igpp departmental examination - longbase strainmeter...
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IGPP
Departmental
Examination
1990
1 Sept. 10, 1990
1990 Geophysics Departmental Examination
General Notes:
• You have four hours. The exam is closed-book.
• The exam is in two parts. The first contains 10 questions, all of which must beanswered. They are brief enough that you should need at most 5 minutes for each(in most instances, much less). You must give an answer for each of thesequestions.
The second part also contains 10 questions. These are somewhat more difficult, andmay require more elaborate answers. You are expected to provide answers to 6 ofthem, but you may attempt to answer all of them, if you feel that you can do a goodjob.
• Although it is important to give correct answers, it is at least as important to showthat you know what you are doing. You will get little credit for results that appearout of thin air, even if they are correct.
• Remember to exercise good sense, and to show that you have a good physicalunderstanding of the subject. Beware of results which are dimensionally wrong, oranswers which are ridiculously off (by many orders of magnitude).
• Please make your answers as readable as you can make them in the time allotted.Brevity is valuable, but not to the point of obscurity. Long rambling essays are notdesired either. You should focus on trying to convey your understanding of thequestions. If the Committee cannot understand what you have done, they willassume that you do not understand either. Merely jotting down disconnectedformulae, without writing what the variables are, does little to show your knowledge.
2 Sept. 10, 1990
Section 1
1. The sea surface is an equipotential. What is the point of measuring gravity on boarda ship?
2. The matter, magnetization and currents inside the earth produce a geomagnetic field
B and a gravitational potential !. Both these fields can be expanded in sphericalharmonics. No proofs are required in what follows.a) What is a spherical harmonic? How many linearly independent spherical
harmonics are there of degree n?
b) Write the expansions for B and ! (of course, you need not give thenumerical values of the coefficients).
c) Where are these expansions valid?d) Why does the expansion for B have no harmonic term of degree 0?
e) Why does the expansion for ! usually have no harmonic terms of degree 1?When might such a term be present?
3. Define “geoid” and “geoid anomaly”. Is there a positive or negative geoidanomaly over a positive mass anomaly? Why?
4. Why is the stress tensor symmetric? What is the most general form of the stresstensor at a free surface with outer unit normal in the positive x3 direction?
5. Give the expressions for P-wave and S-wave velocities in terms of the density ", the
rigidity µ, and the Lamé constant #. Why are P-waves faster than S-waves in theouter core?
6. The Indian and Eurasian plates converge at approximately 50 mm/yr in theHimalayan region. Why is there no well-developed “Benioff zone” along theHimalayan segment of the India-Eurasia plate boundary?
7. If a statistical distribution depends on several unknown parameters, what does itmean to say that an estimator of one of these parameters is (mathematical statementsare acceptable):a) unbiased?b) asymptotically unbiased?c) consistent?d) efficient?e) maximum likelihood?
8. Let x1, ... , xN be independent identically distributed random variables. Suppose we
estimate their mean µ and variance $2 by
3 Sept. 10, 1990
µ = 1
N
xi!i=1
N
!2 = 1
N
(xi - µ)2
!i=1
N
a) Show that µ is an unbiased estimator
b) Show that E(xi xj) = $2%ij + µ2, where E means expected value and %ij is the Kronecker delta.
c) Show that E(xi µ) = $2/N + µ2.
d) Show that E( µ2) = $2/N + µ2..
e) Show that !2 is a biased estimator.
9. Prove the following relations between wavenumber k, wavelength #, phase velocity c,group velocity U, frequency f, and period T:
U = c - !dc
d! U = c + k dc
dk
U = c2 dT
d! U = -!
2 df
d!
10. To describe most earthquake mechanisms, we can specify a fault plane with unitnormal ni, a unit slip vector dj orthogonal to ni, and a scalar moment Mo. Giventhese, the moment tensor of the equivalent double-couple can be written as Mij =Mo (ni dj + nj di).
a) Give an expression for the seismic moment. Define your terms and verifydimensions.
b) In a fault mechanism solution (of the “beach ball” type), two nodal planes aredetermined, the “fault” plane, and the “auxiliary” plane. Find the expressionwhich would give the moment tensor if the earthquake had actually ruptured the“auxiliary” plane instead of the “fault” plane.
4 Sept. 10, 1990
c) Give an example of an earthquake source which cannot be represented in termsof a double-couple. Can this happen in the earth?
5 Sept. 10, 1990
Section 2
1. Give all the evidence you know for the claim that most of the magnetic fieldobserved at the surface of the earth has its sources in a liquid core whose radius isabout half the earth’s radius.
2. The local Gas & Electric company guarantees that all its power is produced in thefrequency band 60±0.03 Hz. You want to estimate the power spectrum of the localvertical geomagnetic field Z between periods of roughly one minute and thirtyminutes. Your grant contains no money for an analogue attenuator of the 60 Hz
signal, so you decide to measure Z every & seconds, choosing & so that the power-line signal will not be aliased into the lowest third of your Nyquist band. That is, iffN is the Nyquist frequency, no power company frequencies alias into frequencies f
satisfying -fN/3 < f < fN/3. What is the largest value you can choose for &, and ifyou choose that value, what is the shortest period at which you can estimate thespectrum of Z?
3. The attached figure shows broadband records of the first arrival of an earthquake atEaster Island (27°S, 108°W) recorded at Piñon Flat Observatory (33.6°N,116.5°W). One component (Z) is the vertical, the other two are geographicallyoriented (NS and EW) horizontals. Identify the directions of these, and explainyour choice. The near-surface P-wave velocity at the station is 6 km/sec; estimate theP velocity at the deepest point of the ray. Is that consistent with what you know ofmantle velocities?
4. The NUVEL-1 rigid plate motion model gives the pole of relative motion betweenthe Pacific and North America plates at 48.7°N, 78.2°W, with an angular velocity of0.78 degrees/Myr.
a) Calculate the relative plate velocity vector at the location of the Loma Prietaearthquake, (37°N, 121.85°W). It this consistent with the geodetically estimated rateof slip on the San Andreas of 16 ± 2.5 mm/yr at that location?
b) In 1906, the San Andreas fault at that location slipped by about 250 ± 60 cm. In1989, it slipped again by about 150±30 cm. Could you reconcile these observationswith the previous results?
c) Assume that the interseismic deformation (between big earthquakes) across theSan Andreas is actually distributed uniformly across a zone extending 40 km oneither side of the fault. What is the rate of strain within the fault zone? If you have a
measurement system capable of measuring strains of 10-7, how much time do youhave to wait in order to have a useful estimate of the rate of slip? Explain youranswer.
5. a) It has been proposed that we prevent stress build-up along a major fault like theSan Andreas by triggering small (Ms = 4) earthquakes in sufficient numbers toaccommodate the tectonic slip rate. Presumably, this would eliminate the risk ofvery large earthquakes (Ms = 8). Assuming that we were able to trigger smallearthquakes at will, how many magnitude 4 events would we have to endure on adaily basis? Comment on the proposed scheme.
6 Sept. 10, 1990
Hints: Assume a magnitude 8 event would happen every 150 years. Thetriggering technique is an irrelevant mystery. Assume the momentmagnitude relation to be log10 M0 = 1.5 Ms + 15.5, where M0 is themoment in dynes cm.
b) Whenever a few naturally occurring magnitude 4 events are felt in California, weinvariably get queries from the press as to whether this means the chances of havingthe BIG one have been thereby reduced. How would respond to such a query?
Hint: Take the b-value of California seismicity to be 1.
6. Sketch the function Q-1(') for an absorption band with a single relaxation peak
centered at '0. Sketch the associated phase velocity c('), and attenuation
coefficient (('), where k(') = ' / c(') - i ((') is the wavenumber. What is the
asymptotic behavior of Q(') at low frequencies ' << '0, and at high frequencies '
>> '0? Is the medium is a good propagator of high frequencies, or a bad one?Why don’t we typically see very high frequencies in seismograms? Justify youranswers.
7. Sketch a springs-and-dashpots representation of a standard linear solid. Denoting
spring constants by G1, G2, etc., and viscous elements by )1, )2, etc., sketch theform of the stress relaxation and strain retardation curves, and label as many features(e.g. slopes, asymptotes) as you can.
8. The oceanic lithosphere is created during seafloor spreading. At the spreading ridgeaxis, hot mantle rock (temperature Tm) comes in contact with cold seawater(temperature To) causing it to cool and strengthen. As this young lithosphere iscarried away from the spreading axis it continues to cool by diffusion of heat. Usedimensional analysis to arrive at approximate formulas for the variations in surface
heat flow and seafloor depth. Don't worry about factors of 2, *, etc; any nontrivialanswer with the proper dimensions is correct. The following parameters will beuseful
(Dimensions M-mass, L-length, T-time):
Tm - To temperature difference across the Kthermal boundary layer
+ thermal diffusivity L2 T-1
t age of the lithosphere T
k thermal conductivity M L T-3 K-1
( coefficient of thermal expansion K-1
"m mantle density M L-3
"w seawater density M L-3
(Temperature, labelled K above for convenience, is actually a dimensionlessquantity)
7 Sept. 10, 1990
a) What are the dimensions of heat flow? (i.e. Mi Lj Tk where i, j, k are exponentsto be supplied). Write the general expression relating heat flow to temperature inthe Earth.
b) According to the boundary layer cooling model, the surface heat flow is
proportional to (*+t)-1/2. Add some the above parameters to this function todevelop a more complete formula for the surface heat flow as a function of age.Show that it has the proper dimensions.
c) As the lithosphere cools, it contracts and increases in density causing thermalsubsidence. Assuming local isostatic compensation, thermal subsidence is
proportional to (+t)1/2. Develop a nontrivial and dimensionally correct formula forthe seafloor depth as a function of time. How would the rate of thermal subsidence
change if + is increased? Why?
d) What other observables are affected by lithospheric cooling?
