Självständigt arbete Nr 44

Seismic Tomography in the Source Region of the May 29 2008

Earthquake-Aftershock-Sequence in Southwest Iceland

Seismic Tomography in the Source Region of the May 29 2008 Earthquake-Aftershock-Sequence in Southwest Iceland

Karin Berglund

Karin Berglund

Uppsala universitet, Institutionen för geovetenskaperKandidatexamen i Fysik, 180 hpExamensarbete C i geofysik, 15 hpTryckt hos Institutionen för geovetenskaper Geotryckeriet, Uppsala universitet, Uppsala, 2012.

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On May 29th 2008 two earthquakes with moment magnitude of Mw ~6 occurred in the southwestern part of Iceland. The second earthquake struck within only seconds after the first, on a fault ~5 km west from the first fault. The aftershock sequence was recorded by 14 seismic stations during the subsequent 34 days. The recorded earthquakes were detected and located with a Coalesence Microseismic Mapping (CMM) technique. The output data from this program has been used as basis for the tomography algorithm PStomo_eq, which simultaneously inverts for both P- and S-wave velocities and relocates the events. Within the study area of 46×36 km the three-dimensional velocity structure has, successfully but not conclusively, been modeled to depths of ~10 km.

The Vp/Vs ratio varies from 1.74 to 1.82 within the study area. The velocity increases with depth starting from 2 km where the P-wave velocity is 4.6 km/s and the S-wave velocity is 2.7 km/s, at a depth of 10 km the P-wave velocity is 6.9 km/s and S-wave velocity is 4.0 km/s. In the horizontal slices a high velocity area is seen in the northwestern part of model. This is interpreted to be caused by a magma body rising up from below and lithifying at high pressure. From cross-sections a large low velocity zone is seen in the western part of model area concentrated above the seismicity. The low velocity anomaly is found between depths of 2 km to 4 km, stretching from 21.5° to 21.2° W. It is interpreted to be caused by high porosity within the area. The depth to the brittle crust is increasing from the western part of the model towards the eastern part, right in the middle of the model it abruptly decreases again. The depth to the base of the brittle crust is increasing from 7 km in west to 9 km in the middle of model.

Självständigt arbete Nr 44

Seismic Tomography in the Source Region of the May 29 2008

Earthquake-Aftershock-Sequence in Southwest Iceland

Supervisors: Ólafur Gudmundsson and Ari Tryggvason

Karin Berglund

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I

Abstract

Seismic tomography in the source region of the May 29th 2008 earthquake-aftershock-sequence in southwest Iceland

Karin Berglund

On May 29th 2008 two earthquakes with moment magnitude of Mw ~6 occurred in the southwestern part of Iceland. The second earthquake struck within only seconds after the first, on a fault ~5 km west from the first fault. The aftershock sequence was recorded by 14 seismic stations during the subsequent 34 days. The recorded earthquakes were detected and located with a Coalesence Microseismic Mapping (CMM) technique. The output data from this program has been used as basis for the tomography algorithm PStomo_eq, which simultaneously inverts for both P- and S-wave velocities and relocates the events. Within the study area of 46×36 km the three-dimensional velocity structure has, successfully but not conclusively, been modeled to depths of ~10 km.

The Vp/Vs ratio varies from 1.74 to 1.82 within the study area. The velocity increases with depth starting from 2 km where the P-wave velocity is 4.6 km/s and the S-wave velocity is 2.7 km/s, at a depth of 10 km the P-wave velocity is 6.9 km/s and S-wave velocity is 4.0 km/s. In the horizontal slices a high velocity area is seen in the northwestern part of model. This is interpreted to be caused by a magma body rising up from below and lithifying at high pressure. From cross-sections a large low velocity zone is seen in the western part of model area concentrated above the seismicity. The low velocity anomaly is found between depths of 2 km to 4 km, stretching from 21.5° to 21.2° W. It is interpreted to be caused by high porosity within the area. The depth to the brittle crust is increasing from the western part of the model towards the eastern part, right in the middle of the model it abruptly decreases again. The depth to the base of the brittle crust is increasing from 7 km in west to 9 km in the middle of model.

Keywords: seismic tomography, Iceland, Mid-Atlantic ridge

Department of Earth Sciences, Uppsala University, Villavägen 16, SE-752 36 Uppsala

II

Referat

Seismisk tomografi på efterskalvssekvensen den 29:e maj 2008 i sydvästra Island

Karin Berglund

Den 29:e maj 2008 inträffade två jordbävningar med magnitud MW ~6 på sydvästra Island. Den första jordbävningen följdes tätt av en andra jordbävning på en förkastning ~5 km väster om den första. Påföljande efterskalvssekvens registrerades av 14 seismiska stationer under 34 dagar efter huvudskalven. De registrerade skalven har detekterats och lokaliserats med en Coalesence Microseismic Mapping (CMM) teknik. Utdata från detta program har använts som grund för tomografin som genomförts med PStomo_eq, en algoritm som inverterar oberoende för både P- och S-vågs hastigheter och samtidigt omlokaliserar eventen. Inom det undersökta området på 46×36 km har en tredimensionell hastighetsmodell, om än inte slutgiltigt, modellerats för djup ned till 10 km.

Vp/Vs kvoten varierar mellan 1.74 och 1.82 inom studieområdet. Hastigheterna ökar med ökande djup, på ett djup av 2 km är P-vågs hastigheten 4.6 km/s och S-vågs hastigheten 2.7 km/s och vid 10 km är P-vågs hastigheten 6.9 km/s och S-vågs hastigheten 4.0 km/s. I den nordvästra delen av modellen återfinns en höghastighetszon. Denna tolkas vara orsakad av en magma kropp som stigit och kristalliserat under högt tryck. De vertikala tvärsnitten visar en låghastighetsanomali i västra delen av modellen, koncentrerat ovan seismiciteten. Denna anomali sträcker sig från ett djup på 2 km ned till 4 km, från 21.5° till 21.2° V. Den tolkas vara orsakad av en hög grad av porositet. Djupet för den bräckliga jordskorpan ökar från väster till öster i modellen, för att i mitten abrupt minska igen. Basen av den bräckliga skorpan ökar från 7 km i väst till 9 km i mitten av modellen.

