx 4
[4]
This is the fundamental equation for
an elastic lithospheric plate under a
continent. We now model the loading
pattern q. We choose a sinusoid to
accomplish this - recall that Fourier
theory states that any arbitrary signal
Figure 2: We assume an equipotential
surface, thus, at the surface, the deflection
appers ‘filled in.’ NOTE: for the purposes of
this paper, the lower lithosphere has the
same properties as the mantle.
[Reprinted from Reference 2]
(such as topography on a continent)
can be built up from a sum of
sinusoidal frequencies10. Thus let us
express the topography height
(compared to the datum) for a
particular wavelength as:
h = h0 sin(2π
x
λ
)
[5]
if we assume constant density within the terrain, we may find the loading pattern due to h:
q( x) = ρ c gh0 sin(2π
x
λ
)
JOHN MOORES
[6]
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Furthermore, since the loading is sinusoidal we know from experience that the response of
the elastic plate will be also, thus:
z ( x) = z 0 sin(2π
x
λ
)
[7]
Substituting [7] into [4] we can derive an expression for the response of the lithosphere to a
given terrain load:
z0 =
h0
ρn
D ⎛ 2π ⎞
−1+
⎜
⎟
ρc
ρc g ⎝ λ ⎠
[8]
4
Note that as λ→∞ we have:
z 0 (λ = ∞ ) =
ρ c h0
ρn − ρc
[9]
If this equation is examined closely it can be shown that cross multiplying the fraction and
multiplying both sides by g yields q(x) = Q(x). That is to say the additional weight of the
terrain is equal to the weight of mantle displaced. Recall that this is the definition of an
isostatically compensated surface feature. As such we may define the degree of compensation
as:
h0
4
ρm
D ⎛ 2π ⎞
−1+
⎜
⎟
z 0 (λ )
ρc
ρc g ⎝ λ ⎠
ρm − ρc
=
C (λ ) =
=
4
ρ c h0
z 0 (λ = ∞ )
D ⎛ 2π ⎞
ρm − ρc + ⎜ ⎟
ρm − ρc
g⎝ λ ⎠
[ 10 ]
It can be seen from graphing members from this family of equations [Figure 3] that there is a
rapid cross-over from C ≈ 0 to C ≈ 1. Since this represents a shift from elastically supported to
isostatically compensated behavior, we define the characteristic wavelength as being the
wavelength corresponding to C(λchar) = 0.5. Substituting these into equation 10 we derive:
⎛ 2π
⎜⎜
⎝ λchar
4
⎞
g (ρ m − ρc )
⎟⎟ =
D
⎠
JOHN MOORES
[ 11 ]
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Finally, substituting for the flexural stiffness using equation 2 and rearranging gives us an
expression for the thickness of the lithosphere in terms of this characteristic wavelength:
t3 =
12 g (1 − ν 2 )( ρ m − ρ c ) ⎛ λchar ⎞
⎟
⎜
E
⎝ 2π ⎠
4
[ 12 ]
For this calculation we will use some typical values for the lithosphere which we shall assume
is both homogenous and isotropic. These values are given in table 1:
TABLE 1:
Typical Values for the Lithosphere and Mantle
Elastic Modulus2
70x109 N/m2
Mantle Density7
3300 kg/m3
7
Crust Density
2800 kg/m3
7
Lithosphere Poission ratio 0.25
Thus, the original problem of directly finding the thickness of the lithosphere has been
reduced to determining the characteristic wavelength that divides compensated from
uncompensated behavior.
To find this value, it is important to remember that if the terrain is elastically supported there
is a surplus of mass in columns which include the raised topography compared to the
surrounding countryside. Conversely, for isostatically compensated terrain, all vertical
columns have the same mass. Thus we would expect that for small-wavelength topography
we would observe a positive gravity anomaly with the same wavelength but no anomaly when
considering compensated long-wavelength topography.
We must be careful, however, since there is an implicit assumption in this view of the theory.
We have assumed that we are measuring the gravity anomaly at a specific constant altitude.
This is commonly referred to as the ‘free-air’ gravity anomaly and is the ideal dataset to use.
Unfortunately, it is much more difficult to obtain free-air gravity readings – which require an
extremely stable platform at altitude1 - then it is to simply walk over the terrain and take
stationary readings every few miles.
This latter scheme of measurement obtains what is called the bouguer gravity anomaly and
also includes an implicit assumption. Specifically, the bouguer gravity anomaly data is not raw
data, but has been processed to compensate for the changing altitude of the observer.
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Typically, this is performed by subtracting out the estimated contribution from the extra mass
between the observer and the datum (i.e. the mountain they are standing on). Thus, the
bouguer gravity anomaly assumes that all terrain is elastically supported. As such, at short
wavelengths of topography we observe no corresponding gravity anomaly.
However, at large wavelengths,
the terrain is compensated and
thus the correction to the raw
data produces an artificial gravity
low at the same wavelengths as
the compensated terrain. This is
sometimes referred to as the
‘hollow mountain’ problem –
since the gravity anomaly
produced by large, compensated
mountain chains is lower then
would be expected if they were
elastically supported. This paper
takes advantage of this artifact of
the data.
