3. Abstract
This report investigates the seismic anomaly produced in the event of a lithospheric de-
lamination. Lithospheric delamination is a convective removal process of the lithosphere
in the subsurface. It’s overall environmental impact is linked to volcanism events and
local topographic variations. As one of the lesser documented and understood earth re-
cycling processes, this report investigate the ability to image a seismic velocity anomaly
indicative of a lithospheric delamination utilizing the van Wijk et al. approach. In or-
der to accomplish this I use a geodynamic model of delamination and convert the model
pressure-temperature structure into seismic velocity. The results demonstrate consistent
seismic velocities deviations to the Isabella anomaly in the Sierra Nevada mountain range
of California. The ability to successfully reconstruct seismic anomaly continues to sup-
port the validity of seismic studies in pursuit of further understanding the delamination
process. Though successful in this approach, future improvements to the start model in
terms of overall complexity is required. The performed sensitivity analysis emphasizes
the impact of attenuation, specifically, in the proper definition mantle water content as
well as grain size. In conclusion I find the van Wijk method a good approximation to
converting pressure and temperature to seismic velocities, however, in order to obtain
better results mantle heterogeneity and anisotropy will need to be further studied.
4. Chapter 1
Introduction
1.1 Motivation
The lithosphere delamination phenomenon is a best perhaps described as a method for
the earth to recycle crustal and continental mantle lithosphere material. The principal
mechanisms for crustal recycling are subduction at plate margins as well as lithospheric
delamination and drip mechanisms [1]. While subduction is a well understood process
that is essential to the plate tectonic engine of planet earth, delamination, on the other
hand requires recycling via convective removal of the lower parts of the lithosphere and
is more difficult to detect [1]. Delamination is a form of vertically and spatially localized
tectonics which often generate amoeba-like or circular surface topography effects that are
regional results of tectono-magmatic processes at convergent plate margins[1]. Simply
put, lithospheric delamination is crustal and lithospheric removal by peeling a mechanism
driven by convective mantle fluids and a density instability. This process is formally
depicted in figure 1.1.
Tectonics events lead to crustal shortening or extention. During shortening, duc-
tile delamination of continental lower lithosphere via Rayleigh-Taylor instabilities can
produce continental magmatism with a range of major- and trace-element compositions
and volatile contents [2]. It doesn’t take much to produce these effects either. Density
contrasts as small as 1% are sufficient to drive gravitational Rayleigh-Taylor instabilities
[2]. These events have been noted to have occurred in the past in the southern Sierra
Nevada. Where the Sierra Nevada is a mountain range located in between the Central
Valley and Basin and Range Province of California [3]. during the Pliocene, at roughly
3.5 Ma, small volumes of high potassium magmas were erupted through the granodior-
ites [4]. Thus, understanding and detecting delamination events may lead to a better
understanding of not only the past but may perhaps increase the prevention of environ-
mental hazards linked to magmatic intrusions. The removal of mantle lithosphere from
beneath continental crust has been called on to explain a number of tectonic features,
including volcanism and anomalous heat flow, upper mantle seismic velocity and gravity
anomalies, extensional tectonism, and both positive and negative topographic transients
[5]. The process may also play a significant role in continental crust becoming more felsic
as denser mafic material recycles back into the mantle [6]. Currently seismic tomogra-
2
5. (a) Earth (b) Lithosphere Asthenosphere Boundary
(c) Convection Driven Removal (d) Complete Lithosphere detachment
Figure 1.1: The Convective Removal Process
phy models show an inclined region of high velocities at roughly 3-8% higher than the
surrounding mantle) [17]. These are velocity deviations are interpreted to represent a
slab of delaminating lithosphere.
1.1.1 Objective
The goal of this project is to further examine the ability to accurately detect delami-
nation events with the use of seismic tomography. Seismic imaging is currently utilized
to study various economical and environmental methods. Its most prominent usage is
currently in the oil and gas and resource mining industries. During my 3 month study
I utilize a numerical model of delamination consisting of pressure (P) and temperature
(T), with the aim of mapping them to seismic velocity such that P,T ⇐⇒ V (P, T).
The hope is that the velocity anomalies seen in our forward model will accurately rep-
resent the delamination process. The conversion will take place following the van Wijk
method [8]. In particular this report looks to image a delamination similar to the Sierra
Nevada’s Isabella anomaly. Figure 1.2 shows the geographical location of the seismic
anomaly (Isabella) of interest.
