Geophysics Final Exam Flashcards
1
Q
Rock Physics Checklist
A
I V P G I M
Identity Volume fraction Physical Properties Geometry Interactions Methods
2
Q
Combining Rock Physics
A
X^a = sum(f*X^a)
3
Q
Inverse Problem
A
Use the measured geophysical response, physics, and prior knowledge to interpret what’s going on in the subsurface
4
Q
Wave-based Imaging Methods
A
GPR (EM) waves
Seismic (mechanical) waves
5
Q
Forward Problem
A
Know the structure, physics, etc. to predict the geophysical response
6
Q
Wave
A
V = wavelength * frequency
7
Q
Amplitude (A)
A
Maximum magnitude of displacement from rest
8
Q
Wavelength
A
Distance required to complete one cycle of displacement (peak to peak or trough to trough)
9
Q
Period (T)
A
Time required to complete one cycle of displacement [time]
10
Q
Frequency (f)
A
The number of cycles completed in a given time period [Hz]
11
Q
Attenuation
A
The loss of energy as the wave propagates
1) Geometric spreading
2) Intrinsic attenuation (e.g. loss of energy to heat)
3) Scattering
12
Q
Wavefront
A
A curve (or surface) connecting points of constant phase (i.e. connecting peaks or troughs)
13
Q
Wavefield
A
The collection of all waves at one point in time (e.g. photo of ripples on a pond)
14
Q
Ray
A
The path traced out by following a fixed point on a wavefront. Perpendicular to wavefront
15
Q
Dispersion
A
When the velocity of a wave depends on frequency
most waves are made up of many frequency components
16
Q
Compression
A
An applied stress causes a change in the volume of the rock
17
Q
Bulk Modulus
A
K = -dP / dV/V
Inverse of compressability
18
Q
Compressability
A
B = -dV/dP / V
19
Q
Poisson’s Ratio
A
u = -e1 / e3
Ratio of perpendicular strain to parallel strain (relative to applied stress)
Related to Bulk: K = E / 3(1-2u)
Related to Shear: G = E / 2(1-u)
20
Q
Shear
A
Applied stress causes deformation (E) (strain) but no change in volume occurs
21
Q
Shear Modulus
A
G = shear stress / shear strain
rigidity
22
Q
Young’s Modulus
A
E = uniaxial applied stress / observed strain
23
Q
Surface Waves
A
Rayleigh wave – elliptical retro-grade motion with exponentially decaying amplitude with depth
Love wave – shear motion polarized in the plane of the surface (side to side)
24
Q
Properties that control Radar
A
1) Electrical conductivity, sigma
2) Magnetic permeability, u
* 3) Dielectric Permittivity, e
25
Dielectric Permittivity (e)
Relates charge separation (polarization) to the applied electric field: p = e*E
e = K*e_o
26
Dielectric Constant (K)
Describes the ability of a material to store energy due to charge polarization. Is the relative permittivity of a dielectric material
K = e / e_o
27
Dielectric Constant (K) vs. Dielectric Permittivity (e)
K = e / e_o
e = K*e_o
28
Dielectric Constant and Velocity
V = c / (K^1/2)
c = speed of light = 3E8 m/s
29
WARR
Wide Angle Reflection and Refraction
| -- move receiver but keep source fixed
30
CMP
Common Midpoint
| -- keep the center fixed, move source and receiver from center
31
COP
Constant Offset Profile
| -- move source and receiver together at same interval
32
P-wave
Primary wave
| Vp = (K + 4/3G / density)^1/2
33
S-wave
Secondary wave
| Vs = (G / density)^1/2
34
Vp / Vs
(K/G + 4/3)^1/2 = (1-u / 1/2-u)^1/2
35
Slope and Intercept
Square the traveltime eq: t^2 = 1/V2 x^2 + 4h^2 / V1^2
Plot x^2 vs t^2 to get a straight line:
slope = 1 / V1^2
intercept = 4h^2 / V1^2
36
Arrivals
Radar: Air first
Seismic: Air last
Both: Direct waves straight, reflection curved to join ground wave, refraction separates from reflection at ground wave
37
t vs. x:
| Radar or Seismic?