There is some debate between
supporters of either bouguer or
free-air anomalies as to which
method is the most accurate.
Both have problems. The free-air
anomaly relies heavily on
Figure 3: Degree of Compensation family of
curves. The degree of Compensation is plotted
against wavelength and frequency (1/λ).
NOTE: crossover occurs at C = 0.5.
accurate altitude keeping and
positional awareness while also
requiring extremely stable inertial
platforms for the gravimeters on
notoriously unstable (when compared to the requirements) aircraft. The bouguer data, while
more readily available then free-air data and potentially more accurate, require data for large
sections of terrain to detect the long-wavelength artifact. Thus, it may be impossible to obtain
accurate local thickness estimates. This is only intended to give a flavor to the situation – the
resolution of this debate is beyond the scope of this paper. Instead, we will use the bouguer
data exclusively.
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In either case, we proceed by performing a spectral analysis on the terrain and gravity maps
in what is known as the Coherence Method5,6. Once these have been transformed into the
frequency domain we compute the power spectrum of both maps as well as the cross power.
These are defined as follows:
PT ≡ (1/n) < Tij* · Tij >
[13-15]
PG ≡ (1/n) < Gij* · Gij >
STG ≡ (1/n) < Tij* · Gij >
Note that the individual power spectrums of the maps will be real (since they are the product
of individual elements of transformed maps multiplied by their complex conjugates) while the
cross-power, in general, is complex. From this we may define the Coherence as:
Coh =
S TG
2
[ 16 ]
PT ·PG
It can be shown that the coherence is 1 when the frequencies in T and G are correlated and 0
when they are uncorrelated. Thus we expect that for bouguer gravity the Coherence will be
perfect at low frequencies (long-wavelength) and fall off to zero at the characteristic
frequency. For the free-air anomaly, we would expect that the Coherence will be zero at low
frequencies and jump to perfect coherence at the characteristic frequency. For perfect
datasets, we would expect both estimations of the characteristic frequency to be identical,
however, in practice this does not occur4.
IMPLEMENTATION
Using the MATLAB programming environment, code was written to calculate the value of the
coherence over the averaged power spectrums. Three geographic regions were investigated –
the Basin and Range, the Appalachians and the Canadian Shield. The source data for the
topography in the form of USGS tiles is displayed in Appendix A.1 with the approximate
extracted regions highlighted with colored overlays. The actual extracted regions are
displayed in Appendix A.2. This source data was obtained at
http://edcdaac.usgs.gov/gtopo30/gtopo30.html and was degraded from its initial 30 arc
second grid spacing to 2.5 arc minutes8 in order to match with the bouguer gravity map.
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Likewise, the bouguer gravity anomaly map is displayed in Appendix A.3 with the specific
extracted regions displayed in Appendix A.4. This was obtained from the binary raster image
available from NOAA at http://www.ngdc.noaa.gov/seg/fliers/se-2004.shtml . Two notes of
caution are required in the context of this dataset. First, in order to better display the map,
the image was created by reversing the bit ordering (given format is 16 bit LSB) to MSB/big
endian in order to exaggerate the differences between elements. For the purposes of coding,
however, the original little endian bit ordering was employed.
Secondly, it was found (though a significant effort) that there is an error in the header file9.
The data is claimed to begin at 0.75°W and end at 180°W but instead was found to be
deviated to the east by close to 10°, thus the original gravity maps of basin and range were
actually being produced from data recorded in the middle of the pacific. It is unknown if this
error is unique to the raster file or if it is replicated in the other data formats in the NOAAprovided package.
It is important to discuss the criteria for the selection of patch dimensions. Table 2
enumerates the locations and dimensions of the selected patches:
feature
Latitude limits
Longitude limits
Maximum λchar
TABLE 2
Patch Sizes and Locations
Patch Name
Basin and Range
Canadian Shield
37°N – 42°N
45°N – 65°N
113°W – 118°W
94°W – 99°W
557km
2230km
Appalachian
45°N -60°N
61°W – 76°W
1670km
The basic criteria are these: the patches must be as large as possible since we are most
interested in correlating long-wavelength topography. However, at the same time we should
attempt to limit the number of different geological regions included in a particular patch - that
is, we assume that the characteristic wavelength is a property of a particular geological region
and is different for adjacent regions. Thus we have an upper limit on the size: for instance the
basin and range patch cannot be larger then the basin and range geological region.
This led to some peculiarly sized patches. In particular, the Canadian shield patch depicted in
Appendix A.1 which attempts to reduce the minimum observable frequency (maximize the
observable wavelength) at the expense of the number of elements to be averaged since the
slice is very thin in longitude. Furthermore, since there is no elevation data for water, patches
containing ocean should be avoided if at all possible. In the case of the Appalachian patch we
have ‘cheated’ a little, and it is expected that this will introduce some error, however, it was
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thought that the selected patch would give a better picture of the region then any individual
smaller slice and that since the surface area of ocean included is small this effect could be
minimized.