3
7. 1.2 Previous Work
In order to simulate the temporal evolution of the delamination, I utilize a previously
built geological model consisting of a dense root with weak zone, mantle lithosphere,crust
and asthenosphere. Listed in table 1.1 are the densities and rheology of the model
shown also below in figure 1.3. The L-shape weak zone controls the asymmetry of the
delamination process.
Earth Zone Density (ρ0)
kg
m3
Rheology
Crust 2800 WQx5
Mantle Lithosphere 3250 WOx5
Root 3350 WOx1
Weak Zone 3250 1019
(Pa · s)
Table 1.1: Model Parameters
Notes:
• WQ: Wet Quartz
• WO: Wet Olivine
• Multiplicity is indicative of dryer and mechanically stronger zones
Figure 1.3: Geological Model
5
8. 1.2.1 Governing Equations
For the given geological model described above. We require solutions to the following
equations.
Please note that ALL variable definitions and values that are defined in Appendix B of
this paper.
Density Equation:
ρ (T) = ρ0 [1 − α (T − T0)] (1.1)
where: α = 3 × 10−5
, T0 = 900K
Mass Continuity Equation:
∂ρ
∂t
+ · (ρu) = 0 (1.2)
Conservation of momentum:
∂ρui
∂t
+
∂
∂xj
[ρuiuj] =
∂σij
∂xj
+ ρgi (1.3)
Conservation of energy (heat equation):
∂Φ
∂t
+ · (Φu) − · k T − H = 0 (1.4)
where: Φ = ρcpT → heat per unit volume
1.2.2 Solution Methodology
These equations were solved using a SOPALE code using a finite element solver for a
2D thermal mechanical evolution [19]. Figure 1.4 demonstrates the model evolution.
6
9. (a) time 0 ma (b) time 0.76 ma
(c) time 1.08 ma
Figure 1.4: Geological Model Evolution
7
10. Chapter 2
Problem Foundation
2.1 Framework
In order to handle the model data a 2D (x-distance, depth) uniform spatial Eulerian
grid was created utilizing Matlab. The original synthetic was not uniformly spaced so
this new grid uses a linear interpolation scheme with grid size of 25x1(km). The final
model space represents a range of 0-400(km) in depth, while laterally, the model covers
0-800(km).
2.1.1 Pressure and Temperature Cross-Section Modeling
For the grid at hand temperature and pressure are both strictly a 1D function of depth
at the start of the model. Both profiles are plotted below in figure 2.1.
8
11. 0 2 4 6 8 10 12 14
0
50
100
150
200
250
300
350
400
Pressure Gradient
Pressure (GPa)
Depth(Km)
(a) Pressure Gradient
200 400 600 800 1000 1200 1400 1600 1800
0
50
100
150
200
250
300
350
400
Temperature Gradient
Temperature (K)
Depth(Km)
(b) Temperature Gradient
Pressure (GPa) Cross Section
Distance (km)
Depth(km)
0 100 200 300 400 500 600 700 800
50
100
150
200
250
300
350
400
2
4
6
8
10
12
(c) Pressure Grid
Distance (km)
Depth(km)
Temperature (K) Cross Section
0 100 200 300 400 500 600 700 800
50
100
150
200
250
300
350
400
400
600
800
1000
1200
1400
1600
(d) Temperature Grid
Figure 2.1: Temperature and Pressure Profiles
2.2 Velocity Conversion
Van Wijk [8] gives an approach for calculating synthetic seismic velocities were they are
predicted from model temperatures and pressures. The approach also includes elastic
and anelastic effects and variations of mineral phase composition with pressure and
temperature [8]. Seismic anomalies are relative to the model horizontal average to a
regional reference [8]. I convert my temperature and pressure models to seismic velocities
utilizing this approach. After converting my velocities I qualitatively infer on potential
error sources and improvement methods. The conversion from pressure and temperature
data will take place in four steps.
1. T and P → Velocity (V ) at infinite frequency V∞
9
12. V∞(T, P) = 4.77 + 0.038P − 0.000378(T − 300) (2.1)
2. V∞ → to Vshear(s) with the addition of an attenuation factor
3. Vs → to Vcompressional(p) utilizing Poisson’s ratio
4. Calculate velocity deviation for Vp from background field Vp → Vdiff
Vdiff =
(Vp − Vref )
Vref
× 100% (2.2)
Figure 2.2 demonstrates the initial velocity conversion at infinite frequency, the model
space shown ignores the crust for simplicity, thus, the depth shown is the depth from
the base of the crust (40km) down. Furthermore I also neglect attenuation effects at
this stage, the figure 2.2 shown represents the converted geological model at time = 0.76
ma.