9. Explain carefully what is meant by the power spectral density (PSD). Under whatcircumstances is it a useful concept? Describe how the PSD is estimated; how isyour method of estimation designed to avoid the pitfalls of the periodogram?
A marine survey in an area of well-developed lineations provides a map of themagnetic anomaly. The one-dimensional power spectral density (PSD) of theanomaly is estimated from profiles normal to the strike of the lineations; the resultis shown in the figure (note that both axes are linear, not log).
a) Explain the general features of this figure and give an equation that roughlydescribes the PSD.
b) What are suitable units for PSD in this case?
c) Supply numerical values for the axes of the figure appropriate for a typicalmarine situation. Explain how you get these numbers.
d) Sketch the one-dimensional PSD that results from analysis of profiles parallel tothe strike.
8 Sept. 10, 1990
10. The special creationists (A fundamentalist religious coalition, some of whom havePhD’s in science) claim that the earth is only 10,000 years old. One of theirarguments is that the electrical conductivity of the core is estimated (from quantum
theory and high pressure experiments) to be about 3 , 105 mho/meter. The electriccurrents in a a solid sphere with this conductivity and the radius of the earth’s corecan maintain only exponentially decaying magnetic fields whose mean lives areshorter than 15,000 years. If the earth were very much older than that, its magneticfield would once have been very much larger than the present field. The rockscontain no evidence for such large paleofields. Discuss this argument as thoroughlyand undogmatically as you can (in less than one page!).
IGPP
Departmental
Examination
1994
Departmental Examination, 1994
This is a 4 hour exam with 12 questions. Write on the pages provided, and continue ifnecessary onto further sheets. Please identify yourself clearly, and number the pagesunambiguously.
You have on average 20 minutes per question, although some will take longer thanothers. Do not spend too long on any single question! Attempt to answer whatever youcan; credit will be given for partial answers.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
1) On the attached map, Figure 1, label 8 plates, 6 spreading ridges, and 4 trenches.(Use names from the following list and add the appropriate type to each name (e.g.,plate, ridge, or trench).
African NazcaAleutian North AmericanAntarctic Pacific
Central Indian Pacific-AntarcticEast Pacific Peru-Chile
Eurasian South AmericanIndian or Indo-Australian Southeast Indian
Java Southwest IndianMid Atlantic Tonga-Kermadec
a. Which of the six spreading ridges has the highest spreading rate?
b. Which has the lowest spreading rate?
c. Give a typical value for full spreading rate and provide units (i.e. within the rangeof observed spreading rates).
d. Choose an example of a ridge, a transform fault, and a trench and sketch on themap the fault-plane solution (‘‘beach ball’’) you would expect to observe at each one.
e. Given the geometry of the Triple Junction shown in Figure 2, determine the direc-tion and rate of motion between plate A and plate B. What kind of plate boundary isA-B? The boundaries labeled ! !"#$ %"!&'()"*$ (!+,%'$ !&-$ !"#$ ',.//.&0$ !%$ 12$ **34#!"$ 5.%6
".06%7,!%#"!,$*)%.)&8
- 2 -
2) For the Northridge earthquake (January 17th, 1994) the unit moment tensor is wellapproximated by
M "
!""# 0
0$1
000
100 %""&
.
The epicentral coordinate system is x!1 North; x!2 West; x!3 Up. Determine the unit nor-mal vectors and the slip vectors.
What geophysical data would you use to resolve the ambiguity between the fault planeand the auxiliary plane? What is the fault plane for the Northridge earthquake?
- 3 -
3) a. Illustrate with figures why we commonly refer to a layered oceanic crust. Whatare the geophysical properties and thicknesses of these layers? How do the layersrelate to the ‘‘geological structure’’ of oceanic crust?
b. How do the geophysical and geological structures of the crust at fast and slowspreading ridges differ from each other? Illustrate with diagrams if possible.
c. There are structural/morphological and magnetic features that allow the directionand rate of plate motions across an oceanic spreading center to be determined relativeto an Euler pole. What are these features and how are they used?
d. An attempt has been made to establish an absolute reference frame for platemotions. What features are commonly used in this context and are they truly fixedrelative to each other or the mantle?
- 4 -
4) Explain the meaning of the following terms: (a) geopotential; (b) reference ellipsoid;(c) geoid; (d) geoid height. Give an approximate relationship between geoid height andthe geopotential on the reference ellipsoid. Estimate the error in this formula.
One frequently reads about such things as the gravity field of spherical harmonicdegree 4, meaning the field that results when only the degree 4 spherical harmonicsare summed, while the rest are dropped. Since the spherical harmonic coefficients allvary with the orientation of the reference axes, such functions are seemingly coordi-nate dependent. Show how to write the degree-l function on the surface of the unitsphere as a single integral of the original function. Hence show the degree-l functiondoes not depend on the choice of axes.
- 5 -
5) Explain what is meant by the term power spectral density (PSD) function. When isit a useful concept?
What techniques are used to make power spectral density estimates consistent? Whatcauses bias in a power spectral density estimate? How can it be reduced?
Sketch the autocorrelation functions and power spectra you would expect to be associ-ated with the time series shown in Figure 3. Don’t forget to label the axes.
At a site well away from the ocean, the ground motion PSD is flat in acceleration from1 Hz to 10 Hz at a level of 10$14 (m s$2)2 Hz$1. What is the root-mean square (RMS)acceleration of the ground in this band? What is the associated RMS ground ampli-tude?
- 6 -
6) Two well-known statistical estimation techniques are least squares and maximumlikelihood estimation. Explain the principle on which each is based and how theydiffer.
You are given a set of measurements of temperature, pi , at various depths,xi , i " 1, . . . , n , in a borehole. p is believed to increase linearly with depth. Find theleast squares estimates of the parameters in such a model and the variance in yourparameter estimates. State any assumptions you find it necessary to make. How wouldyou determine whether the linear model is suitable for your data?
- 7 -
7) Give all the evidence you know for the claim that most of the magnetic fieldobserved at the surface of the earth has its sources in a liquid core and that the corehas a radius about half the earth’s radius. Where does the rest of the field come fromand how do we know this?
How is the North magnetic pole defined? How is the virtual geomagnetic pole (VGP) ofpaleomagnetism defined? What is the equation for the magnetic field of a dipole?Explain briefly how a VGP position is calculated and what observations are needed inthis calculation.
- 8 -
8) What is Fermat’s Principle as applied to pulse propagation in acoustics andseismology?
Suppose that a region of the earth is surrounded by boreholes (not necessarily verti-cally aligned) and the free surface. How would you use Fermat’s principle to conduct aseismic tomography experiment that utilizes both the boreholes and the free surface?
- 9 -
9) A seismic experiment results in the P -wave travel time curve shown in Figure 4.
a. Sketch a P -velocity vs. depth profile which would result in the observed travel timecurve.
b. Sketch a '(p ) curve for this travel time curve.
c. Indicate on the travel time curve where the largest amplitudes should be expected.Consider a homogeneous half-space with P -velocity (#"4 km/s and S-velocity )#"2.5km/s, and an upgoing SV wave incident upon the free surface.
d. For what range of ray parameters will a reflected P -wave "#$ 9#$/")-+:#-;
A vertically-traveling upgoing P -wave with amplitude Ai encounters the velocity vs.depth profile in the near surface shown in Figure 5. The density is constantthroughout the profile.
e. What would be the observed amplitude As at the surface for (i) the high-frequencylimit, and (ii) the low-frequency limit?
- 10 -
10) The constitutive relationship for a perfectly elastic material can be written
'i j "Ci jkl *kl
where '#' is the stress tensor, *#* is the strain tensor and C is a fourth order tensor ofelastic moduli.
a. Why is '#' symmetric?
b. Why is *#* symmetric?
c. Out of a possible 81 coefficients in C, how many are independent? (You should makethe usual assumption < what is the usual assumption?)
d. For an isotropic material, how many independent elements of C are there and why?
e. For a transversely isotropic material, how many independent elements of C arethere and why?
f. Why are reference Earth models often assumed to be transversely isotropic (at leastin solid regions!)?
g. How would you modify the constitutive relationship to include anelasticity?
h. Is it possible for an anelastic solid to exhibit attenuation without physical disper-sion? Explain.
- 11 -
11) Use any appropriate numerical method you like to determine which formula (a),(b), (c) goes with which number (A), (B), (C). You need only get a close enough approx-imation in each case to eliminate two of the three choices < a two- or three-digit resultis quite sufficient. No points for guessing, so give your reasoning.
(a)n"2+,
($1)n ln n (A) 0.239087
Find d f (x -)!dx where f (x -)" 0 in
(b) f (x ) " (x $ cosx )!7 (B) 0.261799
(c)$,.,
4!(64 $ x6)dx (C) 0.225791
- 12 -
12) Devise a suitable question for this examination that has not been asked and pro-vide an answer for it.
IGPP
Departmental
Examination
1995
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IGPP
Departmental
Examination
1996-97
Departmental Exam 1996–1997
NAME:
RELAX! This is a closed-book 4 hour exam with 15 questions. This averages to about 16 minutes per
question although some will take longer than others (the first 10 are thought to be easier than the last
5!). Do not spend too long on any single question! You are not expected to be able to answer all of the
questions but please attempt to answer whatever you can. Credit will be given for partial answers.
Brevity is valuable, but not to the point of obscurity. Correct answers matter, but so does evidence
that you know what you are doing. If you get results that are dimensionally wrong, or clearly off by
orders of magnitude, show that you see this, even if you cannot fix the problem.