A useful MATLAB toolbox, gtopo30, was available to read in and process the data tiles for the
USGS data set, however, since the NOAA data was significantly older it was necessary to
write a separate program to read in and order the data. This file – ggrav.m - is provided along
with the rest of the code in Appendix B.
Once this data was successfully introduced into the MATLAB environment, a bit more
preprocessing was required. In order to eliminate DC power problems and broadening at low
frequencies, the mean was removed and the resulting maps were windowed. Three types of
windows were attempted – Welch, Hanning, and Bartlett11. These are displayed for
comparison in Appendix A.6. While all three were found to be effective, the Welch window
appeared to perform marginally better then its counterparts and thus was used on data from
all three regions.
Following this, the maps were transformed into the frequency domain using the built-in
Matlab 2-dimensional fast fourier transform function (fft2) before being reduced. A bit of
explanation is required regarding the way in which transformed matrices and vectors are
stored in the frequency domain. The transform of a discretely sampled signal has the same
number of elements as the original signal. For a vector input, we may obtain the frequency of
any frequency domain sample according to:
ωk =
2π
2π
k=
k
τ
N∆t
[ 17 ]
Where k is the index of the vector and τ is the spatial or temporal period of the original time
or spatial signal – that is the number of samples (N) multiplied by the temporal or spatial
spacing between each sample (∆t). From discrete fourier theory, we know that the maximum
frequency which we can extract from a time signal has a wavelength of two divisions, that is:
ωn =
2π
2∆t
[ 18 ]
This maximum frequency is called the nyquist frequency. Note that we obtain the nyquist
frequency when k = N/2 – this begs the question: what is stored in the rest of the vector? As
it turns out, this stores the spectrum from the negative nyquist back to zero1 [ Figure 4 ].
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This bookkeeping (since negative frequencies are no less physical) means that we have stored
frequencies in the range of – ωn to + ωn even though the negative portion of the spectrum is
redundant. In fact, for visualization purposes, the function fftshift in MATLAB will reorganize
the vector such that the transform is symmetric about the zero frequency instead of the
nyquist.
This situation is more complex in two
dimensions [ Figure 5 ], but remains
analogous. Here we have four
quadrants in which the first and third
are copies, as are the second and
fourth, however, each set contains
independent information. Thus it is
necessary to perform a superposition
to ensure that all the pertinent data is
extracted. For the purposes of
reducing this data, we superimpose
the first and second quadrants (the
elements of the second are reversed in
Figure 4: Array ordering in a 1-Dimensional
Fourier Transform
ωx, but not in ωy). This ensures that the
averaging routine which follows
(concentric circles) has all the independent information. First, we obtain the power spectra by
employing equations 13-15.
Figure 5: Array ordering of a 2-Dimensional Fourier Transform showing concentric
averaging circles and the effect of fftshift.
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Next we average values along lines of constant frequency. Since we have assumed that the
lithosphere is isotropic this was accomplished by averaging in concentric circles. In order to
keep the routine as general as possible, an elemental averaging was performed with ωx scaled
with respect to the ωy by the spatial period at the middle latitude in the range. Since both
maps provided each have a geographic projection this means that while the absolute size of
the divisions in the latitude, φ, do not change, the size of the divisions in the longitude get
smaller as we approach the poles. As such, scaling by the average is at best an
approximation.
This leaves only the plotting of the coherence values. For all three zones it was decided to plot
averages over 1,2 and 3-elements. This refers to the number of elements in y between
concentric circles in the averaging routine. The resulting coherence patterns have been
plotted in Appendix A.4 on a semi-log scale vs. the corresponding angular frequency.
ERROR CONSIDERATIONS
Before the results of the analysis are discussed, a digression regarding error considerations is
required. First, qualitative concerns will be discussed which will be followed by an attempt to
quantify the statistical error involved in the coherence calculations assuming that the data up
to that point have no intrinsic measuring error – that is they are assumed to be perfect
samples of a stochastic process.
The first consideration has to be for the source data. Since we are dealing with distributions,
we have two potential errors – measurement errors and positioning errors. Generally
speaking, the topographic map appears to be more reliable since it is taken from relatively
few sources, has a greater resolution and is significantly more recent then the gravity
anomaly map8,9. Furthermore, measuring elevation, especially on a large scale, is a far
simpler task then measuring the gravity anomaly.
Additionally, we know that there is at least one systematic error in the gravity raster file
which was detected – the entire map was shifted by close to 10° to the East. The correct axes
were only obtained by referencing easily identifiable locations on the gravity map (even these
were only available since Mexico was a ‘no data’ region and the map could be calibrated by
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locating Yuma and Cancun) Despite these obvious flaws, for the purposes of this paper, we
shall consider both these errors (positional and measurement) to be zero for both topographic
and gravity anomaly maps. Lastly, this is not to say that the maps are not noisy. In fact, the
bouguer gravity anomaly mapping technique was originally devised to search out density
variations in the crust for the purposes of oil, gas and mineral exploration1. Since we are
interested in long wavelength features, these density variations are simply high-frequency
noise and can be ignored.