Nyckelord: Island, seismisk tomografi, mittatlantiska ryggen

Institutionen för Geovetenskaper, Uppsala universitet, Villavägen 16, 752 36 Uppsala

III

Contents

Abstract .................................................................................................................................................................... I

Referat ..................................................................................................................................................................... II

Contents ................................................................................................................................................................ III

1. Introduction .................................................................................................................................................... 1

1.1. General introduction ............................................................................................................................ 1

1.2. Background ......................................................................................................................................... 2

2. Theory ............................................................................................................................................................ 3

2.1. General ................................................................................................................................................ 3

2.2. Ray tracing .......................................................................................................................................... 3

2.3. Inversion .............................................................................................................................................. 5

2.3.1. Earthquake location .............................................................................................................. 5

2.3.2. Velocity perturbations .......................................................................................................... 7

2.4. Solutions to inverse problem ............................................................................................................... 8

2.4.1. Least-squares inverse ........................................................................................................... 9

2.4.2. Damped least squares ......................................................................................................... 10

2.5. Quality of model ................................................................................................................................ 10

2.5.1. Model resolution ................................................................................................................ 10

2.5.2. Model covariance ............................................................................................................... 11

2.5.3. Tuning the damping ........................................................................................................... 12

2.5.4. Model regularization .......................................................................................................... 13

2.6. The tomography method ................................................................................................................... 14

2.6.1. Joint inversion .................................................................................................................... 14

2.6.2. Tuning the damping ........................................................................................................... 15

2.6.3. Checkerboard test ............................................................................................................... 15

2.6.4. Setting up the tomography problem ................................................................................... 16

3. South Iceland Seismic Zone ......................................................................................................................... 17

4. Method .......................................................................................................................................................... 18

4.1. Data acquisition ................................................................................................................................. 18

4.2. Data selection .................................................................................................................................... 18

4.3. Performing the tomography ............................................................................................................... 19

4.4. Model resolution ............................................................................................................................... 23

5. Results .......................................................................................................................................................... 24

6. Discussion..................................................................................................................................................... 33

7. Conclusions .................................................................................................................................................. 35

8. Acknowledgements ...................................................................................................................................... 36

9. References .................................................................................................................................................... 37

1

1. Introduction

1.1. General introduction

Earthquakes occur almost continuously all over the world. Some of the earthquakes are strong enough to damage structures such as buildings and roads, but most of them are too small to be felt by humans. Yet they can be detected with sensitive modern instruments. Seismology is the science that studies the waves that are generated by these earthquakes and propagate through earth. The time it takes a wave to travel, and its path depend on the velocity structure in the medium that it propagates through. This dependence makes it possible to determine the velocity structure of that fraction of earth that the wave crosses on its way from source to receiver. Seismic data are the most powerful source for determining the interior structure of earth, its composition and state.

Seismic tomography is a method for determining the velocity structure of the earth, in-between the source and the receivers. It can be performed in a number of ways using seismic data such as amplitude, waveform, attenuation or travel-times. This thesis focuses only on travel-time tomography.

Figure 1. Map of Iceland where study area is marked in red.

Vatnajökull

km

Reykjavik

2

Due to its tectonic setting, Iceland is known as a very active seismic area. In 2008 May 29th, the southwestern part of Iceland was struck by two earthquakes. The first earthquake with moment magnitude of Mw 5.8 was followed by an earthquake of moment magnitude Mw 5.9 within only seconds after the first one (Brandsdóttir et al. 2010). The seismic stations within the area of ruptures recorded nearly 20 000 events or aftershocks within the 34 days following the main two earthquakes.

The aim of this project was to perform seismic tomography within the area affected by the earthquakes, see study area in figure 1. The recorded data from the aftershock sequence was previously run through a program called Coalescence Microseismic Mapping (CMM), which is an automated algorithm that detects and locates events. On basis of data from these aftershocks, a three-dimensional velocity structure was developed using the inversion algorithm PStomo_eq, a computer algorithm for local earthquake travel-time tomography (Tryggvason 1998).

1.2. Background

Local earthquake tomography was introduced in the mid 1970s and has since then become a well-established method. It can be used for either modeling the planet as a whole or for imaging the shallower crust locally. Earlier tomography techniques only constrained earthquake locations and P-wave velocity since the instrumentation in older networks were mostly vertical.

The algorithm used in this thesis originates from Harley M. Benz at USGS (Benz et al. 1996). It was modified by Ari Tryggvason at Uppsala University to simultaneously invert for earthquake location and velocity structure, and independently invert both P- and S-wave first arrivals. The modification of using S-waves in addition to P-waves, compared to only P-waves as in the original algorithm, significantly improved the resulting models. Even though the S-waves tend to be rather complex and difficult to precisely pick, they are large in number and contain plenty of additional information, which is why it is important to include them in the inversion. A larger set of data contributes to a higher quality model, the S-wave arrival times in addition to the P-wave arrival times also improves the hypocenter locations. Especially the source depths are improved, which is a well-known problem within the event location theory (Gomberg et al. 1990). Earlier problems caused by assuming that P- and S-waves follow the same paths are avoided. The algorithm uses controlled sources or local earthquakes. Controlled sources meaning that source location and origin time is known, local earthquakes meaning unknown locations and origin time for earthquakes within local distances.

The aftershock sequence from the May 29th earthquakes delineates two segmented faults, parallel in the North-South direction (Brandsdóttir et al. 2010). The distance between the two faults is 4-5 km. There are also several smaller North-South striking faults and activity stretching westward. The earthquakes are interpreted to be caused by the accumulation up of local stress due to the transform motion of the divergent plate boundary that intersects Iceland.

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2. Theory

2.1. General

Seismic waves are emitted whenever a fault ruptures within the earth. Among those waves are compressional P-waves and shear S-waves, which are the only ones used in this project. The waves propagate through earth and at some time later they arrive at earth’s surface were they can be recorded by sensitive seismic instruments. A seismic recording contains much information such as arrival time, amplitude, shape, frequency and phase polarity. However, only the arrival times are taken into account in this thesis. Since the arrival time depends on where the rupture started and the structure of the medium that it propagates through, information can be extracted about for example the hypocentral locations and the physical properties of the earth.

Seismic tomography is a method used for determining the velocity structure of the earth using seismic data such as waveform, attenuation, traveltimes. This project is only concerned with traveltime tomography.

The problem of determining velocity structure can be thought of as a forward problem and an inverse problem. The forward problem can be stated as: given the origin of an earthquake and a seismic velocity structure of the earth, when will the rays arrive at the recording station? The inverse problem can be divided in two parts stated as: given observed arrival times of an earthquake, first where and when did the earthquake occur and secondly what velocity structure explains the waves’ observed arrival times?

The local earthquake tomography problem is a highly non-linear problem, which complicates the task of solving it. The non-linearity arises from the fact that measured traveltime anomalies depend on the seismic velocity structure directly but also on the way the waves refract through the earth, which in turn depends on the velocity structure. The non-linearity is handled by either linearizing the problem and iterating or by applying non-linear inversion techniques such as Monte Carlo or Genetic algorithms (not described in this thesis).

2.2. Ray tracing

Solving the forward problem involves tracing the ray, which is normal to the wavefront, along its path as it travels from source to receiver. Given the origin of a particular earthquake and the velocity structure of the earth the rays’ travel time can be predicted.

The path that the ray will take depends on the velocity structure of the earth, see figure 2. Rays will bend from a fast velocity towards a slow velocity, resulting in a concentration of energy within the low velocity anomalies. Also, rays taking a longer path will generally arrive later and have lower amplitude than those arriving earlier.

4

Figure 2. Image demonstrating how rays bend when passing a low or high velocity zone. Z is depth and X is the

distance along free surface.

This thesis only involves earthquakes within local distances and therefore the curvature of the earth can be neglected and the earth can be approximated as flat. Ray theory has its basis in the eikonal equation (for general ray theory see Lay and Wallace 1995). From that it can be further evolved to Snell’s law, which is known from optics and the result for a one-dimensional varying medium is

(1)

Where u is the slowness which is the inverse of the velocity v, t is the travel-time, i is the incidence angle (the ray’s angle away from the gradient direction of slowness of velocity in the medium), x is the displacement and p is the ray parameter which stays constant all along the path.