1) Explain briefly what is meant by the following terms used in geophysics. Give rough orders of
magnitudes of quantities where appropriate.
a) Chandler wobble (period?)
b) Bouger anomaly (typical size?)
c) Westward drift (in geomagnetism) (speed?)
d) Rayleigh wave (speed?)
e) Mechanism of an earthquake (moment? magnitude?)
f) Absolute plate motion (speed?)
2a) Explain what an Euler pole (pole of rotation) is in plate tectonics. How is the location and rotation
rate of the pole determined? Why does specifying the location of this pole, and the rate of rotation,
adequately describe plate motion? Is the pole of rotation fixed? Why or why not?
b) The pole of rotation for the Pacific North-American plates is in eastern Canada; the rate of rotation
is 0.8 degrees per million years. Calculate, at least roughly, the plate motion expected here in San
Diego.
3) State briefly in words: a) Rayleigh’s principle (as applied to standing waves in seismology) and
b) Fermat’s principle (as applied to body-wave seismology). Why are these variational principles
important?
By using Fermat’s principle, show that Snell’s law holds for a ray transmitted through a planar
interface.
4) Use index notation to represent the following, assuming that the deformation is very small, thematerial
behaves isotropically and there is no time-dependent behavior:
a) the stress tensor (!ij) for the case that in each of the three coordinate directions the stress equals the
lithostatic overburden (P )b) the strain tensor in terms of displacement (u)c) the volumetric dilation (! or dilatation)
d) the stress tensor in terms of dilatation, strain tensor and the elastic Lame coefficients.
If a spherical volume rapidly rises from depth to the surface it will dilate. If no external forces act on
the volume at the surface, the principal stresses correspond to the coordinate axis directions:
!rr = (" + 2µ)#rr; !!! = !"" = "#rr;
Recall that the Lame parameters can be written in terms of Youngs Modulus (E) and Poisson’s ratio ($):
1
" =$E
(1 + $)(1 ! 2$); µ =
E
2(1 + $)
Discuss the ratio of the tangential stress to the radial stress for an incompressible material ($ = 0.5) vsa compressible material ($ = 0.3).
5) Explain how you could write an algorithm to simulate measurements of a geophysical time series
whose power spectrum h(f) falls off like 1/f2, where f is frequency.
6) Sketch the “beach ball” P -wave fault plane solutions for slip on a fault striking N40!W and dipping
70! if the fault is a) right-lateral strike slip, b) thrust, and c) normal? Give examples of the tectonicenvironments in which you would expect to find each type of earthquake.
.
7) What are the average continental and oceanic heat flows? Which is higher and why? Estimate the
temperature gradient near the surface (take the thermal conductivity to be 4 Wm"1 K"1). How far into
the Earth can you extrapolate temperature using this surface gradient? Explain your answer.
8) For the surface described by:
x2
a2+
y2
b2+
z2
c2= 1
where a, b and c are constants, find the unit normal vector n̂ and find " · n̂.
9) What is a biased estimator? What is an inconsistent estimator? Give an example of each.
You are given a set of measurements of temperature, Ti, at various depths, di , i = 1, . . . , n, in aborehole. You are told that T is supposed to increase linearly with depth. Find an unbiased, consistentestimate of the parameters in the model and their associated uncertainties. State any assumptions you
find it necessary to make. How would you determine whether the linear model is suitable for your data?
10) What is the frozen flux approximation in geomagnetism? Why is it useful and when is it likely to
be valid?
11a) The spherical harmonic expansion of themagnetic field in the Earth’s atmosphere can be partitioned
into internal and external parts. Write down the equation for the geomagnetic potential showing this,
and give the physical interpretation of the two parts.
b) Observations of the magnetic field vector are made all over the Earth’s surface, which you may take
to be a sphere. Show how the internal and external parts of the field can be determined from such
measurements.
c) Is there an internal part of the geomagnetic field that changes with a period of one day? If so, what
is its magnitude approximately, and what causes it?
12) Discuss what is meant by the term resolution in the solution of an inverse problem. Describe how
the idea can be made quantitative in the setting of a linear inverse problem for an unknown parameter
varying in only one spatial dimension.
Briefly describe how the simple linear 1-D approach must be amended to tackle nonlinear problems
and those involving an unknown property distributed in two or more spatial dimensions.
2
13) Oceanic lithosphere is created during seafloor spreading. At the spreading ridge axis, hot mantle
rock (temperature Tm) comes in contact with cold seawater (temperature T0) causing it to cool and
strengthen. As this young lithosphere is carried away from the spreading axis it continues to cool by
diffusion of heat. Use dimensional analysis to arrive at approximate formulae for the variations in surface
heat flow and seafloor depth. Don’t worry about factors of 2, %, etc. The relevant physical parametersare:
(DimensionsM -mass, L-length, & -time)
(Tm ! T0)/Tm temp. diff. across the thermal boundary layer
' thermal diffusivity L2&"1
t age of the lithosphere &k thermal conductivity ML &"3K"1
( coefficient of thermal expansion K"1
)m mantle density ML"3
)w seawater density ML"3
a) What are the dimensions of heat flow (i.e.,M iLj&k where i, j, k are exponents to be supplied). Writethe general expression relating heat flow to temperature in the Earth.
b) According to the boundary layer cooling model, the surface heat flow is inversely proportional to#%'t. Combine the relevant physical parameters in a formula for the Earth’s surface heat flow as a
function of age. Check that dimensions agree with part (a).
c) As the lithosphere cools, it contracts, increases in density and the plate subsides. Develop a formula
relating seafloor depth to plate age by first writing down the relationship between density and temper-
ature [)(x, z) = f(T (x, z))]. Now, assuming local isostatic compensation, relate a mass column atthe axis to one off-axis at position x. For the boundary layer cooling model, the amount of subsidenceis proportional to
#'t. Use this fact and your mass balance to obtain your final formula. How would
the rate of thermal subsidence change if ' is increased? Why?
d) What other observables are affected by plate cooling?
14) Figure 1 plots the time elapsed since Earth’s magnetic field previously reversed its polarity at the time
of occurrence of each reversal. One model for the time of occurrence of geomagnetic reversals (during
which the field reverses its polarity) supposes that they may be regarded as a stochastic process, in which
the lengths t of successive polarity intervals are independent, and identically distributed, according tothe following probability density function
f(t) = "e"#t for all t $ 0
a) What is the mean time between reversals for such a model? What is the average reversal rate for such
a process? What is the variance in the time between reversals?
b) There are 194 reversals in the time span illustrated. Derive the maximum likelihood estimate for the
reversal rate assuming that these are a realization of a process of the kind described. What is the
standard deviation in your estimate?
c) Write down the autocovariance function for such a process. What would the power spectrum of
interval lengths look like, and what are its units? Is the density function given above likely to be a
good model for the data illustrated below? Justify your decision.
3
15) A seismic experiment results in a P -wave travel time curve which looks like:
a) Sketch a P -velocity vs. depth profile which would result in the observed travel time curve.b) Sketch a &(p) curve for this travel time curve.c) Indicate on the travel time curve where the largest amplitudes should be expected.
Consider a homogeneous half-space with P -velocity ( = 4 km/s and S-velocity * = 2.5 km/s, and anupgoing SV wave incident upon the free surface.
d) For what range of ray parameters will a reflected P -wave not be produced?
A vertically-travelling upgoing P -wave with amplitude Ai encounters the following velocity vs. depth
profile in the near surface. The density is constant throughout the profile.
e) What would be the observed amplitude As at the surface for (i) the high-frequency limit, and (ii) the
low-frequency limit?
4
IGPP
Departmental
Examination
2000
Dept Exam 2000
1. a) Sketch the major plate boundaries on the map and indicate the type of relative
motion at representative locations (divergent, transform, convergent). Write the names of
the major plates on the map and put a ‘C’ in the general region of large, stable cratons.
b) Briefly describe the fundamental differences between oceanic and continental plates
(composition, thickness, age, style of accretion).
2. What are the geomagnetic coordinates of a station at the geographic position (!,")
(colatitude, longitude) if the geomagnetic pole is at (!0,"0)?
3. Given only that a fluid globe satisfies an equation of state of the form P=P(T,V), where
P is pressure, T is temperature and V is specific volume (the reciprocal of density, #), and
that the equation of hydrostatic equilibrium is V gradP + g = 0, where g is the
acceleration of gravity, show that surfaces of equal specific volume (or density), pressure
and temperature all coincide (Hint: take the curl of the equilibrium equation to obtain
information on the directions of the gradients of # and P).
4. How can potential theory be used to derive seafloor topography from satellite altimetry
of the seasurface? Include in your discussion: what is being measured, the governing
equations and relevant boundary conditions.
5. Explain seismic phase velocity and group velocity. You may find the concept of
stationary phase useful in your explanation. Can phase velocity be negative? Can group
velocity be negative?
6. a) The Fourier spectrum of the time series a(t) is A($) and of b(t) is B($). The
convolution c(t) = a(t) * b(t). What is the Fourier spectrum C($) of c(t)?
b) The product p(t)=a(t)b(t). What is the Fourier spectrum P($) of p(t)?
7. a) The temperature gradient near the surface of the continents is about 30°C/km. If this
gradient was maintained throughout the lithosphere what would be implied about the
state of typical rock materials at the base of the plate? What can you deduce about how
the thermal gradient varies with depth throughout the lithosphere? What does this suggest
about the distribution of heat sources within the continental lithosphere?
b) What are the dominant modes of heat transfer in the Earth? Discuss the relative
efficiency of each mode in the different layers of the Earth (e.g. core, mantle, crust).
c) Consider the cooling of an initially molten Earth. Where will solidification begin if the
curve of melting point versus depth is (i) superadiabatic (ii) subadiabatic.