The second consideration results from the projection used and from the spherical nature of
the earth’s surface. For the purposes of this project, it is assumed that the selected regions
are perfectly flat . This is known to be false - especially for the larger plates which can
measure in excess of 2000km on a side - but may not be a bad approximation.
However, the gridding of the data is of much greater concern. The longitudinal period was
estimated by using the value at the median latitude and assuming that the regions were
rectangular. For certain plates this is highly questionable, especially in the case of the
Canadian Shield Patch. Here the period in y is 20° (45°N to 65°N) and in x it is 5°. The
scaling of the longitude domain is proportional to the cosine of the north latitude, thus the
employed scale factor (at 55°N) is 0.5736, however the actual variation from the bottom to
the top of the patch varies from the high of 0.7071 (at 45°N) to the low of 0.4226 (at 65°) –
a change of just over 40%. Naturally this is the worst case – these figures are much lower for
the basin and range patch since it is much smaller and located closer to the equator. This
needs to be taken into account when comparing the reliability of the result obtained from
different patches.
In the special case of the Appalachian patch we must also be mindful of the inclusion of ocean
surface -as was alluded to in the last section.
Even if we neglect all these errors, we must still consider the statistical error in getting from
the various power spectra to the coherence. From basic statistics we may define these errors
(characterized by calculating the standard deviation) as:
δPT =
δPG =
1
2NT
1
2N G
PT
[ 19 ]
PG
[ 20 ]
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δS TG =
1
2 N TG
PT PG
[ 21 ]
Thus by the SSE method we have, for the standard deviation in the Coherence:
δCoh =
∂Coh
∂Coh
∂Coh
δS TG +
δPT +
δPG
∂S TG
∂PT
∂PG
=
2 S TG
1
PT PG
2 N TG
PT PG +
− S TG
2
PT PG
2
1
2NT
PT +
− S TG
PT PG
2
2
1
2N G
[ 22 ]
PG
If desired, the code could be extended to take into account this statistical error, however, it is
provided here purely for illustrative value.
RESULTS AND DISCUSSION
The coherence distribution is given in Appendix A.5. From these graphs, the characteristic
frequencies are obtained for the fall-off where the Coherence is 0.5. This is summarized in
table 3 in which we have assumed the values from Table 1 in order to calculate the elastic
thickness:
TABLE 3:
Summary of Characteristic Attributes of the three
Patch Name
Attribute
Basin and Range Canadian Shield
ωchar
5-8.5x10-3 rads/km
6.5x10-2 rads/km
λchar
96.7km
1300km – 740km
Elastic thickness
3.5 km
110km – 53km
Patches
Appalachian
0.9-1.5x10-2 rads/km
700km – 420km
49km – 25km
The first thing to notice about this data is the roll-off properties exhibited by the figures in
Appendix A.5. The curve for the Basin and Range looks a great deal like the expected
theoretical form of Figure 3. We have a clear plateau and a sharp fall-off. In this case the
most likely curve is the 2-point average, however, all three curves are very similar at the
cross-over coherence.
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The plateau feature is much less pronounced on the graphs for the Appalachians and the
Canadian Shield. This is also expected since these regions were anticipated to produce long
characteristic wavelengths due to large thicknesses. As a comparison, note that characteristic
wavelength for Basin and Range of 96.7km (table three) is only 17% of the maximum value
(table 2), given the patch size while this figure is 57% for the Canadian Shield. Unfortunately,
since the geographic domains of the particular regions are limited it is impossible to increase
the patch size to encompass more of the roll-off curve.
One particularly puzzling aspect of the curves is the presence of a spike in the data just after
the fall-off. This is particularly pronounced in the basin and range data and does not seem to
be eliminated by increasing the averaging width. This suggests that this might be physical. If
this is the case, it is possible that we are seeing local variations within the region. For
instance, if, locally, within the basin and range zone there was a region with a substantially
thinner section of elastic lithosphere (for instance due to volcanism or a hotspot) we could
have compensated geography in a section of the spatial domain at very small wavelengths –
this would give rise to an improved coherence at these higher frequencies.
This could also be accomplished without requiring a change in lithosphere elastic thickness if
the lithosphere were to locally suffer a failure (i.e. to break from loading) and thus lose much
of its flexural rigidity. This is not an altogether impossible situation – note that the thickness
of the lithosphere in the basin and range region is comparable to the height of the topography
suggesting that there may be buckling failures at specific locations within the region which
allow higher-frequency topography to be compensated. Despite this speculation, it is equally
possible that this is simply an anomaly of the data sets used and has no physical significance.
It is interesting to note that the ordering in terms of thickness is what was expected with the
Basin and Range patch exhibiting the smallest elastic thickness and the Canadian Shield
exhibiting the largest. These thicknesses are also compatible with results obtained by Bechtel
et al3 using a coherence method on bouguer gravity data. That group obtained an average
value of 5km for the basin and range and a minimum of 4km which agrees well with the
value obtained here. Furthermore, for the Canadian Shield they found a maximum thickness
of 128km and an average thickness of 82km while for the Appalachian region a thickness of
39km was obtained. These also compare well with the values in Table 3.