Generally the P- and S-velocities increase with depth in earth, so the slowness decrease with depth.

Figure 3. A simplification on how a ray bends in a one-dimensional velocity model. Velocity is v, Z is depth and

X is the distance along free surface. On the right side is a demonstration of how the velocity increases with

depth.

Using simple geometry, see figure 3, one can obtain

(2)

(3)

dz

dx

ds i

X

Z Z

v

5

If the velocity continuously increases, the ray will eventually reach a turning point where . Using the chain rule and integrating from the free surface to the turning point, multiplied by two since the ray path is symmetric about its turning point.

(4)

The slowness is given by

(5)

In a similar way as for the distance, the travel-time can be derived. Using the chain rule and integrating from the free surface to the turning point, times two for symmetry around the turning point.

(6)

These expressions are for a one-dimensional velocity model, in a three-dimensional model the tracing is more complex but it follows the same principles.

2.3. Inversion

A more difficult problem is the opposite of the forward problem, observations of the arrival times are given but no information about the source locations or the medium that the wave propagated through on its way to the receiver. Inversion is applied to the observed travel times to find an earthquake location and a velocity structure that will explain the data. Inversion theory is concerned with problems of making conclusions about physical properties of the earth from such observed data. For general discrete inversion theory see Menke 1989.

The tomography problem consists of two inversion problems, that of locating the earthquakes and that of finding the velocity structure of the earth. The trade-off between those problems, meaning that the locations depend on the velocity structure and vice versa, makes finding a good model very complex.

2.3.1. Earthquake location

In location problems earthquakes are generally considered as point sources, although they are actually ruptures that can be up to hundreds of kilometers in size. The earthquakes are defined by its hypocenter and origin time, where the hypocenter is considered to be the point where the seismic energy first started to radiate. The problem of locating an earthquake includes determining the location and time of the rupture. In order to describe the origin of an earthquake there are four parameters needed, the three parameters for space x, y, z and one for time t.

The forward problem can be stated as

(7)

relating a set of data d, a vector containing measurements, to a set of hypocenter parameters represented by the vector h. Where F is a function that relates the two through a ray-theoretical approximation and is non-linear.

6

To solve the problem, it is linearized around an initial guess and then iteratively improved to better fit the data. So first, a location guess is made and the predicted travel-times are calculated for this first guess.

The very first guess can be made from either just the closest station which has the earliest arrival time. Or the differential travel-time between P- and S-waves can be calculated. Assuming depth is zero and that the discrepancies are proportional to distance, the distance to earthquake can be estimated. Circles are drawn around the stations with radii equal to the distance from earthquake, at least three stations are required to locate the event. Where the circles intersect, the earthquake occurred.

Once a starting guess of the location is found, the arrival times tp are predicted and the discrepancies of predicted and observed arrival times to are calculated. The aim is to modify the initial guess so that this discrepancy is minimized, therefore modifications to the starting model are sought for origin time and location.

Predicted arrival times can be described as a general function of the model hypocentral vector h containing the location parameters

(8)

Where is the event origin time and the second integral term is the traveltime for the ith ray. The

latter term is a function of the assumed or guessed event location and velocity structure. Velocity is v, u is the slowness and ds is incremental path along the ray.

The residuals between the observed and predicted arrival times can be expressed as

(9)

What is sough is the vector h that minimizes the residuals between the observed and predicted arrival times . F is a non-linear function of the earthquake location, the problem is solved by linearizing around a starting model.

If the true is close to the function can be expanded around the solution with Taylor series. Truncating the series and using only the first term will provide the linearization of the problem.

(10)

Where

(11)

Here are the adjustments that move the solution to better fit the data.

(12)

In order to minimize the residuals,

is solved for . This means that if

can be

calculated, a linear inverse problem can be written in matrix from as

(13)

7

Where matrix

(14)

states how a change of the jth hypocenter model parameter modifies the prediction of the i

th arrival time. To find the best fit to this equation a least-squares strategy is generally used (further details in section 2.4.1).

2.3.2. Velocity perturbations

The waves do not only carry information about the earthquake locations, but also about the velocity structure of the earth. The wave’s velocity, and hence its ray path, depends on physical properties of the rock material that it passes through such as rigidity, density and compressibility.

The first step to determine the velocity structure is to assume that the earthquake locations are known, and also an initial reference velocity model. Assuming the reference velocity model vo(z) is known, means that also the slowness model uo(z) is known. The velocity model can be modified so that the predicted travel-times will fit the observed data better. The travel-time t for a ray travelling along path s can be described by the integral

(15)

where u is the slowness and ds is the incremental path along the ray. The purpose of choosing slowness instead of velocity is because the travel-time is linearly dependent on slowness, not velocity.

As described earlier, predicted arrival times can be calculated by tracing the rays in the reference model. Then the discrepancies between those predicted from the initial velocity model and earthquake location and the observed travel time are computed.

(16)

The residuals indicate velocity anomalies relative to the starting model. Positive residuals results from late arrivals indicating low velocity anomalies and negative residuals results from early arrivals indicating a high velocity zone.

The problem is to find a change in the model slowness u that will contribute to reducing the discrepancies between observed and predicted data. Similar to the location problem, one wishes to minimize the discrepancies which in terms of slowness is expressed as

, (17)

where the first term is the effect of changing the slowness without changing the ray paths and the second term is the effect of changing the ray path without changing the slowness. Fermat’s principle states that the ray always chooses to take the path that results in the least travel-time. That principle suggests that the precise ray path is not needed to obtain a correct travel-time, getting close enough should suffice. Therefore, the effect of changing the slowness encountered by the reference ray is a first order effect while the consequence of changed ray path due to slowness perturbations is a higher

8

order effect. If the slowness is perturbed by a small amount δu, the second term can therefore be ignored. The discrepancies can be expressed as

(18)

For simplicity, start with a one-dimensional model and divide earth into different horizontal layers with uniform slowness perturbations. For each of the travel-time residual there is a specific ray path that connects source and receiver. Finding this path is done by the earlier described ray-tracing problem, which will enable the study of slowness changes along the ray path that are caused by the changes in travel-time.

The parameterization is made by dividing the earth into different horizontal layers, then computing the length if the ith ray in jth layer, calling slowness perturbation of the jth layer (see figure 4).

j=1

j=2

j=3

j=4

j=5

j=6

j=7

Figure 4. Demonstration on how the parameterization is made, and how a ray can bend in different layers of the

earth.

Equation (18) can then be discretized as

(19)

Here Gij is simply the distance that the ith ray travels in the jth layer, it’s an operator that predicts data vectors from model vectors. Layers that are not crossed by any ray are not constrained by the data. is the data vector containing the difference between the observed travel-times and the travel-times predicted by some starting model, is the model vector containing the perturbation in slowness from the starting model.

The problem has now been linearized. The one-dimensional case will generally be overdetermined, since there are more data than unknowns. If this is the case, the problem is solved using a least-squares strategy, see section 2.4.1.

2.4. Solutions to inverse problem

The question is; given some observed arrival times, is it possible to find a model that fits them? There are many different techniques to solve this problem, the most important in this project are listed below. In essence, after linearizing the problem it is solved in the similar manner no matter if the sought model is the slowness perturbations or locations of the earthquake.