8. Show that for a uniform sphere of radius a, in which the P-wave velocity is %, the
travel-time curve is given by T=(2a/%) sin(&/2), where & is the epicentral distance.
9. Show that for a flat earth, in which the velocity increases linearly with depth, the
seismic rays are arcs of circles.
10. The central Atlantic displays an essentially continuous record of the magnetic field
polarity epochs throughout the formation of oceanic crust of the basin. Between the Mid-
Atlantic Ridge and the East Coast of the U.S. magnetic anomalies 1-34 and ‘M-series’ 1-
25 have been identified. The oldest anomaly (M25) is estimated to correspond to crust
formed about 160 Myr ago. a) How might this date have been obtained? b) Assuming
each anomaly number indicates paired normal and reverse epochs, make a rough estimate
of the duration of an ‘average’ magnetic epoch. c) You want to determine the period of
time it takes for a geomagnetic reversal to occur. How might you do this?
11. A deep sedimentary basin of great horizontal extent is filled with shale. At the
surface, the density of the shale is 2200 kg/m3. The density increases exponentially to a
value of 2700 kg/m3 at a depth of 5 km and then remains constant for greater depths.
What gravity anomaly would be observed at the center of the basin (very far from its
boundaries)? If the depth of this basin had been estimated from an observed gravity
anomaly with the assumption that the near-surface density (2200 kg/m3) was valid at all
depths how would that estimate compare to the actual depth (H)?
12. An earthquake at the North Pole is recorded by a very sensitive gravity meter at the
South Pole. Assume that the source time function of the earthquake is a delta function.
How long after the earthquake will the recording begin?
IGPP
Departmental
Examination
2001 (joint)
Name:
Departmental Exam: June 2001
start time:
end time:
This is a closed-book 3 1/2 hour exam. You may use a calculator but do not consult any notes,
books, or colleagues for help. The exam has two sections of 7 questions each for a total of 14
questions. Some questions can be done quickly. The questions are not arranges in a special
order so look for the easy ones first and please scan both sections.
Correct answers matter but so does evidence that you know what you are doing. If you get
results that are dimensionally wrong, or clearly off by orders of magnitude, show that you see
the problem even if you cannot fix the problem.
Section 1 - Joint Track
(Seismology and Geodynamics)
1. (a) Give several definitions for the ray parameter p.
(b) Give approximate values for oceanic and continental crustal thicknesses.
(c) About how fast is long term motion along the San Andreas fault?
(d) Arrange the following seismic wave types from fastest to slowest: S, P, Love, Rayleigh
(e) Sketch and label example focal mechanisms (“beach balls”) for a strike-slip fault, a
normal fault, and a reverse fault. Which of these mechanisms best describes the 1999 Hector
Mine earthquake?
2
(f) For a spherical wave front in a homogeneous material, how does the amplitude of the
wave vary as a function of the radius?
2. Consider a simple velocity model consisting of:
LAYER 1: a surface 2-km thick layer with a P velocity of 4 km/s
LAYER 2: a 4-km thick layer with a P velocity of 6 km/s
LAYER 3: a half-space with a P velocity of 8 km/s
(a) Sketch approximate T(X) and X(p) curves for surface-to-surface P waves that travel
through this model. Label the axes with appropriate units. Label the prograde and retrograde
branches of the T(X) curve.
(b) What is the vertical two-way travel time for a P-wave that bounces off the top of Layer 3?
(c) Using the approximation that the P-wave velocity is 1.7 times faster than the S-wave
velocity, how long would it take an S wave to travel 10 km (surface-to-surface horizontal
distance), assuming that it reflects off the top of Layer 2?
3
3. Given the rotation pole between the African and South American plates (pole; latitude=62.5°,
longitude=320.6°, rate= 5.58 x 10-9 radian/yr), calculate the spreading rate at a point on the
northern Mid-Atlantic Ridge (lat= 30°, lon= 319°). (provide relevant equations)
4. (a) On the attached map label 8 plates, 6 spreading ridges, and 4 trenches. Use names from
the following list and add the appropriate type to each name (e.g., plate, ridge or trench).
African Nazca
Aleutian North American
Antarctic Pacific
Central Indian Pacific-Antarctic
East Pacific Peru-Chile
Eurasian South American
Indian or Indo-Australian Southeast Indian
Java Southwest Indian
Mid Atlantic Tonga-Kermadec
(b) Which of the 6 spreading ridges has the highest spreading rate?
(c) Which has the lowest spreading rate?
(d) Give a typical value for full spreading rate and provide units (i.e. within the range of
observed spreading rates).
4
5
5. A sill is injected at the base of the oceanic crust. Approximately how long will it take for
the heat pulse to reach the top of the crust. (Assume heat diffusion with a thermal diffusivity
! of 10-6 m2s-1). (order of magnitude calculation)
6. Given a typical oceanic crustal thickness and a typical ocean depth of 4 km, estimate the
thickness of the continental crust at sea level and beneath the Tibetan Plateau (5 km
elevation). Assume a crustal density of 2900 kg m-3 and a mantle density of 3200 kg m-3.
Are these reasonable thicknesses for continental crust? Briefly describe other means for
measuring crustal thickness.
7. (a) Write the expression for the moment tensor of 1) a double-couple, 2) an implosion.
6
(b) What is the expression of the scalar moment of an earthquake (in terms of slip, area and
shear modulus) and how is the scalar moment related to the magnitude (moment-magnitute
relation).
(c) Earth's heat flow is about 25.5 Tw (1 Tw = 1012 watts). What are the sources of the heat?
What is the contribution of earthquakes to the heat flow? (Hint - a magnitude 8.3 earthquake
releases about 1.6 x 1018 Joules.)
IGPP
Departmental
Examination
2001 (geoph)
1. A gravimeter is placed on the floor of the North Sea and records the acceleration of
gravity every second for several days resulting in a time series, g1, g2, . . . gn. The
object of the experiment is to measure a value of g, the gravitational acceleration at
the site. The experimentalists make an estimate of the power spectral density from
their data.
(a) Will there be periodic components in the gravity signal? If so, how would they
appear in the spectrum and what is the best way to correct the data for them?
(b) Is there likely to be a continuous spectrum in addition to periodic components? If
so what are its causes and what would the spectrum look like?
(c) How can you calculate the covariance matrix of the noise in the gravity signal
from the power spectrum? Explain very briefly how you would use the covariance to
make a good estimate of the mean and variance of g? What statistical assumptions
are necessary to make the analysis you suggest and how would one go about verifying
them?
IGPP
Departmental
Examination
2002 (required)
1
Name:
Departmental Exam: June 2002
Please spend 15 minutes reading the exam before entering the start time.
start time:
end time:
This is a closed-book 3 1/2 hour exam. You may use a calculator but do not consult any
notes, books, or colleagues for help. The exam has two sections: geophysics and other.
Correct answers matter but so does evidence that you know what you are doing. If you
get results that are dimensionally wrong, or clearly off by orders of magnitude, show that
you see the problem even if you cannot fix the problem.
Departmental Exam: Geophysics Section – June 2001
ATTEMPT ALL QUESTIONS IN THIS SECTION
1 (a) Evaluate the fourier transform of the boxcar function
!(x) =1 x < 1/ 2
0 x > 1/ 2
" #
$
where
the forward transform is defined as
F(k) = f (x)e! i2"kx
dx!#
#
$ .
(b) Plot the result. What happens at k approaches zero and k approaches infinity.
2 (a) Briefly explain the processes of upward and downward continuation of a harmonicfunction. Explain how this is done when you are given a spherical harmonicexpansion of the Earth’s geomagnetic field.
(b) The observed geomagnetic field at the Earth’s surface is commonly downwardcontinued to the surface of the core, but there is a fundamental restriction on theresolvable length scales there. Why? What is smallest resolvable scale? Why is itpossible to downward continue the magnetic field to the core, but not possible for theEarth’s gravity field?
(c) A lineated magnetic anomaly at the surface of the ocean is found to have awavelength of about 100 km and an amplitude of 500 nT. What is the approximateamplitude of the anomaly on the seafloor, 4 km below, and what is its amplitude atthe altitude of Magsat, 400 km above the surface?
3 Consider the low-order spherical harmonic expansion for the Earth’s gravitationalpotential, V. For each term in the list below, draw a sketch of the shape of thecorresponding geoid perturbation from a sphere.
(a) Degree zero:
Yo
o
. What property of the Earth does this harmonic measure?
2
(b) Degree one:
Y1
m
. The coefficients of these spherical harmonics are all exactlyzero. Why?
(c) Degree two:
Y2
o
. The (fully normalized) coefficient of this harmonic is nearly 300
times larger than any other besides that for l =0. Explain why.
(d) Degree two.
Y2
!1, Y
2
1. These two coefficients are almost exactly zero, and too small
to measure. Why are they small?
(e) Degree two:
Y2
!2, Y
2
2. Sketch the shape of the geoid in the equatorial plane for
which these terms are responsible. Draw the two principal axes of inertia, A and B on
your sketch.
4 (a) Define the autocorrelation function for a stochastic process.
(b) What is the relationship between autocorrelation and power spectral density?
(c) Sketch the autocorrelation functions and power spectra you would expect to be
associated with the following time series (5000 data points). Don't forget to label the
axes. What is the approximate value for the peak in the autocorrelation function.
3
(d) What is the most appropriate computer language(s) for the following task?
i) You have a data series of 10,000 samples and need to compute mean,
variance, trends and spectra.
ii) You are managing a geophysical network delivering 20 Hz data from 3-
channel sensors at 15 stations. You wish to publish hourly means and
variances on the web.