Vinf, t=0.76 ma
Distance (km)
Depth(km)
0 100 200 300 400 500 600 700 800
100
150
200
250
300
350
400
Velocity [Km/s]
4.35
4.4
4.45
4.5
4.55
4.6
4.65
4.7
Figure 2.2: Velocity at infinite frequency Seismic Model (Note: Depth is Depth from
the base of crust ie below 40km
2.2.1 S-Wave Conversion
Most of the seismological observations are made at low frequencies (ω ≤ 1Hz) and con-
tributions from anelastic (including viscoelastic) effects become important, particularly
for shear waves, when temperature is high [9]. At this point I introduce a seismic qual-
ity factor Q. Here Q−1
will represent the anelastic attenuation effect with respect to
the shear wave seismic velocities. As source energy propagates into the earth it loses
10
13. energy to the medium through which it is propagating, while changing mediums, energy
is transmitted and reflected at the interface. This transition then acts to further loses
in energy. This loss in wave energy is most often lost nowas heat [10]. At this point
I’ve neglected the oscillatory behavior of seismic waves and therefore neglected effect of
frequency of elastic waves. In order to account for the thermodynamic effect of energy
loss I will use equation 2.3 which includes anelastic attenuation.
Vs = Vinf(T, P)[1 − F · Q−1
(ω, T, P, COH, d)] (2.3)
Q−1
(ω, T, P, COH, d) = Bd−pQ
ω−1
exp −
(EQ + PVQ)
RT
α
(2.4)
The overall effect of anelasticity is to significantly increase the temperature derivative
of seismic wave velocities. This implies that the temperature anomalies associated with
low velocity anomalies should be significantly smaller [9].
B = B0d
PQ−PQref
Qref
COH
COH(Qref)
rQ
exp
(EQ + PQref VQ) − (EQref + PQref VQref )
RTQref
(2.5)
Shear Vel With attenuation Factor Q t=0.76 ma
Distance (km)
Depth(km)
0 100 200 300 400 500 600 700 800
100
150
200
250
300
350
400
Velocity [Km/s]
4.35
4.4
4.45
4.5
4.55
4.6
4.65
Figure 2.3: S-Wave Converted Seismic Model with Attenuation at model time 0.76 ma
2.2.2 P-Wave Conversion
In order to convert the velocities from S-waves to P-waves I use Poisson’s relationship.
Poissons ratio is known to be sensitive to lithology and Vp/Vs ratio is used in exploration
11
14. geophysics. In this case Poisson’s ratio ranges from 0.23 to 0.25 and is calculated as
function of compressional and shear wave velocity Poissons ratio is calculated from the
seismic velocity by the well-known formula 2.6. The Poisson’s ratio used to convert to
compressional velocity for the rest of this report is ν = 0.25 which is the upper bound
defined for an decompressed mantle (0.23 ≤ ν ≤ 0.25) as proposed by Poirier et al. [11].
ν =
(Vp/Vs)2
− 2
2(Vp/Vs)2 − 2
(2.6)
The values of and Vp and Vs are taken from various recent regional seismological
models.
Vp = Vs
1 − ν
(1/2) − ν
(2.7)
Utilizing equation 2.7 I convert a shear wave at a model time of 0.76 ma. The
attenuation parameters assume a dry mantle which is 50
H
106Si
with a grain size of
1(cm) as suggested by Ducea and Saleeby [14].
Pwave Vel, t=0.76 ma with Poisson, =0.25
Distance (km)
Depth(km)
0 100 200 300 400 500 600 700 800
100
150
200
250
300
350
400
Velocity [Km/s]
7.6
7.7
7.8
7.9
8
8.1
Figure 2.4: P-Wave Convertion of attenuated shear wave Seismic Model
12
15. Chapter 3
Results
3.1 Sensitivity Analysis
In order to investigate potential sources of error I conduct sensitivity tests isolating
parameters in order to view their overall effect on seismic velocities. In particular I
investigate the effect of water content, grain size.
3.1.1 Water Effect
In the attenuation parameter Q−1
I include factor B which introduces the effect of
water content in the mantle. We notice that an increase in mantle water content greatly
affects the seismic velocity. The overall effect is that an increase in water content acts
to slow seismic velocities, specifically it enhances the attenuation term in equation 2.5.
In figure 3.1 we notice the effect of mantle wetness has a greater impact to the change in
velocity than the effect of grain size variation. This result is consistent with the studies
found by (Behn et al. [12]) in which he notes that difference in water content has a
greater effect on Vs than does the calculated change in grain size. The most important
effects to seismic velocity accuracy are (i)enhancement of anelasticity, leading to higher
attenuation of seismic waves and lower seismic wave velocities and (ii) modification
of lattice preferred orientation, leading to changes in seismic anisotropy [13]. Overall,
water enhances anelasticity and that the effects of water on anelasticity can be quantified
through the effects of water on creep that modifies the relaxation time. It follows that
Q−1
depends on temperature, pressure and water content [13].