For every raypath the distance traveled and the travel-times in each block are calculated using a starting model. Since this model is laterally homogenous it is easy to calculate the predicted travel-times. By subtracting the predicted travel-times from those observed, the travel-time residuals are obtained. The travel-time residuals form the data vector that is inverted using the generalized inverse

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to find slowness changes and location adjustments that predict the travel-time residuals as precisely as possible.

2.4.1. Least-squares inverse

Because of errors in observations and theory approximations, a location that will precisely predict the observed arrival times can never be found. Instead a solution that will provide the best prediction is sought. Therefore, the solution must have the very smallest residuals, that is the difference between the predicted and observed data.

If there are four observed arrival times and four unknown the problem can be solved with regular Gaussian elimination to get the inverse of G. Unfortunately, this is never the real case scenario. Most often there are a lot more observations than unknown parameters, so the problem is overdetermined. So in a least-squares strategy the error vector e is to have the smallest possible sum of squares.

(20)

A measure of the misfit can be defined as by the sum of the squared prediction errors Q

(21)

The inverse problem is posed to find that model which minimizes Q. The model estimate will predict data according to

(22)

The prediction error will look like

(23)

Where is the observed traveltime data and is the predicted data. Finding a hypocenter model estimate that is a linear combination of the data

(24)

The sum of the prediction errors is hence

(25)

Minimizing this using the penalty matrix, differentiating with respect to the elements of matrix A and equating with zero defines how to best choose the elements of A so that Q is minimized

(26)

After some entertaining manipulations using Einstein’s notation, this leads to

(27)

Which is the least-squares inverse of G.

The model estimate can be calculated

10

(28)

It is now possible to update the initial model and calculate a new model, this is done iteratively until the fit to data stops improving.

In the same way, the slowness perturbation is estimated. In a one-dimensional model, assuming that matrix GT

G is not singular, the slowness perturbation will look like

(29)

2.4.2. Damped least squares

For a three-dimensional tomographic problem the parametrization is almost always chosen so that there are more data than model parameters, hence overdetermined. However, the problem is seldom overdetermined as generally some of the cubes have few or no rays passing through them making them constrained by data, resulting in a very ill-conditioned sparse G matrix. Any given ray only passes a small number of cubes, which makes the problem underdetermined. In essence the three-dimensional problem is solved the same way as the one-dimensional, but in order to handle the underdetermined parts of the problem one has to modify the solution.

Even though the model area will generally be underdetermined, some parts of it will still be overdetermined since many rays are crossing the model in a small area. So the problem will then be mixed-determined. Solving the inversion for such mixed-determined problem, the least-squares solution cannot be used as in the location problem. One approach to handle the ill-posed least-squares is to impose additional constraints on the problem, a process referred to as regularization. One type of regularization is the damped least-squares solution. The weighting parameter λ controls the degree of damping.

The solution is very similar to the least-squares solution and now looks like

(30)

The damping parameter λ is a free parameter that is chosen to tune the inversion. If λ is chosen to be large the fit to data will be very poor but the uncertainty will be small, if λ is small the fit to data will be large but the solution will be unstable with high uncertainty. More information about how to find a good damping parameter in section 2.5.3.

2.5. Quality of model

2.5.1. Model resolution

Since the problem is not linear, the linearization only provides an approximation of the problem. By iteratively updating the model one handles the non-linearity of the problem. But the solution is very sensitive to how close to the true solution the initial guess is. So how can this be tested for, how good was actually the linearization? How well are the parameters determined and how uncertain is the solution?

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As in the problem for slowness perturbations, the solution is

(31)

Where the resolution matrix R is defined as

(32)

As stated in equation 31, each value of is the inner product of one row of the resolution matrix R and the true model .

Figure 5. Plot of the ith

row of resolution matrix R against jth

column. Hopefully there will be a peak of the ith

row of resolution matrix R at i=j.

The function can be described as a kind of window through which the model construction sees the true model. The matrix results in a peak in the ith row of the matrix Rij, hopefully correlated with the i=j column, see figure 5. The width and location of this peak is a measure of how the model estimate averages the true model.

If an undamped least-squares strategy is possible to apply

(33)

then the parameterization defines the resolution, this means that the parameterization is perfect. But this is generally not the case in structural seismology (see further information section 2.4.2) where a damped least-squares generally are used.

For the case of damped least squares it is obvious that the resolution matrix will not be the identity matrix, and resolution will not be perfect.

(34)

The resolution matrix reveals how well the model can be reconstructed if the data and model parameterization are perfect. Usually the resolution matrix reveals streaking between adjacent blocks where the ray coverage is inadequate to isolate the anomaly uniquely in each block, the resolution is a measure of the models non-uniqueness.

2.5.2. Model covariance

The model covariance provides an estimate of the uncertainty of model parameters and how they may be dependent. The uncertainties in the data can be characterized with the covariance matrix .

(35)

j

ith row

of Rij

i = j

Resolution

kernel

12

Where E is the statistical expectation operator, which quantifies the probability that the variables will take a given value. The probability function is describing the probability that a particular random measurement of the random variable will have a value in the neighborhood of . The expectation operator is the balancing point, or mean, of the probability distribution.

Equation 35 is a complete description if the data uncertainties are normally distributed. The off-diagonal elements describe the covariances of pairs of data and the diagonal elements describe the variances of the individual data values. The assumption can often be made that uncertainties are independent and therefore the matrix will reduce to a diagonal matrix.

The model covariance matrix , which is a description of the uncertainties in the model, can be defined from the data covariance.

(36)

2.5.3. Tuning the damping

In order to damp the equations as in the damped least-squares solution, it is important to find a representative damping parameter . To find a suitable value for it, the information in the resolution and model covariance matrix can be used. Many different are chosen and for each of the values the model covariance and the resolution matrix are computed.

The resolution matrix describes the consequences of incomplete sampling of the model space, and also the non-uniqueness of the model. The over-all resolution length is defined from the width of the resolution kernel (seen in figure 5) and by taking their average over all model parameters. The resolution length will be small if the resolution kernel is well focused on desired location in model space, and resolution length will be large if it is not. So the sum of the inverse diagonals of the resolution matrix will be high if resolution length is small, and vice versa. The sum of the diagonal elements in the model covariance matrix is a measure of the models uncertainty, since it is an average of the variance of the individual parameters.

For each chosen value of , the model uncertainty against resolution length are plotted. The resulting curve will have the shape similar to an L shaped curve, when damping parameter is varied according to figure 6.

Figure 6. Tradeoff-curve for model uncertainty and resolution length, optimal damping parameter is selected in

the middle of the L shaped curve.

Resolution length

Mo

del

un

cert

ain

ty

small

optimal

big

13

According to this curve one can see that choosing a that is very small the model will be very unstable, its uncertainty will be high and the solution is getting closer to the least-squares solution. Which might be very bad in the case where the G matrix is singular. However, little has been sacrificed in terms of smoothing or averaging and the resolution length will be small. If is very big it will prevent modifications in the model. Choosing a that is too large results in the solution that will have a large resolution length because large damping will prevent fitting the data. Therefore the solution will be very stable, but it might be difficult to interpret anything out of the model since only the broadest features are matched. Hence, the optimal choice of damping parameter should be somewhere around the middle of the curve in figure 6, the model would have a reasonable fit to the data and the model is fairly smooth.