5 A vertically propagating upgoing plane wave is incident on a sediment-basement
interface where the seismic velocity drops from 4 km/s to 2 km/s. A cylindrically
shaped depression (with radius of curvature r = 1 km) focuses (approximately) the ray
paths to a focus point within the sediments. Solve for the height of the focus point
above the lowest point of the interface.
4
6 Assume there is a mantle plume directly under the island of Hawaii. As the Pacific
plate passes over plume, the temperature at the base of the plate is suddenly
increased.
(a) Given the plate thickness of 60 km and a thermal diffusivity
! = 10"6
m2s
-1,
approximately how long it will take for the heat flow anomaly to appear on the
surface.
(b) The velocity of the Pacific plate relative to the plume is 100 mm/yr. Design
(sketch) an experiment to detect the anomalous heat flow caused by the plume.
Where would you go and what would you do? Do you see any problems performing
this experiment?
7 Determine the value of the following integral using the method of complex contour
integration.
x sin(x)
1+ x4
0
!
" dx
(a) What is an appropriate integral to consider in the complex plane?
(b) Choose a suitable contour and explain the reason(s) for your choice.
(c) Use the Cauchy Residue Theorem to evaluate the contour integral and indicate the
value that results for the original integral.
(Hint: What are some of the important properties of the numerator and
denominator?)
8 You are given a set of temperature measurements Ti at depths zi, i=1, . . . , N, in a
borehole and you want to estimate the temperature gradient. Assume a linear
variation in temperature with depth and initially assume there are no errors in the
depth measurements. The temperature measurements have errors with a constant
standard deviation.
(a)Derive the least-squares solution for the model parameters.
(b) How would you modify this approach if the standard deviations in the
temperature measurements were not all the same value? (How does this change
!2
?)
(c) Assume now that there are no errors in the temperature measurements but that
there are errors in the depth measurements. How would you solve for the model
parameters in this case? Would you get the same answer for the temperature
gradient as you estimated from part (a)?
5
9 You make broad-band recordings of body waves from an earthquake at a variety of
distances from the source along a great-circle path. The ratio of short period energy
to long period energy decreases with increasing epicentral distance - why?
10 For an isotropic elastic solid, the relationship between the applied stress Tij, and the
resulting strain eij, is given by Hooke's law,
eij =1
E1+ !( )Tij " !#ijTkk[ ]
where E is the Young modulus, ! is the Poisson ratio, and
!ij is the Kroneker delta.
(a) What is a typical value of the Poisson ratio for rocks.
(b) Suppose an elastic solid undergoes a change in pressure with no shear stress.
Relate e11 to the change in pressure. (compression is negative)
(c) What happens when ! approaches a value of 0.5?
(b) Derive the inverse relationship expressing stress in terms of strain.
11 (a) Express the P-wave velocity, Vp in terms of the elastic moduli and the density ".
(b) Given the relationship in part (a), explain why seismic rays curve upward in the
mantle.
(c) Imagine a solid, homogeneous, isotropic planet where the material is perfectly
Hookean (i.e., the elastic moduli do not vary with radius). Assume that the planet
compresses from self gravitation. Sketch how seismic waves would travel through
such a planet.
IGPP
Departmental
Examination
2003
Geophysics Departmental Exam (Written) June 2003
RELAX! This is a closed book 4 hour exam with 16 questions. This averages to 15 minutes a question, butsome questions are easier than others. Attempt all the questions if you can, but understand that it is unlikelythat you will be able to answer all of them – do your best.
Show enough of your working to indicate you know what you are doing. Don’t forget to include units. Ifyour answer is obviously wrong, indicate that you are aware of this even if you can’t fix the problem.
1)What is your name?
2) Please provide short answers to the following questions:
a. Name the five most abundant elements on Earth.
b. What is the age of the oldest part of the ocean floor?
c. What is the currently accepted age of the Earth?
d. What is the age of the core?
e. Name 3 radioisotope systems used to study the evolution of the Earth.
f. When was the last ice age?
g. What are Milankovitch cycles and how are they controlled?
h. What are the driving forces for plate tectonics?
i. What is the Nyquist frequency for 200 Hz data sampling? What does this mean?
j. When was the most recent geomagnetic reversal?
k. What is the radius of the Earth? Core?
l. What is the average p-wave velocity in the upper mantle? S-wave?
m. What is the mean ocean depth?
n. What type of earthquake focal mechanism would you expect at subduction zone? Transformfault?
3) Consider the motion of the Southern Chile triple junction, shown below, in which South America is
1
overriding the Chile Ridge. The triple junction alternates between a trench-ridge-trench and a trench-transform-trench configuration. Assume that the rate of spreading on the Chile Ridge (Antarctic-Nazca)is 60 mm/yr and that the Nazca and South American plates are converging in an east-west direction at 80mm/yr.
a) Determine, by drawing the velocity triangle, the motion of Antarctica with respect to South America.
b) Determine, by including velocity lines, the motion of the triple junction relative to South America whenit is in the trench-ridge-trench configuration shown in the sketch.
c) Determine the motion of the triple junction when the triple junction is in a trench-transform-trenchconfiguration.
80 mm/yr
South
America
Nazca
Antarctica
60 mm/yr
4)What is the approximate value of the geomagnetic field at the north pole, in nanoteslas? Under the axialdipole approximation, what is the corresponding dipole moment of the Earth? Calculate the magnitude ofthe field at the equator: at the Earth’s surface; at the altitude of the satellite Oersted; at the surface of theouter core. Justify your method. You will need to supply numerical values for the various distances.
At the ocean surface near the equator a marine magnetic anomaly is observed with wavelength of 50 km andamplitude of 500 nT. Where are the sources for this field? Compute the corresponding field at the seafloor,4 km below. Compute the rough magnitude of this anomaly field at satellite altitude, and at the surface ofthe core.
5) Write down the equation for the spherical harmonic expansion of a gravitational potential with interiorsources.
Suppose the geoid height is known everywhere at the surface of the reference ellipsoid surface, a sphere forour purposes. How are the coefficients in the spherical harmonic expansion of gravity computed from this
2
function?
Suppose instead the local vertical acceleration of gravity is known. How does one obtain the expansioncoefficients in this case?
Finally, suppose values of the acceleration are known at two hundred points distributed as evenly as possibleover the Earth. Explain how these observations can be used to estimate the expansion. What is the highestspherical harmonic degree at which the results of this last method are likely to be reliable?
6) Answer ONE of
(a) What are the mineral orientations you would look for to determine the refractive indices of a biaxialmineral, using index oils and a crushed sample? For each index of refraction give the grain orientation, theindices you can measure, and explain how the specific grain orientation can be recognized.
OR
(b) The Earth is divided into a core-mantle-crust system. How do we estimate the chemical composition ofthese major subdivisions?
Describe how a (normal) mid-ocean ridge basalt (MORB) differs from an ocean island basalt (OIB) in termsof either trace elements or isotope systematics.
7) What problem is solved by the method of Steepest Descents? Explain how the method works. Describebriefly the modification introduced by the method of Conjugate Gradients.
Show how a linear system of equations,Ax = b
where A ! "n#n and x, b ! "n can be solved by either Steepest Descents or Conjugate Gradients. Underwhat circumstances would this approach be advantageous over a standard numerical method, such as LUdecomposition or QR factorization?
8) Write down Fourier transforms for three of the following functions, and prove your result for one ofthem:
f1(x) = e$!x2
f2(x) =sin(!x)
!x
f3(x) = e$a|x|, a > 0
f4(x) =
!1, |x| % 1/20, |x| > 1/2
3
f5(x) =1
|x|1/2
.
All of the above functions are even in x. What does this imply about their Fourier transforms? What is thecorresponding property for odd functions?
9) State the Convolution Theorem for Fourier transforms in one variable. Is there an analogous result in twoand three independent variables? Why is the theorem useful for large scale numerical calculations?
A stationary stochastic process X(t) is smoothed by convolving it with a function "(t). How is the powerspectral density [PSD] of the smooth series, P"&X(f ), related to PX(f ), the PSD of X? A long time seriesis smoothed by taking the running average of N terms. Explain why is a bad idea.
No proofs are required in this question, just statements of well known results.
10) A 1-Hz harmonic SH plane wave is propagating within a whole space at 6 km/s in the x $ z plane asshown in the figure. The ray vector makes an angle of 30' to the x-axis.
(a) What are values of the horizontal and vertical slowness for such a wave?
(b) Write an equation for the displacement uy as a function of x, y, and t.
30°
x
z
wavefront
11) Name the 4 seismic phases plotted in the figure and label the three layers shown in the earth.
4
Source
12) A strike-slip earthquake in the Mojave desert has produced a surface rupture having length of 50 km,and average slip of 4 meters. Estimate
a) change in strain caused by the earthquake in the ambient crust (order of magnitude).
b) The corresponding static stress drop (for this, you will need to use a shear modulus of the crustal rocks; ifyou don’t know a typical value of the shear modulus on top of your head, try to infer it from other quantitiesyou may know, e.g., Vs and density of the crustal rocks).
c) The corresponding scalar seismic moment. Assume that the earthquake has ruptured the entire brittlelayer (use your best guess for the down-dip fault size).
13) Give your definition of viscosity (in as much mathematical detail as you deem necessary). What is atypical value (range of values) of viscosity of the upper mantle? What measurements were used to inferthese values? Define a time scale over which the mantle rocks will behave like a fluid.
14) It has been proposed that the entire lithosphere of Venus subducted and was replaced by hot mantle ma-terial 700 million years ago. Since then there has been no tectonic activity. Estimate the current lithospheric
5
thickness. The thermal diffusivity is 10$6 m s$2.
15) Explain the what is meant by an adiabatic temperature gradient. Suppose the temperature gradient isless than the adiabatic gradient. What happens when a parcel of material is displaced vertically – either upor down – from its initial position? Suppose the temperature gradient is greater than the adiabatic gradient.What happens when a parcel of material is displaced vertically – either up or down – from its initial position?