While this is encouraging, it is important to recognize that Bechtel et al possessed
bathymetric data for Hudson’s bay and included this in their average over the Canadian
Shield. Furthermore, since the code provided here is intended to deal with rectangular
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patches parallel to the latitude and longitude axes the patch used for the ‘Appalachians’
actually extends over three geological regions, namely the Canadian Shield, Appalachians,
and Grenville regions [ figure 6 ]. However, Bechtel’s Appalachian average is composed of
only the latter two regions and also uses Bathymetric data for the Gulf of St. Lawrence to
obtain a patch of sufficient size.
The largest discrepancies
appear to occur for the
Canadian Shield and
Appalacian region. This is not
surprising – as discussed in
the section on error we have
more reason to distrust our
values from these patches
then the Basin and Range
region. As well, it should also
be noted that Bechtel’s data
has been re-projected on a
Lambert conformal projection
and avoids the error of
Figure 6: Geological Map of North America
[Reprinted from Reference 3 ]
skewed patches at northern
latitudes to which the code
described in this paper is
subject. As such, the method outlined here could be made more precise by re-gridding the
data to a Cartesian plane – this is the simplest extension of the code.
It would also be beneficial to examine free-air gravity anomaly maps. Should these be
available the current code would be able to analyze the coherence with the simple addition of
a utility to read in the data for processing. Not only would this give us a lower bound for λchar
but might be obtained for smaller patches. This is due to the roll-off properties of the free-air
data which occurs at lower wavelengths then for the bouguer data4 (of course, given perfect
data these would be identical, however, experiment suggests that the bouguer gives an upper
bound while the free-air method gives a lower bound) and the fact that we have more highfrequency, small-wavelength data then low-frequency data suggests that the roll-off might be
more pronounced then the bouguer graphs in regions with greater elastic thicknesses.
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CONCLUSIONS
Using a coherence method, values were obtained for the elastic thickness of the lithosphere in
the Basin and Range, Canadian Shield and Appalachian regions of North America under the
assumption of an isotropic lithosphere. The values obtained for the characteristic wavelength
were 3.5km, 110-53km and 49-25km, respectively, in each region. These values agree well
with the data obtained by Bechtel et al especially considering the concerns about the method
which we have used at higher latitudes on larger patches. Furthermore, they support the
hypothesis that the lithospheric thickness is related to the age of the surface features
observed.
This code could easily be extended and the accuracy of the results improved substantially by
re-gridding the data on a Cartesian projection and by comparing the characteristic wavelength
to that obtained by using a free-air method.
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REFERENCES
[ 1 ] Bailey, Richard. (2003) PHY 495 Course Notes. University of Toronto.
[ 2 ] Turcotte,D. L. and G. Schubert (2002) Geodynamics. Second Edition. Cambridge
University Press.
[ 3 ] Bechtel T. D. et al (1990) “Variations in effective elastic thickness of the North American
Lithosphere.” Published in Nature vol 343 pp 636-638.
[ 4 ] McKenzie, D. and Fairhead D (1997). “Estimates of the effective elastic thickness of the
continental Lithosphere from bouguer and free-air gravity anomalies.” JGR-Solid
Earth paper 97JB02481, vol 102 n°12, pp.27523.
[ 5 ] Forsyth, D.W. (1985) “Subsurface Loading and estimates of the flexural rigidity of
continental lithosphere.” Published in the Journal of Geophysical Research, vol 90,
n°B14 pp.12623-12632.
[ 6 ] Macario, A. et al (1995) “On the robustness of elastic thickness estimates obtained using
the coherence method.” Published in the Journal of Geophysical Research, vol 100,
n°B8 pp.15163-15172.
[ 7 ] Sandwell, D.T. (2002) Gravity/Topography Transfer Function and Isostatic Geoid
Anomalies. Notes.
[ 8 ] United States Geological Service (1998) gtopo30 supporting documentation. [online]
http://edcdaac.usgs.gov/gtopo30/gtopo30.html.
[ 9 ] National Oceanic and Atmospheric Administration (1989) Bouguer gravity anomaly
supporting documentation. [online] http://www.ngdc.noaa.gov/seg/fliers/se2004.shtml .
[ 10 ] Haberman, Richard (1998) Elementary Applied Partial Differential Equations with fourier
series and boundary value problems. Prentice Hall Publishing
[ 11 ] Bourke, Paul (1998) Windows. Swinburne centre for astrophysics and supercomputing.
[online] http://astronomy.swin.edu.au/~pbourke/analysis/windows/.
BIBLIOGRAPHY
[ 1 ] Serway, R.A. (1996) Physics for Scientists and Engineers. Volume 1. 4th Edition.
Saunders College Publishing.
[ 2 ] Van Loan, C.F. (2000) Introduction to Scientific Computing: A Matrix-Vector Approach
Using Matlab. 2nd Edition. Prentice Hall Publishing.
[ 3 ] Nyborg, D. (1998) The Fast Fourier Transform and its use in Spectral Analysis of Digital
Audio. Published Online by www.Mathtools.net.