So far, it is assumed that all data are weighted equally. One way of improving the data fit, is to start with a high damping parameter chosen somewhere in the corner of the L shaped curve in figure 6. This way the largest outliers will be removed. Then, one can allow the damping parameter to fade which will result in a better data fit than having a constant value of the parameter through all iterations.

The resolution matrix and the model covariance matrix provide means of appraising the model estimate, meaning that they evaluate how the model estimate sees the true model through a window of finite width and how data uncertainty propagates through the calculation causing uncertainty of the model estimates and how they are correlated. In a way it is unimportant what precise value are chosen for the damping parameter as long as the result is presented with its uncertainty and resolution length.

2.5.4. Model regularization

The sampling of model space is limited, therefore there is no unique solution to the inverse problem. Resolution will be less than perfect and an undamped or unregulated solution will be unstable. One way of dealing with this is by damping a least-squares solution. Damping involves minimizing a combination of data fit and model variance or model size. If a smooth model is sought, a large damping parameter is chosen implying that one wishes the model to be inconsistent with the data and so maximizes the data misfit. Smoothing the model means that one is trying to remove any structure that might exist and therefore the remaining structures are very likely to be real. Using smoothing hinders the model from fitting the data, but it adds stability to the solution.

It is also possible to use Vp/Vs ratio as a regularization parameter in local earthquake tomography (Tryggvason 2002). When inverting for both P- and S-wave slowness, it is useful to damp the Vp/Vs ratio in order to prevent strong variations in it. In this way make the ratio more stable, since it is expected that the two will vary together on physical grounds. Higher temperature will decrease both while changes in composition may increase one and reduce the other. Increased porosity will cause both to decrease, while changes of pore fluids may have opposite effects on the two. A suitable average Vp/Vs ratio is selected for the area of interest and the regularization parameter can be varied according to what degree the ratio is allowed to deviate.

Another option is to regularize by means of the cross gradient (Tryggvason & Linde, 2006). The cross-gradient is simply the vector product of the gradient of variations in P- and S-wave slowness. When both vary in the same or opposite direction the two gradient vectors are parallel and their vector product will be zero. The cross gradient is largest when the two gradient vectors are perpendicular, that is when the two vary in two different directions that differ by a maximal angle. Therefore,

14

minimizing the cross-gradient forces the two slowness fields to vary in similar directions. Even though the petrophysical relationships are not known between the two fields, it is reasonable to assume that any anomaly should appear in the same position in both. By minimizing the cross-gradient between the two fields one can regularize them and force them to change in either the same or opposite direction.

2.6. The tomography method

2.6.1. Joint inversion

For simultaneous inversion for P- and S-wave velocities along with earthquake locations, the travel-time equation is linearized (see section 2.3) about the initial guess and can be written in diagrammatic form as

(37)

for earthquake i. Superscript P and S are for the P- and S-wave model respectively; δt is the vector of travel-time residuals, matrix A is the travel-time partial derivatives of travel-time with respect to hypocenter location, matrix B is the partial derivatives of travel-time with respect to slowness perturbations, the vector h is the hypocenter perturbations and vector u is the slowness perturbations.

There is a coupling between the hypocenter locations and the P- and S-wave velocities. Solving the system of equations makes it difficult to determine what part of the misfit that is due to the slowness model and what part is due to hypocenter locations. Therefore the problems are solved by joint inversion.

For solving the coupled system, without decoupling, it is solved by separating out the hypocenter locations dependence (Pavlis & Booker, 1980). To increase the stability of the solution, three smoothing equations are added.

(38)

Here δtt and B

t is the travel times and their partial derivatives with respect to the slowness after transformation for the hypocenter parameters by separation of variables. The first added term is for smoothing of the slowness perturbations, which is controlled by minimizing the Laplacian (L) of the slowness perturbation field and λ is the parameter that controls degree of smoothing. In the second added term R is the relation of the Vp/Vs ratio and c is the weighting parameter controlling the degree of regularization. In the third added term C is the smoothing equation for controlling cross-gradient variation and k is the weighing parameter controlling degree of regularization (Tryggvason and Linde 2006).

15

For solving equation 38 for P-and S-wave slowness perturbations the LSQR solver is used. It is a conjugate gradient solver for sparse linear equations and sparse least squares. When matrix G is ill-conditioned, the LSQR algorithm has turned out to be a fast and reliable solver (Paige & Saunders, 1982).

2.6.2. Tuning the damping

When the resolution matrix and the covariance matrix are difficult to evaluate, since tomographic problems can be quite large the computer resources are not enough, the tradeoff curve presented in figure 6 will be impossible to produce. It is then useful to use proxies for resolution length and model uncertainty.

In the case where damping parameter is chosen to be small the model uncertainties will be large and the resolution length will be small. Therefore the model will be more variable on a small scale and the model roughness can be used as proxy for model uncertainty. In the case where damping parameter is large the resolution length becomes large and fitting to data is prevented, so data misfit will be large. Therefore, data misfit can be used as a proxy for the resolution length. Plotting model roughness against model misfit, see figure 7, will therefore produce a similar curve as presented in figure 6. The curve presented in figure 7 can now be used to find the best choice of damping parameter.

Figure 7. Plot of model roughness against model misfit. Optimal is in the middle of the L shaped curve.

2.6.3. Checkerboard test

A backhand of iterative methods is that it becomes impossible to calculate resolution and covariance matrices, then one turns to synthetic tests. The basic idea is to check the validity for the specific source-receiver geometry of the experiment and how good the linearization is. A synthetic test may also be used as a tool to find of optimal setup of smoothing, weights and model grid size.

The test used in this thesis is the standard checkerboard test. A three-dimensional synthetic model is created by adding positive and negative velocity perturbations to the starting model, in a checkerboard pattern. Synthetic data are generated by computing travel-times using the same source-receiver geometry as the real data. Noise proportional to the estimated data errors of the real data are added to the synthetic data. The synthetic data is inverted based on the same criteria as the real data, same damping parameters and number of iterations.

The resulting model is examined to see how well the initial synthetic model is recovered. The degree of smearing in the checkerboard pattern will vary with position in the model, indicating different

Mo

del

ro

ugh

nes

s

Model misfit

small

optimal

big

16

resolution in different areas. Regions in the reconstructed model that are in agreement with the starting synthetic model are considered well resolved. Since the primary factors controlling the robustness of specific model is the source-receiver geometry, the crustal structure for the real model is interpreted to be true in areas where the test shows good recovering.

2.6.4. Setting up the tomography problem

An initial earthquake location is estimated, or given. An initial one dimensional velocity model is selected. From this the starting velocity model can be derived. The parameterization is made by dividing the model area into small blocks.

Tracing the rays and computing arrival times are, in the PStomo_eq, made by computing the timefield from a source to all cells in the model. The travel-time from source to receiver is computed based on these timefields. Then the ray tracing is performed backwards, perpendicular to the isochrones of the timefield. If many ray paths crosses in one region it is possible to resolve structure and the slowness variations can be mapped.