16) Explain all the evidence you know for the existence of a self-sustaining geodynamo in Earth’s core.What is the role of the inner core in sustaining the dynamo? How would you go about estimating the age ofthe geodynamo? What are the current estimates for this?
6
2004
Geophysics Departmental Exam: 2004Part 1
This section is 90 minutes, closed book, and consists of questions designed to test your knowledge of factsand figures in the geosciences. The focus will be on the five geophysics core courses, with additional ques-tions on other relevant areas. You are expected to answer all questions. It is better to be brief and to thepoint than to try to write a long answer, and better to admit ignorance than to be in error.
1. What is the evidence that much of this Earth’s core is fluid, and the rest solid? About howbig are these two parts? What are reasonable values for the Poisson’s ratio of each part?
2. How is a random variable described?
3. Suppose
A =
!""#
3 4 1
0 2 3
5 7 2
$%%&
b =
!""#
2
1
3
$%%&
Then, assuming the summation convention, what are Aii and A jib j?
4. Give two examples of seismic low velocity zones in the Earth.
5. What is the ray parameter of a seismic wave?
6. What is the principle of isostasy?
7. Why are elements with periodic number less than and equal to that of iron much more abun-dant than elements with higher periodic number?
8. Derive formulas for the acceleration of gravity and pressure inside a constant-density planet.Assume hydrostatic equilibrium and zero surface pressure.
9. What is the average thickness of (a) the continental and (b) oceanic Earth’s crust? What arethe thickest and thinnest values for each? Give an example of where these extremes occur.
10. Describe, briefly, earthquake magnitude and moment in terms of data that would be used toestimate them. How are they related?
11. How many material constants are needed to completely characterize static deformation of anisotropic elastic solid?
12. What is a FIR filter? Give one thing it can be used for.
13. Estimate the maximum shear stress that can be supported by crustal rocks at depth of 10 km(order of magnitude).
14. What is the expected value of a random variable? What do we mean when we say that a sta-tistical estimator is unbiased? What do we mean when we say that a statistical estimator isrobust?
15. What is the difference between the Earth’s crust and the lithosphere?
16. What is convolution? What is the convolution theorem?
17. What is a triple junction? What is the only type of one that is stable?
18 Calculate the Fourier transforms of the following functions:
(a) box(x) ' ()*
1
0
|x| + 12
otherwise
(b) sinc(x) 'sin ! x
! x
8 July 2004
-2-
19. Explain the meaning of the following terms: (a) geopotential; (b) reference ellipsoid; (c) geoid;(d) geoid height. Give an approximate relationship between geoid height and the geopoten-tial on the reference ellipsoid. Estimate the error in this formula.
Figure 20
20. Figure 20 shows a Q-Q plot of some data (NS distance between two GPS points). How wasthis produced? What does its shape tell you about the distribution of the data, and what doesthat distribution tell you about how best to estimate the ‘‘true’’ value of the data?
21 What is the Earth’s flattening? Why does it occur?
22. What are the main types of plate boundary? Give one geographically specific example of eachtype. What class of earthquake focal mechanism predominates on each type?
23. Suppose there is a large earthquake in a viscoelastic half space where the viscosity is 1021 Pa-s and the shear modulus is 6 ,1010 Pa. About how long will it take for the stress from theearthquake to decay to 1/e of the value just after the earthquake?
24. Define the Lowes spectrum of the geomagnetic field (sometimes also referred to as the Lowes-Mauersberger spectrum). Give a physical interpretation. From what kinds of measurementsis it derived, and how is it calculated?
25. Why is it permissible to downward continue the geomagnetic field down to the core, but nofurther, and what are the limitations of this calculation? Why is it not permissible to do thesame thing for the Earth’s gravitational field?
26. A time series of ocean wave height would be in meters. What would the units (or dimensions)be for the power spectral density of these data?
27. Most of the energy in ocean swell occurs at periods of 5-20 seconds. How rapidly should wesample records of swell to make a digital series?
28. Explain what is meant by an adiabatic temperature gradient. How does this relate to convec-tion in a fluid?
29. Write down a constitutive relationship between tesnor stress and strain for a Maxwell (serialspring and dashpot), and derive an expression for the Maxwell relaxation time (assume notemporal variations in strain).
8 July 2004
-3-
30. What is the difference between basalt and granite? Between granite and rhyolite? What pro-portions of the bedrock of the Earth’s surface is composed of each of these rock types? Arough but reasonable guess is fine.
31. Suppose a seafloor spreading rate is given as 30±3 mm/yr. What does the 3 mean? Whatdoes it mean to say that the 95% confidence limits for the rate are 25 and 35 mm/yr?
32. Define, briefly, what elastic, plastic, and viscous behavior are for materials. Give an examplefor the Earth of each kind of behavior.
Part 2This section is two hours, open book, and consists of problems that will require some amount of rea-soning and equation solving. Answer at least 8 problems of your choice. Even if you cannot solvethe equations, you will get credit for posing the problem correctly, and describing a path for getting asolution. This will test your ability to approach a research issue. While this portion is ‘‘open book’’, itmay not be an advantage to rely on this: books can be very distracting and time-wasting. It is betterto demonstrate an ability to think the problem through, perhaps from first principles. Rememberthat use of equations is often better (and briefer) than lots of words; but the equations must beappropriately explained to show why you have written them.
Figure A
A. The varying rotation rate of the Earth is reported as a time series of the change in the lengthof the day, -T(t). A century of such data were analyzed and the (one-sided) PSD estimated, asimplified version of which appears in the Figure.
(a) Give an interpretation of the power spectrum, and based upon it calculate the variance ofthe LOD record, and the standard error of the clocks used to measure -T . If the highest fre-quency in the graph is the Nyquist frequency, how many measurements were used to calcu-late the PSD? Estimate numerical quantities by eye from the figure.
(b) The day is slowly increasing in length at a rate of about 2 milliseconds per century. Wouldyou expect to be able to detect this signal in the record that was used to generate the PSD?Why is simple least-squares estimation a poor way to estimate the secular change in thiscase?
(c) From the given PSD of length of day changes find the power spectra of the Earth’s angularvelocity " , and of the angular acceleration d" /dt.
B. One frequently reads about such things as the gravity field of spherical harmonic degree 4,meaning the field that results when only the degree 4 spherical harmonics are summed,while the rest are dropped. Since the spherical harmonic coefficients all vary with the orien-tation of the reference axes, such ‘‘degree-n’’ functions are seemingly coordinate dependent.
8 July 2004
-4-
Write the degree-n function on the surface of the unit sphere as an integral of the originalfunction, and use this result to show that a degree-n function does not depend on the choiceof axes.
Figure C
C. Figure C shows the displacement for a harmonic plane wave at t = 0, traveling in the x direc-tion at 5 km/s. Write down an equation for this wave that describes displacement, u, as afunction of x and t. What is the maximum strain, and acceleration for this wave? Suppose ittravels 1000 km through a medium with Q of 100; what will its amplitude then be?
D. Within a spherical shell the magnetic field B can be written as the sum of two parts:B ' S+T, where S is called the poloidal part, and T is the toroidal part. Define these terms.What are the distinguishing features of the toroidal part? In which of the following regionswould you expect the magnetic field to possess a significant toroidal part: (a) The outer core;(b) the upper mantle; (c) the Ocean; (d) the atmosphere below 50 km; (e) the ionosphere.Explain your answers.
E. A downgoing P wave in a medium with a P velocity of 6 km/s travels through a ‘‘corner’’shaped structure as shown, with incident and exiting angles as shown in Figure E. Write aformula relating the various angles and the velocities. What is the P velocity within the cor-ner shaped medium? If the angle of incidence is 45°, what is the possible range of angles forthe outgoing wave, given an arbitrary velocity contrast.
F. Given the rotation pole between the African and South American plates (pole; latitude =62.5°, longitude = 320.6°, rate = 5. 58,10.9 radian/yr), calculate the spreading rate at a pointon the northern Mid-Atlantic Ridge (lat = 30°, long = 319°). Use vector methods, not spheri-cal trigonometry.
G. Assume the lithosphere of Venus has evolved to a steady-state temperature profile. Given acurrent heat flow of 4,10.2 W m.2, a surface temperature of 450°C, a mantle temperature of
1500°C and a thermal conductivity of 3. 3 W m.1 C.1 calculate the thickness of the lithosphere.
H. Abyssal hills, which form at seafloor spreading ridges, have a characteristic spacing of 4 kmand have peak-to-trough amplitude of 400 m. Assume they are infinitely long in the ridge-parallel direction. What is the amplitude of their gravity anomaly at the surface of the
8 July 2004
-5-
Figure E
ocean? What is the amplitude at the altitude of a satellite (400 km)?
I. Stress measurements in a borehole have revealed a north-sourth compression T NN , an east-west compression T EE, and a shear stress T NE on a vertical plane striking east-west. Assum-ing that the stress state is two-dimensional, find the magnitude of the principal stresses.
J. Determine the total heat output on the creeping section of the San Andreas Fault in Watts.Assume the shear stress varies with depth as # = f $ gz, with a coefficient of friction ( f ) of 0.6.The fault is 25 km deep, 100 km long, and creeping at 35 mm/yr. How would you find the pro-file of heat flow at the ground surface? What assumptions could you make to simplify findingthe solution? Be specific and use equations.
K. For the postglacial rebound problem, suppose we have an ice load whose depth is described byh(x1, x2,t): that is, it is a function of position and time (making the Earth flat). What would bethe equations needed to solve for the deformation of a uniform, self-gravitating Earth undersuch a load, assuming that the Earth is elastic? How does this change if it is Maxwellian?Most data on rebound are measurements of sea level, not absolute displacement. Why wouldthese differ, and how would you include this difference in the equations?