3.1.2 Grain Size Variation
In order to properly define our attenuation parameter we account for the effect of grain
size in the mantle. For simplicity I assume a homogeneous mantle with respect to grain
size of 1cm as analyzed from xenoliths in the Sierra Nevada [14]. Although ignoring
the effect of grain size simplifies our problem by greatly reducing computational re-
quirements they also preclude effects such as shear localisation and transient changes
in rheology associated with phase transitions, which have the potential to fundamen-
tally change flow patterns in the mantle [15]. It is therefore inferred that much of the
seismic wave attenuation in the upper mantle may be attributed to grain-size-sensitive
13
16. 400 600 800 1000 1200 1400 1600
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
Attenuation Effect Due to Mantle Water Content, grain size = 0.01(m)
Temperature (K)
ShearWaveVelocity(km/s)
Vinf
Coh=50
Coh=500
Coh=1000
Coh=3000
(a) Grain size = 0.01 (m)
400 600 800 1000 1200 1400 1600
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
Attenuation Effect Due to Mantle Water Content, grain size = 0.0075(m)
Temperature (K)
ShearWaveVelocity(km/s)
Vinf
Coh=50
Coh=500
Coh=1000
Coh=3000
(b) Grain size = 0.0075 (m)
400 600 800 1000 1200 1400 1600
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
Attenuation Effect Due to Mantle Water Content, grain size = 0.005(m)
Temperature (K)
ShearWaveVelocity(km/s)
Vinf
Coh=50
Coh=500
Coh=1000
Coh=3000
(c) Grain size = 0.005 (m)
400 600 800 1000 1200 1400 1600
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
Attenuation Effect Due to Mantle Water Content, grain size = 0.001(m)
Temperature (K)
ShearWaveVelocity(km/s)
Vinf
Coh=50
Coh=500
Coh=1000
Coh=3000
(d) Grain size = 0.001 (m)
Figure 3.1: The effect of Grain Size
diffusional processes occurring in the absence of melt, notably elastically and diffusion-
ally accommodated grain boundary sliding [16]. The effect of varying grain size in my
particular case be seen in figure 3.1 the effects of decreasing the grain size are consistent
with the expected theory as decreasing the grain size leads to a decrease in the shear
wave velocity.
3.2 Qualitative Validation
Since the inversion data for the Isabella anomaly is not currently available to me for direct
analysis it is not possible to conduct a rigorous quantitative data misfit. However, I am
able to make first order qualitative error analysis of the velocity anomalies. I compare
to the delamination in the Isabella anomaly using the paper from Craig Jones [16] which
provides P-wave tomography of the Sierra Nevada’s. Figure 3.2 demonstrates a velocity
14
17. contrast on the order of 3-7% and a size of roughly 220(km) from surface.
Figure 3.2: E-W Tomography Sierra Nevada (Isabella Anomaly) [16]
Looking deeper in the inversion the ray path’s 3.3, there are zones of noticeable
sparsity which leads to further questioning of the actual tomography results. Various
inversion methodologies also lead to greatly varying results. So there are multiple sources
that should be investigated prior to determining a proper reference model. Figure 3.3
demonstrate p-wave inversion tomography along with its ray path’s which most closely
match the rough shape of my delamination model in the 0.86 ma frame from figure 3.5.
15
18. Figure 3.3: E-W Tomography Sierra Nevada ray path’s [16]
In the paper by Jones et. [16] al they utilize a reference velocity as prescribed by the
Incorporated Research Institutions for Seismology (IRIS), model IASP91. The iasp91
reference model is a parametrised velocity model that has been constructed to be a
summary of the travel time characteristics of the main seismic phases [17]. When I used
this as my background I observe an inconsistent anomaly utilizing equation 2.2, this
error is due to the fact that there is an inconsistency in the velocity model in terms
of depth as shown in figure 3.4. In order to utilize the IASP91 reference model a bulk
shift correction with respect to velocity would need to be applied to the model. It is not
discussed and would have been intriguing to see how it was applied in the Jones et.al
paper [17].