Inversion for slowness modifications are made with the conjugate gradient solver LSQR. The velocity model is adjusted for a new, better model with improved data fit. The earthquakes are relocated in the new updated velocity model. The rays are traced in the new model for the relocated events and the problem can be solved for another update of the velocity model.

These steps are repeated until the data fit is satisfactory or until it no longer improves. The iteration process helps reducing the errors resulting from not knowing exactly where the velocity perturbation is situated.

Since earthquakes are generally located using a reference one-dimensional velocity model one could expect the locations to change given a three-dimensional velocity model. It is a tradeoff between velocity anomalies and earthquake locations. This problem is addressed by the iterative process where the tradeoffs between earthquake location and velocity structure will be minimized.

17

3. South Iceland Seismic Zone

Iceland is situated right on top of the Mid Atlantic Ridge. The setting causes intensive volcanism as a result of the spreading of the Eurasian and the North Atlantic tectonic plates which proceeds in a rate of ~1 cm/yr. What makes it even more complex is that a hot spot is situated underneath Iceland in the eastern region. The rift axes are moving westwards relative to the hot spot, which makes the spreading ridge prone to jump eastward back towards the hot spot.

New crust is generated in four segments across Iceland, the Northern Volcanic Zone (NVZ), South-Eastern Volcanic Zone (SEVZ), the Western Volcanic Zone (WVZ) and the Reykjanes Volcanic Zone (RVZ), (see figure 8). The youngest of the zones is the South-Eastern Volcanic Zone. This zone started to replace the Western Volcanic Zone about 2 million years ago which is still ongoing. In between the Reykjanes and the South-Eastern segment is the South Iceland Seismic Zone (SISZ). This transform zone is strongly affected by the motion of the rift zones. The study area, Ölfus, lies within the SISZ. The ages of the rocks are zero at the rift zones and increases with distance from them. The western part of the study area is within the RVZ, implying that surface rocks are younger in the western part of the study area since new crust is created here.

Large earthquakes have struck the SISZ repeatedly in historical time. Earthquake sequences starting with a larger earthquake (M ~7) in the east and proceeding to the west with smaller events are typical for this area. These earthquakes are separated by intervals of quiescence lasting for 45-112 years. The latest earthquakes occurred in 1896, 1912, 1987 and 2000 (Einarsson and Björnsson, 1979).

Figure 8. Tectonic settings in Iceland. The volcanic zones are marked as well as the spreading ridge. The study

area is outlined with a red rectangle.

NVZ

SEVZ

WVZ

RVZ

SISZ

km

18

4. Method

4.1. Data acquisition

A stationary network of seismometers is located within the SISZ, the SIL network. Three of these seismometers are located within the study area, which is a 46×36 km rectangular area. Within two days after the two main earthquakes on May 29th, 11 portable seismometers were also installed within the area, the LOKI network. All 11 of the temporary stations were used in addition to 3 of the stationary SIL stations, together the sensors made up a dense network which contributed to reduce earthquake location uncertainty and also the detection threshold. The stations are distributed as shown in figure 9. The aftershock sequence was recorded by these 14 seismometers and the network recorded a total of 19449 earthquakes within the following 34 days. One of the temporary stations was exchanged during these days, station HOLL was destroyed and was therefore rebuilt as INGO nearby (Brandsdottir et al, 2010).

For detecting and determining the earthquakes location an automated picking program was used, the Coalescence Microseismic Mapping (CMM) technique. The CMM algorithm performs an exhaustive search in space and time for events, and incorporates travel-time inverse theory in imaging earthquake locations.

Figure 9. Map showing the seismic network that recorded the aftershock sequence. SIL stations are marked in

red and the LOKI stations are marked in green.

4.2. Data selection

As discussed earlier, the seismicity is concentrated around the two faults that ruptured (see figure 10). The two major faults are parallel in North-South direction, there are also several smaller North-South striking faults and some activity spreading westwards.

To get a higher quality of the model only the best data are used in the tomography. The events and their phases are selected subject to certain carefully chosen criteria.

km

19

Figure 10. Map of the total seismicity, prior to selection, around the two faults that ruptured. The parallel

North-South faults are easy to spot as well as the seismicity stretching further to the west.

First the CMM catalogue is converted to a format that fits the tomography program. The threshold for travel-time residuals is set to 0.45 s for P-waves and for S-waves, i.e. all data having a larger residual than the preset threshold were thrown out of the data catalogue. This threshold is a very high but it will still help by throwing out extremely high residuals. The first threshold application using this filter resulted in all of the 19449 earthquakes being kept.

Before performing the tomography, the data were also filtered to exclude poorly determined events. This is performed by, based on certain criteria, picking the very best event in each bin of size 0.5×0.5×0.5 km where all included events are considered as one group. The best event is selected based on the highest number of phases recorded, events with fewer than 7 recorded phases were discarded. Events further away than 1.5 focal depths from the closest station were also discarded. Furthermore, phases with travel-time residuals bigger than 0.2 for P-waves and 0.3 for S-waves were discarded. After these selections the filter transforms the data into rectangular coordinate system. Reduction based on these criteria resulted in 3850 selected events out of the original 19449. Hence, the dataset was reduced by 80.2 %. This selection resulted in 89122 P and S phases used as an input for the tomography algorithm.

4.3. Performing the tomography

The seismic velocity models are derived using the inversion algorithm PStomo_eq (Tryggvason 1998). It is a program for simultaneous inversion of P- and S-wave velocities and earthquake location.

The initial earthquake location guess is given by the CMM output data catalogue.

The model area is chosen to be a rectangular 46×36 km large area containing all the events and seismic stations that recorded the events.

km

20

Before inverting for a three-dimensional model, a good starting model is needed. The starting model is derived inverting for a simple one-dimensional model (Tryggvason et al, 2002). The average velocity is computed in each layer and the resulting P- and S-wave starting models are shown in figure 11. Different one-dimensional starting models were tested to verify that this was the best starting model considering that the residuals were most centered around zero comparing to the other models.

Figure 11. The one-dimensional velocity model used as a starting model in the tomography. The black

curve is for P-velocity and the red curve is for S-velocity.

The algorithm computes the time-field from a station to all cells within the model. From these time-fields it calculates the travel-times for all of the events. The ray tracing is then performed backwards, perpendicular to the isochrons. The ray tracing and travel-time calculation is computed on a 0.5 km uniform grid.

A P- and S-wave travel-time residual criterion was also introduced after tracing the rays in the starting model. To compute synthetic travel-times for the starting model, the inversion was run with only one iteration. The residuals were plotted and are seen in figures 12 and 13 for S and P residuals respectively. A perfect residual would obviously be zero, and therefore it is desirable to have the residual as close to zero as possible. The residual plots indicate a wide range of residuals and show some strange effects interpreted to be some kind of artifacts caused by the automated program. It is therefore justified to introduce a threshold for the residuals, preventing the strongly deviating residuals from causing further problems in the model. The maximum P residual was set to 0.20 s and the S

21

residual to 0.25 s, these thresholds were kept constant through all the iterations in the inversion process. The thresholds are seen as red lines in figures 12 and 13.