8 July 2004
IGPP
Departmental
Examination
2006
Departmental Exam 2006 - Part 1
RELAX! This section is 90 minutes and consists of questions designed to test your knowledge of factsand figures in the geosciences. The focus is on the geophysics core courses, with some additionalquestions to address other relevant areaas. You are expected to answer all questions. Credit will begiven for partial answers.
Brevity is valuable, but not to the point of obscurity. Correct answers matter, but so does evidencethat you know what you are doing. If you get results that are dimensionally wrong, or clearly o! byorders of magnitude, show that you see this, even if you cannot fix the problem.
1) What distinguishes continental and oceanic crust? What is a typical age for each and explain whythere is a di!erence.
2) Inside the Earth, there are several depths at which there are abrupt changes in properties. At whatdepths do these changes occur and what is the physical cause of each one?
3) What is the primary evidence that slabs subduct to at least 670km. Given that brittle failure canonly occur at shallow depths in the Earth, what might be the mechanism involved for deep earthquakes.
4) When an oceanic volcano is discovered, it is labelled either an island, a seamount, a guyot, or anatoll. What is the basic characteristic of each of these 4 types of oceanic volcanoes?
5) What is the geological significance of the K-T boundary and how long ago did this occur? Is itunique in Earth history?
6) What is the temperature at the center of the Earth? How do we know this?
7) Describe the di!erence between the group velocity and phase velocity of a surface wave. (You mayfind the concept of stationary phase useful in your explanation.) Which is typically faster?
8) Sketch the travel time curves for the first P -wave arrival and the first S-wave arrival as a functionof epicentral distance from 0! 180 deg. Explain the features you have drawn in terms of the e!ectsof the major structural divisions of the Earth.
9) Sketch a cross-section of the Earth indicating the inner core, the outer core and mantle. Sketch onyour diagram the ray paths for the following seismic phases: PKP , ScSScS, SP , SKKS, and sP .
10) Sketch focal mechanisms (beach balls) for a) an E-W striking right-lateral, vertical strike-slipfault b) a reverse fault on a fault plane striking N-S and dipping 45 degrees to the W c) a typicalearthquake on the SAF (San Andreas Fault) d) a typical earthquake in the Basin and Range e) atypical earthquake in the Andes subduction zone
11) What would happen if you drained the ocean and waited 100,000 years for the Earth to rebound?The density of seawater is 1000 kg m-3 and the density of the mantle is 3300 kg m-3.
1
12) The pole of rotation for the Pacific North-American plates is in eastern Canada; the rate of rotationis 0.8 degrees per million years. Calculate, at least roughly, the plate motion expected here in SanDiego.
13) Use index notation to represent the following, assuming that the deformation is very small, thematerial behaves isotropically and there is no time-dependent behavior: a) the stress tensor (!ij) forthe case that in each of the three coordinate directions the stress equals the lithostatic overburden(P ) b) the strain tensor in terms of displacement (u) c) the volumetric dilation (! or dilatation) d)the stress tensor in terms of dilatation, strain tensor and the elastic Lame coe"cients.
14) What geophysical evidence suppports the idea that the Earth’s magnetic field is being maintainedby dynamo action in the Earth’s core? Briefly describe how you think this is happening. What roledoes the inner core play in all of this?
15) Describe and justify the assumptions used when determining the paleomagnetic pole position fromthe declination and inclination of a remanent magnetisation. The measured declination and inclinationof the paleomagnetic field in Upper Triasic rocks at 41.5deg N and 72.7deg W are D = 18 deg andI = 12deg . Determine the paleomagnetic pole position.
16) An attempt has been made to establish an absolute reference frame for plate tectonics. Whatfeatures are commonly used in this context and are they truly fixed relative to each other or themantle?
17) What are the average continental and oceanic heat flows? Which is higher and why? Estimatethe temperature gradient near the surface (take the thermal conductivity to be 4 Wm!1 K!1). Howfar into the Earth can you extrapolate temperature using this surface gradient? Explain your answer.
18) For the surface described by:
x2
a2+
y2
b2+
z2
c2= 1
where a, b and c are constants, find the unit normal vector n̂ and find ! · n̂.
19) The Moon has a mean density which is 3/5 of the Earth’s mean density and a radius which is 1/5of the Earth radius. At what radius inside the Earth does the central pressure of the Moon occur?Give your answer as a percentage of the total radius of the Earth and you may assume that both theMoon and the Earth have uniform densities. How would your answer change if a more realistic densityprofile for the Earth were used?
20) Consider the triple junction sketched below where a ridge with an azimuth of 135deg is migratingalong a north-south trench. Plate B is being subducted beneath plate A at 5 cm/yr in the directionshown. Calculate the relative motion of plate C to plate A (velocity and direction) and determine the
2
rate and direction of migration of the triple junction relative to plate A.
21) Given the following parameters, develop characteristic times for the following processes.
! - thermal di!usivity (m2s!1) E - Young’s modulus (Pa) " - dynamic viscosity (Pa s) # - density (kgm!3) g - acceleration of gravity (m s!2) D - layer thickness (m) $ - wavelength of surface deformation(m)
a) heat di!usion - Describe this timescale in terms of an experiment or process. b) Maxwell viscoelasticrelaxation - Describe this timescale in terms of an experiment or process. c) glacial rebound viscosity- Describe this timescale in terms of an experiment or process.
22) Earth’s magnetic field is often described in terms of its spatial power spectrum as a functionof spherical harmonic degree (often termed the Lowes’ spectrum). Give a definition for the Lowes’spectrum, sketch the structure expected for the geomagnetic field, and explain its origin. How wouldyou expect the spectrum to di!er for the planet Mars which currently has no internal magneticdynamo?
3
Departmental Exam 2006 - Part 2
This section is 2 hours, closed book and consists of nine problems that will require some amount ofreasoning and equation solving. Even if you cannot solve the equations, you will get credit for posingthe problem correctly, and describing a path for gettin a solution. Brevity is valuable, but not to thepoint of obscurity. Correct answers matter, but so does evidence that you know what you are doing.If you get results that are dimensionally wrong, or clearly o! by orders of magnitude, show that yousee this, even if you cannot fix the problem.
1) There have been a number of theories suggesting that the constant of gravitation, G, has changedwith time, and that this would cause a change in Earth radius. For a small proportional change !G(that is, !G/G << 1), find an expression for the change in radius, assuming linear elasticity applies andthat the Earth is spherically symmetric. You should be able to include (though not of course evaluate)the variation in relevant parameters with depth. Make as much use as you can of the symmetry of theproblem (You should not need any complicated expressions in spherical coordinates, actually). Tryto find a numerical solution, even if a rough one, assuming a uniform Earth, and making use of thetravel time of a P wave through the Earth.
2) A strike-slip earthquake has produced a surface rupture with length of 50 km, and average slip of4 meters. Estimate: a) the change in strain caused by the earthquake in the ambient crust (order ofmagnitude); b) The corresponding static stress drop (for this, you will need to use the shear modulusof the crustal rocks; if you don’t know a typical value of the shear modulus on top of your head, tryto infer it from other quantities you may know, e.g., Vs and the crustal density of the crustal rocks);c) The corresponding scalar seismic moment. Assume that the earthquake has ruptured the entirebrittle layer (use your best guess for the down-dip fault size).
3. Write down an equation describing the main geomagnetic field in terms of spherical harmonics.What are the Gauss coe"cients in this equation? Why does the IGRF model expansion usuallyterminate at degree 14 or less? What is the wavelength on the Earth’s surface that corresponds tothis degree? Are fields of shorter scale observed at the surface, and if so, where do there come from?What is the corresponding wavelength on the core’s surface of the highest degree component in anIGRF model? Is there reason to believe there are fields with finer scales at the core-mantle boundary?
A rocket with a total field magnetometer crashes on a distant planet, falling on a straight line path.The magnetometer records field values exactly described by
B(r) = c(a
r)3, r > a
where r is the distance to the planetary center of mass, and c is a constant, 60µT, and a is the planet’smean radius. What can you say about the Gauss coe"cients of the planet’s magnetic field? Is itpossible that there are nondipole components present?
4
4). Discuss what is a statistical estimator. Explain the meaning of the terms bias and variance asproperties of estimators. Can an estimator have zero bias? Give an example, and prove it is unbiased.Can an estimator have zero variance?
Describe briefly what form does bias take in estimators of the power spectrum based on the discreteFourier transform. What strategy is used to reduce the bias in practical estimation methods? Howcan the bias-reducing method be extended to reduce the variance of the estimates?
5). Define the following terms: geopotential, geoid, geoid anomaly, reference ellipsoid. Derive therelationship between geoid anomaly, N , and the variation of geopotential on the reference ellipsoid.
Suppose the mantle of the Earth was vertically stratified, without lateral density variations, and thatall long wavelength topographic undulations were isostatically compensated. What relationship doesthis model predict between the geoid anomaly and the topography? Is this relationship observed inthe real Earth? What do you conclude from the success or failure of the model?
6) An earth model consists of a stack of 5 horizontal layers, each 2km thick, overlying a halfspace.Starting at the top, the layers have seismic velocities of 4,5,6,7,and 8km/s respectively and the half-space has a velocity of 9km/s. A downgoing ray starts from the surface at an angle of 40 degrees tothe vertical.
a) What is the ray parameter for this ray? b) What is the maximum possible depth that it penetrates?c) Assuming a reflection point at the maximum depth, what is the total travel time and horizontaldistance travelled of the re!ected wave at the surface? d) What is the delay time ! for this ray? e)If the pulse shape starts out looking like a broadened delta function, sketch the shape of the reflectedpulse.