16
19. Velocity Contrast to IRIS with Attenuation, and Poisson = 0.25
Distance (km)
Depth(km)
0 100 200 300 400 500 600 700 800
100
150
200
250
300
350
400
Velocity Deviation [%]
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
Figure 3.4: IASP91 Reference Velocity Anomaly
Figure 3.5 shows the velocity deviation from the background field using a simple
percent deviation as prescribed in equation 2.2. The background or reference veloc-
ity field in these is the p-wave converted seismic velocity at time 0. More specifically
the attenuation parameters utilized include the grain size at 1 cm and water content
50
H
106Si
representative of a dry mantle. We notice that at time 0.86 ma we have the
best agreement with the Isabella anomaly with a velocity contrast of about 4.3% and
slab size of roughly 135 km from the base of the crust. Given the simplicity of our model
the ability to retrieve such successful results is quite encouraging. Increasing the mantle
water content from the dry 50
H
106Si
case (COH) = 500,100,3000
H
106Si
, yields
respective velocity contrast anomalies of 4.9%,6.2%,12.3% and size of 142 km, 151 km,
169 km. The grain size variation from 1 cm to 0.75 cm, 0.5 cm and 1 mm result in
velocity contrast anomalies of 4.4%,4.5%,5% and size of 135 km, 135 km, 135 km. These
results confirms that mantle water content has a much more substantial impact on the
size and velocity contrast of our delamination.
The initial geological model parameters could be ”tweaked” in order to more accu-
rately replicate the velocity anomaly. For example the size weak zone and dense root
could be modified such that the slab size is more consistent to the Isabella slab observed.
Accurately reconstructing the model parameters would be a crucial step for further in-
terpretation of accuracy in the conversion. At this point increases imaging the anomaly
would require a shift to quantitative error analysis. However, qualitatively agreement
of the model provides optimism for more in-depth analysis, as I already can observe a
17
21. Chapter 4
Conclusion
In summary, this report demonstrates the ability to utilize a generic model of delam-
ination and at least, to first order, fit the observed velocity anomalies for the Sierra
Nevada mountain range in California. The results demonstrate a removed lithosphere
with a velocity contrast of 4% for a dry mantle and 1 cm grain size, with its size on the
order of about 130 km. This supports the idea that the Isabella anomaly may represent
lithosphere that is in the process of delamination.
4.1 Future Works
In order to refine the accuracy of this method it would be beneficial to apply a correction
or bulk shift to the IRIS reference velocity in order to use a consistent reference velocity
model. Having access to the ray path’s as well as the inversion scheme used by Jones
et al. [17] may also provide improvement to my results. The problem can be progressed
by adding complexity to the model in the form of anisotropy which would then require
additional sensitivity analysis. In converting my shear wave velocity to compressional
velocity I assumed a constant 0.25 Poisson’s ratio and grain size 1 cm for the entire
model space. Ideally I would like to extend my Poisson’s ratio usage to a function
of the form ν(T) such that a compressional velocity for a non uniform Poisson’s ratio
could be obtained. To extend the model to a more realistic state which would include a
heterogeneous and anisotropic mantle and lithosphere in the next step for this study.
4.2 Acknowledgements
The work of Huilin Wang for the geological model used in this project is greatly ap-
preciated. I would also like to acknowledge Dr. Claire Currie for her mentorship and
supervision during my project.
19
23. Appendix A
Matlab Codes
Appendix A contains the codes utilized to complete this project. All codes where made
by Karl-Yvan Mome using MATLAB.
A.1 Temperature and Pressure Mesh
Figure A.1: Eulerian Grid Part 1
21
32. Appendix B
Parameter Definitions and Values
Variable Description Value Units
ρ density see table 1.1
Kg
m3
α non dimensional constant 3 × 10−5
unitless
T temperature N/A Kelvin (K)
u velocity N/A
m
s
t time N/A s
x distance N/A m
σ stress N/A Pa
g gravity 9.81
m
s2
H heat source N/A Kelvin(K)
P Pressure N/A GPa
F constant (1/2) cot(πα/2) s
Q−1
attenuation factor N/A s
w frequency 1/8.2 s
COH olivine water concentration N/A
H
106Si
d grain size N/A m
dQref reference grain size 1.24 × 10−5
m
TQref reference temperature 1538 Kelvin(K)
PQref reference pressure 0.3 GPa
COH(Qref) reference water content in olivine 50
H
106Si
EQref reference activation energy 505
kJ
mol
VQref reference activation volume 1.2 × 10−5 m3
mol
PQref reference grain size exponent 1.09 unitless
Bo prefactor for Q for ω = 0.122Hz 1.28 × 108 mpQ
s
EQ activation energy 420
kJ
mol
VQ activation volume 1.2 × 10−5 m3
mol
pQ grain size exponent 1 unitless
rQ water content exponent 1.2 unitless
30
33. Chapter 5
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