Radar = no groundroll
Seismic = air last
- - units
- - calc velocities
38
Frequency vs. Resolution
Higher frequencies:
- - better resolution
- - shallower
- - more attenuation
Lower frequencies:
- - poor resolution
- - deeper
- - less attenuation
39
Absorption
Loss of E due to (a) intrinsic attenuation (heat loss) and (b) scattering
Ii=I1e^(-q (ri-r1) )
40
Reflection Strength
Depends mostly on:
1) Reflection coefficient
2) Attenuation
41
Radar Reflection Coefficient (R) (normal incidence)
R = amplitude of reflection / amplitude of incidence
R = V2 - V1 / V2 + V1 = sqrt(K2) - sqrt(K1) / sqrt(K2) + sqrt(K1)
42
Seismic Reflection Coefficient (R) (normal incidence)
R = p2 V2 - p1 V1 / p2 V2 + p1 V1 = z2 - z1 / z2 + z1
43
Energy Density / Intensity (I)
I = E/S = E / 4pi r^2
44
Direct wave
t = x / v
45
Reflection
t = 2 / Vrms ( x^2 /4 + h^2)^1/2
x^2 - t^2
46
Refraction
t = ( 2 h1 (V2^2 - V1^2)^1/2 ) / (V1V2) ) + x/V2
47
Diffraction
t = 2/V ((x - xs)^2 + h^2)^1/2
48
Traveltime (t)
t = 2a/V1 = 2(x^2 / 4 + h^2)^1/2 / V1
49
Dix Equation
Vn^2 = ([Vrms^2]n tn - [Vrms^2]n-1 tn-1) / (tn - tn-1)
50
Resistivity of Rocks
Surface charge:
-- clays = conductive
Porosity:
-- high = LOW resistivity, HIGH conductivity
51
2nd Layer Velocity
V2 = sqrt( (Vrms(dt1 + dt2) - V1^2 dt1) / dt2)
52
2nd Layer Thickness
h2 = dt2 V2
53
Thickness (h) vs. Wavelength (w)
h >> w:
distinct reflections apparent
h = w:
bottom layer is mirrored
h = w/2:
not as clear
h = w/4:
looks like 1 reflection overall
54
Reflection Coefficient (R)
Quantifies the fraction of energy returned to the surface by a contrast in properties
55
Mulitples
Primary reflection: tn = 2h/V1
1st multiple: tm1 = 2t_p = 4h/V1
2nd multiple: tm2 = 3t_p = 2h/V1
56
Snell's Law
Sin O2 = V2/V1 sin O1
```
O1 = angle of incidence
O2 = angle of transmission
```
57
Critical Angle (Oc)
Oc = sin^-1 (V1/V2)
58
Critical Distance (Xc)
Distance at which a refracted wave is generated
Xc = 2h / [(V2/V1)^2 - 1]^1/2
59
N Layer Refraction Traveltime
tn = X/Vn + 2/Vn sum( (hi (Vn^2 - Vi^2)^(1/2)) / Vi )
60
Ohm's Law
V = I*R
```
V = voltage
I = current
R = resistance
```
61
Resistivity vs. Conductivity
resistivity = 1 / conductivity
conductivity = 1 / resistivity
62
Resistance (R)
R = L / A * p
```
p = resistivity
A = cross sectional area
L = length
```
p = A/L V/I = K V/I
63
Archie's Law
nonclays:
Peff = a * O^-m * Sw^-n * Pw
clays:
Peff = [(a * O^-m * Sw^-n * Pw)^-1 + o_surface]^-1
```
O = porosity
Sw = saturation
Pw = fluid resistivity
a = 0.41 - 2.13
m = 1.64 - 2.23
n = 1.1 - 2.6
```
64
Fluid Conductivity
Ow = 10^3 * F * sum( abs(Zi) * Ci * ui
```
F = 9.648E4 C/mol
Zi = valence of ion
Ci = concentration (mol/L)
ui = ionic mobility ((m/s)/N)
```
65
Voltage
at point p:
Vp = pI/2pi *( 1/r1 - 1/r2)
p = dVmeas/I * 2pi[1/r1 - 1/r2 - 1/r3 + 1/r4]^-1
66
Depth of Resistivity Arrays
Wenner: h = a/2
Schlumberger: h = L/3
Dipole-Dipole: h = na
67
Electrode Array Geometric Factors (K)
Wenner: K = 2pi*a
Schlumberger: K = pi/a^2 [1 - b^2/4a^2]
Dipole-Dipole: K = pi * n * (n + t) * (n + 2) * a
Square: K = pi*a *(2 + sqrt(2))
68
Direct Current (DC)
Current flows continuously in direction of applied voltage
69
Applied Current (AC)
Current switches direction with applied voltage