7) Derive the following relationship between the rate of increase in seafloor depth with age, "d/"t andthe di"erence between the surface and basal heat flow (qs ! qb):
"d
"t=
#
Cp($m ! $w)(qs ! qb)
You will need Fourier’s law, energy conservation, and isostasy as follows:
q = k"T
"z
"T
"t=
k
$mCp
"2T
"z2
5
d(t) = ! !"m
("m ! "w)
L!
0
T dz
where
L - asymptotic lithospheric thickness and also the depth of compensation (m) d - seafloor depth (m)q - heat flow (W m!2) ! - coe!cient of thermal expansion (K!1) Cp - heat capacity (J kg!1 K!1)"m, "w mantle and seawater density (kg m!3) k - thermal conductivity (W m!1 K!1)
8) The ratio of parent to daughter isotopes is used to determine the age of a rock. The rate of decreasein the number of parent atoms is equal to the number of parent atoms times the decay constant. Startwith No parent atoms at time zero.
a) Setup the di"erential equation and initial condition. What are the units of the decay constant? b)Solve the di"erential equation to determine the number of parent atoms at some later time. c) Solvefor the number of daughter atoms as a function of time.
too easy??
9) In many geophysical problems it is common to suppose that measurements can be written as linearfunctions of the parameters of interest. For example in seismic tomography one can write a vector oftravel time residuals d using the linearized equation
d = Gm
a) Briefly explain what the model vector m and the design matrix G represent in this equation. Whatis the principle of least squares? Derive the least squares solution for the model vector m, and giveits solution in terms of the so-called normal equations. Under what circumstances might you regardthe least squares solution as optimal?
b) What does it mean when these equations are described as ill-conditioned? How can one avoidthe pitfalls associated with poorly conditioned equations? Explain what is meant by a damped leastsquares solution for m, and how it would modify the normal equations you derived in (a). What e"ectwould you expect this to have on the resulting model?
6
IGPP
Departmental
Examination
2008
Departmental Exam 2007-2008
This year’s test does not separate the short and long questions into different sections. It would be a good idea to skim the entire test before beginning so that you can budget your time sensibly. You have 3 ! hours, plus an extra half hour if you want. You can either write your answers on the test itself or on separate sheets of paper. Let us know if you need a bathroom break, but please remember that the exam is closed book.
You should try to answer all the questions. Credit will be given for partial answers, but not for material irrelevant to the question. Brevity is appreciated, but not to the point of obscurity. Correct answers matter, but so does evidence that you know what you are doing. If you get answers that are dimensionally wrong, or clearly off by orders of magnitude, show that you can see this, even if you cannot fix the problem.
(1) a. What is the most abundant mineral in Earth's upper mantle?
b. What is the average density of Earth's crust?
c. What is the mean ocean depth? Deepest depth?
d. What is the P-wave velocity in ocean water? S-wave velocity?
e. My GPS instrument collects 100 independent measurements of height, from which I compute a mean of 700 m and a standard deviation of 50 cm. What is the standard error in the mean?
f. I test my GPS data against a normal distribution using a Kolmogorov-Smirnov (K--S) test and get a probability of 0.90. What does this mean?
g. At what depth is the upper mantle/lower mantle boundary?
h. What range of age is appropriate for Carbon-14 dating? Potassium-Argon?
i. How strong is Earth's magnetic field? Mars'?
j. Would you characterize Earth's geomagnetic field spectrum as red, white, or blue?
k. What would be a typical surface heat flow? Geothermal gradient?
l. What is the typical thickness of oceanic crust? Continental crust?
m. Give a definition of earthquake b-value.
n. Define scalar seismic moment M0
o. Sort from fastest to slowest: Love waves, Rayleigh waves, P waves, S waves
p. Draw a sketch defining strike, dip and rake for an earthquake fault.
(2) Why does ocean depth vary approximately as one over the square root of seafloor age?
(3) What are the three main types of plate boundaries and what type of earthquake focal mechanism typically occurs on each?
(4) Sketch a hypsometric curve for the earth (i.e., fractional area versus elevation and depth). Label the elevation/depth axis but don’t worry about an absolute scale on the area axis. Label and describe the main features of this histogram.
(5) Assume the lithosphere of Venus has evolved to a steady-state temperature profile. Given a current heat flow of 4x10-2 W m-2, a surface temperature of 450˚C, a mantle temperature of 1500˚C and a thermal conductivity of 3.3 W m-1 C-1, calculate the thickness of the lithosphere.
(6) The decay constant for radioactive decay of 87Rb to 87Sr is 1.42x10-11 yr-1.
(a) Set up the differential equation for the rate of change in 87Rb atoms versus the current number of 87Rb atoms.
(b) Solve the differential equation for the number of atoms as a function of time assuming an initial number of No.
(c) What is the half life?
(d) If the Earth started with No atoms of 87Rb, how many are remaining today?
(7) Abyssal hills on the seafloor have a characteristic wavelength of 10 km and produce a gravity anomaly amplitude of 5 mGal at the bottom of the ocean. What is the amplitude of the gravity anomaly at the sea surface where the mean ocean depth is 3 km?
(8) (a) What is the orbital radius for a satellite in geostationary orbit (i.e. orbit period = 1 day)?Note that: G = 6.67 x 10-11 m2 kg s-2 Me = 5.98 x 1024 kg
(b) J2 is about 10-3. At the surface of the Earth the gravitational effect of the J2 term is about
10-3 times the total acceleration of gravity. What is this ratio for the satellite in geostationary orbit?
(9) For inaconformal map projections (such as the Mercator) the scale is not the same everywhere (which is why Greenland looks so big), but shapes are preserved in any very small region. What kind of strain does this correspond to? Which tensor components, or combinations of components, are zero in this case? What about an equal-area map projection, in which the scale in each small region is the same, but shapes are distorted?
(10) Suppose we have a planet made of incompressible material of density . Write down and! solve the equations to give g and the pressure as a function of radius; take the radius of the planet to be R. What is the Poisson’s ratio of the material? Would you expect there to be any kinds of seismic waves that could propagate, and if so what would they be?
(11) The above figure shows data relating earthquake magnitude to fault rupture length. From these data, how would you go about finding the most likely value of rupture length for a given magnitude? How about the other way around (i.e., finding the most likely magnitude for a given rupture length)? What kind of estimator is involved, and what assumptions make it valid (if it is)?
(12) The local Gas & Electric company guarantees that all its power is produced in the frequency band 60 ± 0.03 Hz. You want to estimate the power spectrum of the local vertical geomagnetic field Z between periods of roughly one minute and thirty minutes. Your grant contains no money for an analog attenuator of the 60 Hz signal, so you decide to measure Z every x seconds, choosing x so that the power-line signal will not be aliased into the lowest third of
your Nyquist band. That is, if fN is the Nyquist frequency, no power company frequencies alias
into frequencies f satisfying fN/3 < f < fN/3. What is the largest value you can choose for x, and if you choose that value, what is the shortest period at which you can estimate the spectrum of Z?
(13) (a) Explain the term isostasy.
(b) What is meant by local isostatic compensation? Explain the difference between the Pratt and Airy mechanisms.
(c) What is regional compensation? What property of the lithosphere can be estimated from a regional compensation model if appropriate quantities are measured? What are those quantities?
(d) Which of the two, local or regional compensation, is at work at mid-ocean rises?
(e) A broad mountain range is locally compensated. What is the effect of the range on the free-air gravity anomaly? What is the effect on the geoid height? Explain your answer.
(14) (a) State the induction equation governing the evolution of the magnetic field in a moving, electrically conducting fluid.
(b) The magnetic Reynolds number Rm gives the ratio of the magnitudes of two terms in the equation. What processes do the terms represent? Derive the usual order of magnitude
expression for Rm based on typical parameters of the system.
(c) Make an estimate of the value of Rm for fluid motions in the Earth’s core? You will need to assume values for some properties of the core. What consequences for the study of the
magnetic secular variation follow from the magnitude of Rm?
(15) Gauss showed that the geomagnetic field could be represented by the gradient of a scalar potential ! with spherical harmonics composed of two distinct parts, an interior part and an external part.
(a) Write down this spherical harmonic expansion and explain what the symbols mean. What are the major physical sources of the internal part of the field, and which group of spherical harmonic coefficients is the largest? What are the major physical sources of the exterior field? In addition to magnitude, what other important characteristic distinguishes the exterior field from the interior one? What is the spherical harmonic degree of the terms that dominate in the exterior field?
(b) The gravitational potential is also derived from a scalar potential, V. It too has a spherical harmonic expansion, with internal and external parts. Write out the expansion. Which terms are present here, but not in the magnetic potential and why? What causes the exterior part? Which spherical harmonic terms are the largest in the exterior part?
(16) The topography h, defined by elevation above sea level, in a region can bemodeled by an isotropic stochastic process with Power Spectral Density (PSD)given by
Sh(k) =a
2(b2 + k2)3/2(1)
where a and b are constants, and k is the horizontal wavenumber magnitude.
How would you find the root-mean-square (RMS) deviation from the meanvalue, based on the PSD, Ph?
Show how to calculate Ph(kx), the PSD of elevation along a straight road acrossthe region. There is no need to actually perform the detailed calculations.
How is the autocovariance of the topography on the road found from the PSD?
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The autocovariance of the topography along the road can be shown to be
Rh(x) =!a
bexp(!2!bx), x " 0 (2)
If a = 0. 2 m, and b = 0. 0005 m!1, what is the RMS deviation from the meanelevation?
With the same constants, calculate the RMS elevation di!erence between points100 m apart on the road. Would you regard this road as very hilly, or justsmoothly undulating? Explain your answer.
2