[ 4 ] Carrol, B.W. (1996) An Introduction to Modern Astrophysics. Addison Wesley Publishing
Inc.
[ 5 ] French, A.P. (1971) Vibrations and Waves. Published by WW Norton & Company.
[ 6 ] Karner, G.D. and Watts, A.B. (1983) “Gravity Anomalies and Flexure of the Lithosphere
at Mountain Ranges.” In the Journal of Geophysical Research, vol 88 n°B12. pp.
10449-10477.
[ 7 ] Uncredited Reading. Chapter 6, sections 6.6 to 6.13.
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APPENDIX A
CONTENTS: GRAPHICAL
Topographical Maps (USGS Tiles)
Topographical Maps (Generated Patches)
Main Gravity Map (NOAA Raster Image)
Gravity Maps (Generated Tiles)
Coherence vs. Angular Frequency
Windows
A.1 Topographical Maps (USGS TILES)
Basin
2
& Range
Region
Canadian Shield
Region
1
Appalachian
Region
3
Counterclockwise from Above:
1. USGS Tile w100n90
2.USGS Tile w140n90
3.USGS Tile w140n40
Designation refers to top left hand corner of the tile. NOTE: while the Appalachian
and Canadian Shield regions derive solely from w100n90, Basin and Range
stretches over tiles w140n90 and w140n40. Approximate areas as shown.
A.2 Topographical Maps (GENERATED PATCHES)
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A.3 Main Gravity Map (NOAA RASTER IMAGE)
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A.4 Gravity Maps (Generated Tiles)
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A.5 Coherence vs. Angular Frequency
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A.6 Windows
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APPENDIX B
CONTENTS: CODES
Matlab Code – Core program
Matlab Code – Supporting Functions (general)
Matlab Code – Supporting Functions (region-specific)
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MAIN PROGRAM CODE
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CANADIAN SHIELD ZONE
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%
% Lowercase letters indicate space domain quantities
% Uppercase letters indicate frequency domain quantities
%
% The code is identical for all zones except different loaders are used
% (APtopo for the Appalachians and BRtopo for Basin and Range)
clear all
CStopo
%% Script which loads the two matrices into the Matlab Environment
%% map = topographical map , gamap = gravity anomaly map
xRange = lonlim(2)-lonlim(1);
yRange = latlim(2)-latlim(2);
nx = length(map(1,:));
ny = length(map(:,1));
nxq = floor(nx/2)+1;
nyq = floor(ny/2)+1;
%% nyquist elements in x and y
%GLAZIER’S SECTION – WINDOW PREPARATION
for i = 1:nx
%% welch window processing
for j = 1:ny
welch(j,i) = (1-(((i-1)-(nx-1)/2)/((nx-1)/2))^2 ) *(1-(((j-1)-(ny-1)/2)/((ny-1)/2))^2 );
end
end
for i = 1:nx
%% hanning window
for j = 1:ny
hanning(j,i) = sin((i-1)*pi/(nx-1))*sin((j-1)*pi/(ny-1));
end
end
for i = 1:nx
for j = 1:ny
%% bartlett window
if(j <= floor(ny/2) )
if( i <= floor(nx/2))
bartlett(j,i) = (i-1)*(j-1);
else
bartlett(j,i) = (nx-i)*(j-1);
end
else
if( i <= floor(nx/2))
bartlett(j,i) = (i-1)*(ny-j);
else
bartlett(j,i) = (nx-i)*(ny-j);
end
end
bartlett(j,i) = bartlett(j,i)/((ny-floor(ny/2))*(nx-floor(nx/2)));
end
end
map = map.*welch;
gamap = gamap.*welch;
%% Applying the window
%figure
%surf((1:nx),(1:ny),map)
%% visualization
MAP = fft2(map);
GAMAP = fft2(gamap);
%% 2D Fourier transform
%figure
%surf((1:nx),(1:ny),abs(MAP))
M = shifty(MAP,nx,nxq,nyq);
G = shifty(GAMAP,nx,nxq,nyq);
%% shifty superimposes quadrants 1 and 2
%figure
%surf((1:nxq),(1:nyq),abs(M))
PM = M.*conj(M);
PG = G.*conj(G);
SMG = M.*conj(G);
%% topographic map power
%% gravity anomaly map power
%% cross power
tau = 6378*2*pi*(latlim(2)-latlim(1))/360;
omega = 2*pi/tau;
%% Period of measurements (N*delta) in the y-direction
%% frequency domain
%% AVERAGING PROTOCOL
ea = 3;
[mB,mS] = mAVG(PM,ea,latlim);
[gB,gS] = mAVG(PG,ea,latlim);
%% matrix elements per average
%% concentric circular averaging (compensated for latitude)
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[xB,xS] = mAVG(SMG,ea,latlim);
C = (abs(xS).^2)./(mS.*gS);
Figure
semilogx(((mB-1).*ea+1).*omega,C,'r')
%% coherence calculation
%% graphing of 1,2 and 3 element averages
ea = 2;
[mB,mS] = mAVG(PM,ea,latlim);
[gB,gS] = mAVG(PG,ea,latlim);
[xB,xS] = mAVG(SMG,ea,latlim);