70
Pure Capacitor (C)
Ability to hold a charge
C = Q / V
```
C = capacitance
Q = charge
V = applied voltage
```
71
Chargeability
= Vp / Vo --> difficult to get Vp accurately
Apparent = A / Vo
A = integral (area) of Vp over a specified time window
72
Spectral (Frequency-Domain) Induced Polarization (SIP)
Apply a sine wave of a particular frequency. Compare the amplitude and phase (i.e. time shift) of the observed voltage relative to the applied current
73
Complex Conductivity (sigma = o)
o(f) = o'(f) + io''(f)
```
o' = real conductivity --> energy loss (conduction)
i = sqrt(-1)
o'' = imaginary conductivity --> energy storage (polarization)
```
74
Electrical Double Layer
Mineral surfaces are usually charged.
| Causes an imbalance of charge near the pore walls
75
Membrane / Electrolytic Polarization
Pore throat constriction - charge on mineral surface leads to a build-up at pore throats
Constriction by clay particles - ions accumulate on either side of a charged particle in the pore space
76
Self Potential (SP)
Redox reactions = electrochemical
Groundwater flow = electrokinetics
77
Tomography Traveltime (t)
through each cell (tj):
tj = sum( Lj / Vj ) = sum( Lj sj )
```
sj = 1 / Vj = slowness of cell j
Lj = length of ray in cell j
```
78
Least-Squares Data Fitting
Steps:
1) Collect data --> (x,y)
2) Define a model --> y=mx+b
3) Define a measure of error --> "least squares"
4) Find parameters of the model that minimize error
79
Tomography Forward Problem
t = L s
d = G m
d=data
G=design matrix
m=vector with all model parameters
80
Tomography Inverse Model
m = ( G^T G + a I )^-1 ( G^T y - a mo )
```
a = trade-off/regularization parameter
I = MxM identity matrix
mo = best guess of m
```
Not as good (will blow up):
m = ( G^T G )^-1 G^T y
81
Newton's Law
Fg = G m1 m2 / r^2
G = 6.672E-11 m^3/kgs^2
Fg = gravitational force = 9.8 m/s^2
m1, m2 = masses of objects
r = distance between the centers of the objects
82
Units of Gravity
1 Gal = 0.01 m/s^2
g = 981 Gal
83
Absolute Gravity (g)
The actual value of acceleration due to gravity measured at a point in space
84
Relative Gravity (dg)
CHANGE in gravity from a background value
=> useful for measuring variations in density
85
Spherical Earth vs. Non-spherical Earth
Spherical: r1=r2, g1=g2
Non-spherical: r1 x=x r2, g1 x=x g2
Polar radius is ~21km shorter than the equatorial radius
86
Forward Model for Gravity Anomaly
dg = G (4/3 pi a^3) (dp) z / ((x-xs)^2 + (z-zs)^2)^3/2
G = 6.672E-11 m^3/kgs^2
xs, zs = position and depth of "sphere" center
a = radius of sphere
dp = p2 - p1 = density difference between inclusion & background
87
Inverse Problem
- - Fit data using prior knowledge
- - Collect additional data
- - Reduce the problem to remove non-uniqueness
88
Half Max Gravity Anomaly (dg_1/2max)
dg_1/2max = G (4/3 pi a^3) dp z / ( X_1/2max^2 + z^2)^3/2
89
Corrections to Gravity Observations
B I L T F I T
1) Bouger correction
2) Instrument drift
3) Latitude
4) Terrain correction
5) Free-air correction
6) Isostatic correction
7) Tides
90
Bouger Anomaly
Reported gravity anomaly
> signal actually related to ground inclusion
dg = g_obs - sum( corrections - g_base )
91
Gravity Anomaly (dgz)
dgz = G SSS dp(x,y,z) z/r^3 dxdydz
92