Figure 12. Plot of the spread of the P-wave travel-time residuals. P-threshold marked with red line.

Figure 13. Plot of the spread in the S-wave travel-time residual. S-threshold marked with red line.

The inversion problem is solved by the LSQR solver. The number of LSQR steps varies up to 90. The three-dimensional inversion grid size was set up to be 1×1×1 km. The number of iterations of the inversion are set to 5, since no improvement in data misfit is obtained iterating further (see figure 17).

The model regularization is controlled in three different ways. The first option is to weight the smoothing equations. The misfit versus the model roughness is plotted in order to choose a reasonable weighting parameter. The first smoothing parameter is chosen to be as high as possible, according to the plot, and for later iterations the smoothing weight is allowed to fade. The variation of smoothing weight can be seen in figure 14, the solid line. At the first iteration the smoothing weight is set to 200, fading to 10 in the last iteration.

S r

esid

ua

l (s

) P

re

sid

ua

l (s

)

22

The variation of the Vp/Vs ratio is controlled by a damping parameter, preventing the ratio from differing too drastically from a chosen reference value. The value of the Vp/Vs ratio, which the models are damped towards, is chosen to be 1.78 which is a normal ratio for the Iceland region (Tryggvason et

al, 2002). The regularization parameter for the Vp/Vs ratio is set to 50 in the first iteration, decreasing to 10 in the fourth iteration and in the last iteration the ratio is allowed to vary freely. The damping of the Vp/Vs ratio over iterations can be seen in figure 14, the stippled black curve.

Figure 14. The upper diagram The black solid line is the weighting of smoothing, the dotted line is the

damping of the Vp/Vs ratio.

The parameter controlling the cross-gradient is set to 10 000 and kept constant through all the iterations, except in the last iteration where it is not used.

Innumerable tests have been made with different choices of regularization parameters to find the combination that result in the best model. The final setup resulted in a stable model with no oscillations in velocity in adjacent cells. In addition to trying out the different parameters on the real dataset, it is also done on the synthetic dataset.

The algorithm traces rays in the initial one-dimensional model and calculates residuals between observed and predicted travel-times. It inverts the residuals to minimize them and finds an update of the velocity model that better fits the data. The velocity model is updated and earthquakes are relocated in the new model. The rays are traced from the new location in the new velocity model and so on…

The number of allowed travel-times, see figure 15, is increasing in each iteration.

Figure 15. Diagram shows how the number of phases increases as the velocity model and earthquake

location is getting better.

23

4.4. Model resolution

The model reliability was tested with a synthetic reconstruction test, a standard checkerboard test.

The checkerboard test is performed with ±10 percent velocity perturbations in the starting model, see figure 16 (figure 32 for cross section). The transition from a low-velocity block to a high-velocity block was performed over 3 model blocks. Synthetic travel-times are computed and noise added as to fit the same amount of noise as in the true data.

The inversion was then repeated, with the same regularization parameters to observe how well the added velocity anomalies were reconstructed.

A small hand-picked set of data was also available, with ~600 picks. This was used as another test for the CMM catalogue. The best model achieved from tomography on the CMM catalogue was used as a starting model in the tomography with the hand-picked data. This test was done to see what anomalies the hand-picked data supports and which it disagreed with.

Figure 16. Checkerboard added to study area.

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5. Results

The root-mean-square residual reduction can be seen in figure 17. Comparing the travel-time fit for the earthquake locations in the initial one-dimensional model with the final earthquake locations in the three-dimensional velocity model gave a reduction of the residual from 0.082 to 0.066 (20%) for the P-waves and 0.117 to 0.084 (28%) for the S-waves.

Figure 17. The RMS data fit reduction curve for each iteration. The solid black lines represents the P-velocity

model and the solid blue line represents the S-velocity model. The stippled lines are the maximum allowed

residual, for P-velocity models in black and S-velocity models in blue. Note that the last iteration is only the ray-

tracing in the updated velocity model.

A horizontal slice through the models at depth 4-5 km shows a high-velocity anomaly in the upper western part of the model, the S-velocities showing a larger and more distinct increase than the P-

velocities, see figures 18 and 19. The P-velocities are approximately 6.3 to 6.5 km/s at this depth, the high-velocity area is 6.8 km/s. In the S-velocity model the high-velocity area is clearer with a velocity of 3.8 km/s compared to the surrounding velocities of 3.5-3.6 km/s.

The relative velocity variations in figure 20 and 21 show reduced velocities, relative to the starting model, mainly concentrated in areas with high seismicity. The increased velocities in the S-model are more distinct than in the P-model. Also in these figures, it is possible to see the raised velocities in the upper western part of model.

The Vp/Vs ratios are reduced in most parts of the horizontal sections to as low as 1.74, but in the south part of the model there is an increase up to 1.82. Overall, a variation of 1.74 to 1.82 is not very strong.

Figure 18-34, clarification. The models viewed with relative velocities are the velocities relative to the starting model. Areas with high resolution, according to the checkerboard test, are outlined in red. The velocity scale is calibrated according to Vp/Vs ratio of 1.78. The grey areas in the models are simply the unsampled blocks. The seismicity in each layer in the models is shown as black dots. The white triangles are the stations.

25

Figure 17. The Vp/Vs ratio

variation at depth 4-5km.

Lower ratio in the northern

part while in some areas as

high as 1.82 in the southern

part. Although, overall a very

insignificant variation.

Figure 18. The S-wave velocity

[km/s]. A relatively large high

velocity area in the northwest

part of model area.

Figure 19. The P-wave

velocity[km/s]. Also a high

velocity area in northwestern

part of model, although not as

sharp as for the S-velocity.

Vp/Vs ratio

S-wave velocity

[km/s]

P-wave velocity

[km/s]

26

Figure 20. Relative S-

velocity [%]. The higher

velocity region in

northwestern part can

be seen.

Figure 21. Relative P-

velocity [%]. Lower

velocities concentrated

around the seismicity.

Relative P-wave

velocity [%]

Relative S-wave

velocity [%]

27

Figure 22. The Vp/Vs ratio. A zone of low ratio is concentrated along the seismicity in the lower

region.

Figure 23. The S-wave velocity. Starting at 2.7 km/s at a depth of 2 km increasing to 4.0 km/s at a

depth of 10 km.

Figure 24. The P-wave velocity. Starting at 4.6 km/s at a depth of 2 km increasing to 6.9 km/s at a

depth of 10 km.

Vp/Vs ratio

S-wave velocity [km/s]

P-wave velocity [km/s]

Distance (km)

Distance (km)

Distance (km)

Depth

(km

) D

epth

(km

) D

epth

(km

)

28

Figure 25. The S-velocity relative to the starting model.

Figure 26. The P-velocity relative to the starting model.

In figures 23 and 24 the velocities are shown in vertical cross-sections. The P-velocity at 2 km depth is 4.6 km/s or higher. Then there is a rapid increase to a depth of 3 km where the velocity is 6.0 km/s. From 3 km to 10 km the velocity increases and at 10 km the velocity is as high as 7.0 km/s. The resolution at a depth of 10 km is poor and the information should not be completely trusted. The S-

velocity at depth of 1 km is 2.6 km/s or more, and at 2 km depth the velocity is 2.7 km/s. At a depth of 3 km the velocity has increased to 3.4 km/s. At a depth of 10 km the velocity has increased to 4.0 km/s.