C = (abs(xS).^2)./(mS.*gS);
hold
semilogx(((mB-1).*ea+1).*omega,C,'b')
ea = 1;
%% averaging (matrix elements per average)
[mB,mS] = mAVG(PM,ea,latlim);
[gB,gS] = mAVG(PG,ea,latlim);
[xB,xS] = mAVG(SMG,ea,latlim);
C = (abs(xS).^2)./(mS.*gS);
semilogx(((mB-1).*ea+1).*omega,C,'g')
grid on
title('Canadian Shield Region - compensated for lattitude')
xlabel('Angular Frequency (rads/km)')
ylabel('Coherence')
legend('3-point average','2-point average','1-point average')
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CONCENTRIC AVERAGING
%
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function [ B,S ] = mAVG(T, bn,latlim)
%
%
%
%
%
%
%
%
Calculates concentric circular averages of matrices treated by shifty.m
(i.e. assumes an overlapping 1st and 2nd quadrants). Furthermore assumes a geographic
projection with the y-axis of the matrix running North-South and the x-axis running East West
designed to divide the matrix along concentric circles centered at the first element
into several bins averaged over bn matrix elements
Modified to accept non-square matrices and to compensate the x-coordinates for the
change in period at the average y lattitude
%clear all
%T = [ones(50,100); zeros(50,100) ];
%bn = 1;
%% sample input matrix (used only for test cases)
%% averaging (bin) width – elemental, but does not have to be an integral number
a = 1/cos((latlim(2)+latlim(1))/2*pi/180);
%% lattitude scale factor applied to x-domain
sx = length(T(1,:));
sy = length(T(:,1));
k = 0;
while( bn*k < (sqrt((sx*a)^2 + sy^2)+bn) )
R(k+1) = (bn*k)^2;
B(k+1) = k+1;
S(k+1) = 0;
N(k+1) = 0;
k=k+1;
end
%% Creates vector of widths
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for y = 1:sy
r = ((1-1)*a)^2+(y-1)^2;
while( r < R(k) )
k = k-1;
end
for x = 1:sx
r = ((x-1)*a)^2+(y-1)^2;
if( r >= R(k+1) )
k = k+1;
end
%% selects correct contour to be one BELOW current value
%% at BEGINNING of matrix row (thus all other moves are greater by one contour)
%% Advances contour by one if appropriate
S(k) = S(k) + T(y,x);
N(k) = N(k) + 1;
end
end
S = S(1:length(S)-2);
B = B(1:length(S));
N = N(1:length(S));
%% concatenation to remove extra elements
S = S./N;
%figure
%plot((B-1).*bn,S)
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function [ M ] = shifty(MAP,nx,nxq,nyq)
%%
%%
%%
%%
%%
Function to reorganize MAP into M by concatenating
to a matrix of size nxq by nyq since we require only
information from one of quadrants (1,3) and one of (2,4)
we select quadrants 1 and 4. The elements of 4 are reversed
and added to the elements of quandrant 1 (for averaging purposes)
M = zeros(nyq,nxq);
M(:,1) = MAP(1:nyq,1).*2;
for j = 2:nxq
if( nx-(j-2) > nxq )
M(:,j) = MAP(1:nyq,j)+MAP(1:nyq,nx-(j-2));
else
M(:,j) = MAP(1:nyq,j).*2;
end
end
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NOAA GRAVITY RASTER READER
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function [ G, xlong, ylat] = ggrav(ilat, ilong, dlat, dlong )
%%
%%
%%
%%
%%
%%
function which extracts the pertinent block of data
from the gravity anomally file and returns it as a matrix A
ilat,ilong are the initial lattitude and longitudes respectively
of the SOUTHWEST corner element (top left) and the size of the patch
is given by the number of points to be taken, dlat and dlong
NOTE!! dlat and dlong must be multiples of 2.5/60 degrees
dtheta = 2.5/60;
xmin = -180;
ymax = 81+1/6;
%%Overall Grid Parameters
tic
%% DATASET READING
%% since the dataset is relatively small (7.85 million elements)
%% we use a brute force method to read in the entire dataset (this never took more then 15 seconds)
fstar = fopen('dgrav.rst', 'r', 'l');
[ A , count ] = fread(fstar, [4302,1824] , 'bit16=>double');
fclose(fstar);
%% Reading in the entire dataset
t = toc;
disp(t)
tic
A = A';
t = toc;
disp(t)
ylat = ilat:dtheta:(ilat+dlat);
xlong = ilong:dtheta:(ilong+dlong);
%% orienting the coordinate axes
%% vectors containing lat and long distr
y = floor(abs((ylat - ymax)/dtheta -1));
x = floor((xlong - xmin)/dtheta +1);
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tic
k=1;
l=1;
for i=y(1):sign(y(length(y))-y(1)):y(length(y))
for j = x(1):sign(x(length(x))-x(1)):x(length(x))
G(k,l) = A(i,j);
if(G(k,l) == -9999)
G(k,l) = 0;
end
l =l+1;
end