dgi
| 1 block
dgi = A/r^3 + B/2r^5 ( (5z (3z^2 - r^2)/r^2) - 4z ) + 3C(x^2 - y^2) / r^5
r^2 = x^2 + y^2 + z^2
x, y, z = distances between measurement point and block i
A = 8G dp_i abc
B = A (2c^2 - a^2 - b^2) / 6
C = A (a^2 - b^2) / 24
2a, 2b, 2c = lengths of block i in x, y, z direction
93
Resistivity Array Choice
Depends on:
1) Type of structure to be mapped
2) Sensitivity of the resistivity meter
3) Background noise level
Things to consider:
1) Depth of investigation
2) Sensitivity of the array to vertical & horizontal structures
3) Data coverage
4) Signal strength
94
Chargeability
= Vp / Vo > difficult to get Vp accurately
Apparent = A / Vo
A = integral (area) of Vp over a specified time window
95
Spectral (Frequency-Domain) Induced Polarization (SIP)
Apply a sine wave of a particular frequency.
| Compare the amplitude and phase (i.e. time shift) of the observed voltage relative to the applied current
96
Complex Conductivity (sigma = o)
o(f) = o'(f) + io''(f)
o' = real conductivity > energy loss (conduction)
i = sqrt(-1)
o'' = imaginary conductivity > energy storage (polarization)
97
Self Potential (SP)
Redox reactions = electrochemical
Groundwater flow = electrokinetics
98
Least-Squares Data Fitting
Steps:
1) Collect data > (x,y)
2) Define a model > y=mx+b
3) Define a measure of error > "least squares"
4) Find parameters of the model that minimize error
99
Relative Gravity (dg)
CHANGE in gravity from a background value
> useful for measuring variations in density
100
Spherical Earth vs. Non-spherical Earth
Spherical: r1=r2, g1=g2
Non-spherical: r1 x=x r2, g1 x=x g2
Polar radius is ~21km shorter than the equatorial radius
101
Inverse Problem
-- Fit data using prior knowledge
-- Collect additional data
-- Reduce the problem to remove non-uniqueness
102
Center of Mass (xs, zs) and Mass of Inclusion (m)
xs = get from peak of dg curve
zs = 1.305X_1/2
? m = 255dgmax(X_1/2)^2 ?
103
Gravimeter
-- Sensitive - can detect change of 0.01 mGal
-- Rely on a mass pivoted on a beam attached to a spring
-- Buildings/mtns/etc. influence gravity, measurements
104
Gravitational Potential (u)
Represents the WORK ( / energy ) required to bring a unit mass from infinity to a position, r, away from the Earth
105
Gravity Forward Model
dg = M dp
[dg1; dg2;...dgN] = [K11, K12,...K1M; K21, K22,...K2M...] * [dp1; dp2; dpN]
dp = (M^T M + I a) M^T dg
K = constant that depends on model properties (grid size) and measurement location
106
Spectral (Frequency-Domain) Induced Polarization (SIP)
Apply a sine wave of a particular frequency.
| Compare the amplitude and phase (i.e. time shift) of the observed voltage relative to the applied current
107
Tomography Forward Problem
t = L s
d = G m
d=data
G=design matrix
m=vector with all model parameters
108
Tomography Inverse Model
m = ( G^T G + a I )^-1 ( G^T y - a mo )
```
a = regularization parameter
I = MxM identity matrix
mo = best guess of m
```
Not as good (will blow up):
m = ( G^T G )^-1 G^T y
109
Forward Model for Gravity Anomaly
dg = G (4/3 pi a^3) (dp) z / ((x-xs)^2 + (z-zs)^2)^3/2
G = 6.672E-11 m^3/kgs^2
xs, zs = position and depth of "sphere" center
a = radius of sphere
dp = p2 - p1 = density difference between inclusion and background
