Module 43

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Module 4-3
The Floating Lithosphere 3. Cross Section of
North America
Quantitative Concepts and Skills Weighted
sums Manipulating equations Trial and error
modeling Fitting lines to data Using the slope of
the fitted line
Calculating the pressure at the depth of
compensation.
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PROBLEM
This cross section, published some 50 years ago,
was a milestone presentation of geophysical data.
The section shows the depth to the Mohorovicic
Discontinuity, which is defined by seismic
velocities, at many continental and oceanic
localities. Are the vertical columns in
isostatic equilibrium? Is pressure constant at
the depth of compensation? We will focus on the
dark yellow columns, for which the data are
tabulated in the original publication.
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This close up from Slide 2 shows the thirteen
columns we will consider. The numbers on the
columns are P-wave velocities 4-5 km/sec for the
sedimentary rocks of the mid-continent 6 km/sec
for the continental crust 8 km/sec for the
upper mantle (the antiroot).
Depth in km
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PREVIEW Slides 5-7
The spreadsheet of Slide 5 is a tabulation of
the data presented diagrammatically in Slide 3
elevation of the land surface, depth of the Moho,
and P-wave velocities of the two or three layers
at each column. The three layers are (1)
sedimentary rocks (in two of the columns), (2)
continental crust, (3) antiroot. Slide 6
calculates the pressures at the depth of
compensation in the simplest possible way (1) it
assumes that the depth of compensation lies below
the variable thickness of the crust (at 40 km,
specifically), (2) it converts the elevations
from feet to km, (3) it calculates the layer
thicknesses at each column (being given the
thickness of the sedimentary layer as new data)
(4) and it assumes uniform density for each layer
(2.5 g/cm3 for the sedimentary layer, 2.67 g/cm3
for the continental crust 3.27 g/cm3 for the
antiroot). This last assumption means that we
assume that the continental crust and upper
mantle are homogeneous. The slide completes the
calculation of pressures using the SUMPRODUCT of
densities and thicknesses and presents the
results with some summary statistics (upper right
box) and a graph of pressure vs. locality across
the continent. Slide 7 examines the consequences
of changing the depth of compensation.
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Here are the basic data elevation (in ft), depth
to the Moho (in km), and the velocity of the
P-waves in each of the three layers.
Recreate this spreadsheet, Then think about how
you can amend the spreadsheet to calculate the
pressure at the depth of compensation.
First you will need to convert the elevations
from feet to meters. Then you can add rows to
give the thickness of each of the layers. You
can work out the thickness of the continental
crust from the elevation data and the depth to
Moho. You can work out the thickness of the
antiroot from the thickness of continental crust
and the (assumed) depth of compensation (and for
two columns, additional information about the
thickness of the sedimentary rocks). The next
slide shows a solution for a depth of
compensation of 40 km.
See next slide for a solution
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These are standard values.
Line graph showing pressure vs. location.
Modify your spreadsheet to duplicate this. Then
change the depth of compensation to be 100 km.
What do you think will change? In particular,
which numbers in the Results Box will change?
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Unchanged. Why?
Different. Why?
What about the thicknesses? How have they
changed?
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PREVIEW Slides 9-10
In Slides 6 and 7, we assigned a single density
value to each of the three layers
(homogeneous-layers assumption). But look at the
velocities in Slide 3. The velocities for a
particular layer vary from column to column,
suggesting that the homogeneous-layers assumption
may be an oversimplification. The variation is
especially apparent for the continental crust
(compare Utah to Minnesota). Might the variation
in velocity reflect density heterogeneities that
would make the pressures more nearly equal along
the depth of compensation? The spreadsheet of
Slide 9 develops a correlation between velocity
and density. The regression lines are of
particular interest. We can use these lines to
convert the velocities of Slides 3 and 5 to
densities, and then use the densities in our
SUMPRODUCT calculations. Slide 10 amends and
modifies the spreadsheet of Slide 7 to do the
calculation.
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Copy this part from Slide 6.
Click for help.
Copy this part from Slide 5.
Coefficients from Slide 9
Calculate the densities from the velocities and
the regression-line coefficients, and the
pressures from the densities and thicknesses.
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Heterogeneous crust
PREVIEW Slides 12-13
If you look again at the data slides (Slides 3
and 4), you will notice that the velocities of
the continental crust fall into two groups. One
group is in the range of 5.5-5.8 km/sec. The
other group is in the range 6.0-6.2 km/sec.
Perhaps we can assign modified standard block
densities one for continental crust 1 (the
standard of Slide 5), and the other for
continental crust 2 (with the lower velocity).
Slide 12 works out a block density for
continental crust 2. The calculation uses the
slope of the regression line from the 0.6-GPa
data in Slide 9. The calculation is based on the
fundamental relationship that allows
interpolation and extrapolation. Two ways of
expressing this relationship are where
dy/dx is the slope of the line. Slide 13 uses
the new modified block density for the
continental crust (y2) in a spreadsheet modified
from Slide 6 to calculate a new set of pressures
at 40 km.
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Heterogeneous crust, 2
Copy this part from Slide 5 and separate the two
types of crust.
Click for help.
From Slide 6
From Slide 9
Use in Slide 13
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Heterogeneous Crust, 3
14
Heterogeneous Crust
Heterogeneous Upper Mantle
Depth of Compensation At 100 km


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End of Module Assignments

• Answer the questions in Slide 7.
• What depths correspond to the 0.2, 0.6, and 1.0
  GPa pressures of Slide 9?
• 3. Pool the density vs. velocity data in Slide
  9 and calculate one regression line to cover the
  complete set of pooled data. Use the new
  regression coefficients, instead of the three
  sets of coefficients, to revise the spreadsheet
  in Slide 10. Hand in that spreadsheet.
• 4. Based on the results in Slide 14, what do
  you think of isostasy? Would you say that North
  America is in isostatic equilibrium?
• 5. Recreate the spreadsheet of Slide 14 and
  change the depth of compensation to 120 km. Hand
  in that spreadsheet.
• 6. List all the variables that affect the
  assessment of isostatic equilibrium by comparing
  pressures at the depth of compensation. How does
  variation in each variable affect the
  comparison.? The calculation is most sensitive
  to which variable? The calculation is most
  robust with respect to which variable?