l = 1;
k = k+1;
end
%% EXTRACTION OF PERTINENT SECTIONS
%% use NaN for visualization, but 0
%% requried to perform operations on Matrix (mean,sum)
t = toc;
disp(t)
% CONCATENATED GRID
%for i = 1:dlat*2.4
% for j = 1:dlong*2.4
%
anom(i,j) = G(10*i-9,10*j-9);
%
xa(j) = xlong(10*j-9);
%
ya(i) = ylat(10*i-9);
% end
%end
%% Visualization aid routine - creates a concatenated grid with larger divisions to aid in visualization of large patches
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MAP LOADING SCRIPT
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BASIN and RANGE ZONE
%
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%% Runs as a script to import the maps into the Matlab environment
%% for use by the main program
tic
latlim = [37 42];
lonlim = [-118 -113];
%% We must bridge the tiles
latlim1 = [40
latlim(2) ];
latlim2 = [latlim(1) 40
];
lonlim1 = lonlim;
lonlim2 = lonlim;
%% B&R char [37 42]
%% B&R char [-118 -113]
%% upper
%% lower
[map1,maplegend1] = gtopo30([],5, latlim1,lonlim1);
[map2,maplegend2] = gtopo30([],5, latlim2,lonlim2);
%% upper map w140n90
%% lower map w140n40
map = [map2;map1];
map(isnan(map)) = -1;
t=toc;
disp(t)
L = size(map);
x = linspace(lonlim(1),lonlim(2),L(2));
y = linspace(latlim(1),latlim(2),L(1));
%figure
%surf(x,y,map)
%title('Topographic Map of Basin and Range')
%% visualization of topography
[ gamap, xlong, ylat] = ggrav(latlim(1), lonlim(1)+10, latlim(2)-latlim(1), lonlim(2)-lonlim(1) );
%figure
%surf(xlong, ylat, gamap)
%title('Gravity Anomaly Map of Basin and Range')
%% visualization of gravity anomaly
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%% REMOVE MEAN FROM DATA
mapMean = mean(mean(map));
gaMean = mean(mean(gamap));
map = map - mapMean;
gamap = gamap - gaMean;
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MAP LOADING SCRIPT
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CANADIAN SHIELD ZONE
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%% Runs as a script to import the maps into the Matlab environment
%% for use by the main program
tic
latlim = [45 65];
lonlim = [-99 -94 ];
[map,maplegend] = gtopo30([],5, latlim,lonlim);
map(isnan(map)) = -1;
t=toc;
disp(t)
%% NS elongated [45 65]
%% NS elongated [-99 -94 ]
L = size(map);
x = linspace(lonlim(1),lonlim(2),L(2));
y = linspace(latlim(1),latlim(2),L(1));
%figure
%surf(x,y,map)
%title('Topographic Map of Canadian Shield')
%axis equal
%view(-76,88)
[ gamap, xlong, ylat] = ggrav(latlim(1), lonlim(1)+10, latlim(2)-latlim(1), lonlim(2)-lonlim(1) );
%figure
%surf(xlong+10, ylat, gamap)
%title('Gravity Anomaly Map of Canadian Shield')
%% REMOVE MEAN FROM DATA
mapMean = mean(mean(map));
gaMean = mean(mean(gamap));
map = map - mapMean;
gamap = gamap - gaMean;
JOHN MOORES
DETERMINATION OF THE ELASTIC THICKNESS OF
THE CRUST USING SPECTRAL ANALYSIS ( 2003 )
PAGE
46
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PHY 495S
ASSIGNMENT #2
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The Elastic Thickness of the Earth’s Crust
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by: John Moores (c) February - April 2003
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Supervisor: R. Bailey
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MAP LOADING SCRIPT
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APPALACIANS ZONE
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Runs as a script to import the maps into the Matlab environment
%% for use by the main program
tic
latlim = [45 60];
lonlim = [-76 -61 ];
[map,maplegend] = gtopo30([],5, latlim,lonlim);
map(isnan(map)) = -1;
t=toc;
disp(t)
%% typical value [43 62]
%% typical value [-80 -61]
L = size(map);
x = linspace(lonlim(1),lonlim(2),L(2));
y = linspace(latlim(1),latlim(2),L(1));
%figure
%surf(x,y,map)
%title('Topographic Map of Appalacians')
[ gamap, xlong, ylat] = ggrav(latlim(1), lonlim(1)+10, latlim(2)-latlim(1), lonlim(2)-lonlim(1) );
%figure
%surf(xlong, ylat, gamap)
%title('Gravity Anomaly Map of Appalacians')
%% REMOVE MEAN FROM DATA
mapMean = mean(mean(map));
gaMean = mean(mean(gamap));
map = map - mapMean;
gamap = gamap - gaMean;
JOHN MOORES
DETERMINATION OF THE ELASTIC THICKNESS OF
THE CRUST USING SPECTRAL ANALYSIS ( 2003 )
PAGE
47