The relative velocity in the vertical cross-sections shows a large low-velocity anomaly in the western part of the profile, concentrated in the upper 4 km right above the seismicity. See figures 25 and 26.

The seismicity in the vertical cross-sections shows a sharp edge at the base deepening from west to east. It is concentrated along two vertical clusters which are the faults where the main occurred ruptures started. The Vp/Vs ratio decreases with depth and tends to follow the bottom edge of the seismicity. The ratios range between 1.75 to 1.80 (see figure 22).

The vertical cross-sections through the velocity model stretches from 21.5° to 20.8° W, and from 0 km to 10 km depth. The uppermost 1 km is not well resolved, the upper 2 km are resolved moderately well according to the checkerboard test. Areas considered well-resolved are outlined in red. The resolution for the P- and S-velocity models are very similar.

Relative S-wave velocity [%]

Relative P-wave velocity [%]

Distance (km)

Distance (km)

Depth

(km

) D

epth

(km

)

29

Figure 27. Result of

the checkerboard test,

for S-velocity.

Figure 28. Result of

the checkerboard test,

for P-velocity.

Figure 29. The initial

checkerboard model.

S-wave velocity

perturbation [%]

P-wave velocity

perturbation [%]

Velocity

perturbation [%]

30

Figure 30. Results of checkerboard test in a vertical cross-section, for S-velocity reconstruction.

Figure 31. Results of checkerboard test, for P-velocity reconstruction.

Figure 32. The initial checkerboard model.

S-wave velocity perturbation [%]

P-wave velocity perturbation [%]

Velocity perturbation [%]

Depth

(km

) D

epth

(km

)

Distance (km)

Distance (km)

Distance (km)

Depth

(km

)

31

Figure 33. Velocity model made

with handpicked data, using an

CMM velocity model as starting

model. For S-wave velocities.

Figure 34. Velocity model made

with handpicked data, using an

CMM velocity model as starting

model. For P-wave velocities.

The velocity model resulting from the handpicked data, using an starting velocity model resulting from tomography based on CMM data in figure 33 and 34. The high velocity zone is also here visible in the north western part of model.

S-wave velocity

[km/s]

P-wave velocity

[km/s]

32

Figure 35. The locations of the

earthquakes before relocating

them in the tomography algorithm.

The two parallel North-South

faults are seen, as are the

seismicity stretching westwards.

Depth is 3-4 km.

Figure 36. The locations of the

earthquakes after being relocated

in the tomography algorithm. The

seismicity has lined up in a more

clear pattern than prior to the

relocation process. Depth is 3-4

km.

The relocated earthquakes are seen in figure 36. Comparing these locations with the initial ones in figure 35, it is clear that the seismicity has concentrated and lined up towards the faults and has a more regular pattern after relocation.

33

6. Discussion

The shallowest part of the crust, the upper most 1 km, is not well resolved in the model. This is due to the strong heterogeneity in the near-surface crust, it is difficult to construct the model parameterization finely enough to handle such heterogeneity. Also, there are not many rays crossing in the uppermost part which is needed for resolution.

Anomalies in velocity reveal information of variation in porosity, temperature and composition. Porosity is especially important at shallow depths. A higher porosity results in lower velocities. A lower temperature results in a higher velocity than a higher temperature, and increasing the temperature often results in a slightly larger decrease in velocity of the S-waves than the P-waves.

The further you go down into the earth, the higher the pressure is. This is the reason why the velocities increase with depth such as in figures 23 and 24.

In figures 18-21 a relatively large high velocity area can be seen in the north-western part. One possible cause of the zone could be that a magma body cooled and lithified at very high pressure, causing the body to be very compact and therefore resulting in the high velocities.

In figures 20 and 21 there are regions of lower velocity, mainly in the center of figure and in the southern and eastern part. These are mainly concentrated in the areas with high seismicity. Outside these areas there are regions with higher velocity, the higher velocities are caused by unfractured rock. The lower velocities are interpreted to be the result of porous material caused by the faults. Although, it can be discussed whether or not the porosity is the reason for why the earth ruptured in that very area or if it is the result of the ruptures.

The larger low-velocity anomaly in the upper most 6 km in the western part of figures 25 and 26 is probably the result of high porosity, which often is the case in the shallow crust. Within the area of low velocity, the seismicity is also very low. If the earth is highly deformable, little stress is built up and therefore there will be little release of stress and hence no earthquakes.

Rocks fail by brittle failure at shallow depth where temperatures are low. At a greater depth rock deforms due to ductile flow which occurs at a higher temperature. Therefore, earthquakes mostly occur above the transition zone between these two zones. The base of the brittle crust is hence the cut-off depth of the seismicity, where the cut-off depth is defined as where a sharp drop in number of events occur. The thickness of the brittle crust increases from 7 km in west to 9 km in the middle of the model. The reason could be that the spreading ridge is situated at the western part of the model area and so the crust is younger and hotter here.

According to the checkerboard tests, the resolution is good from 2 km down to 9 km. The S- and P-model´s resolution is in good agreement.

Comparing the velocity model resulting from pure CMM data with the velocity model resulting from handpicked data (using the CMM velocity model as starting model), roughly the same structures are seen. The high-velocity anomaly in the north-western part of the model area is enhanced indicating that the anomaly is likely to be real.

The lower Vp/Vs ratios within the northern part of the model, are interpreted to be caused by a geothermal area.

34

Plotting the raw CMM data in a Wadati diagram, reveals that the Vp/Vs ratio used in the CMM algorithm was 1.75. This may have caused later problems since the true Vp/Vs ratio for Iceland is closer to 1.78 than 1.75.

Figure 37. A Wadati diagram indicating that the Vp/Vs ratio used in

the CMM algorithm is 1.75.

As expected, some errors occur. The errors might come from timing or picking, which has its origin in both instrumental and manmade mistakes. Obviously the errors for picking ought to be larger for the S-waves compared to the P-waves since they arrive within the P-wave coda. Another thing to keep in mind is that there might be errors due to mislocation of the stations.

Errors may also be generated during the forward part of the tomography. Those might be caused by chosen parametrization and dizcretization. Errors can originate also from selecting a starting velocity model and initial earthquake locations. These are after all only crude estimates since the true velocities and locations are unknown. If the initial assumed solution is not close enough to the true solution the minimized misfit might just have reached a local minimum, not a global minimum. The method explores the region concentrated around its initial guess, which is why it is very important to have a good first guess of the solution and to evaluate different starting models.

There are many different ways, not presented here, of solving the problem stated in this thesis. Different approaches to the problem yield different solutions. There is no unique solution to the problem. The solution depends on the choices that are made upon what data are used and the model that is sought. With the newly gained knowledge, all these choices are based upon what I thought, with guidance from my supervisors, seemed reasonable. Therefore, I have no intention in claiming that this is neither a final nor true result. It is simply my version of a solution. The result presented in this thesis is not my final result, it is as far as I proceeded during the work of this thesis.

35

7. Conclusions

The output data catalogue given from CMM from the aftershock-sequence from May 29th is run through the tomography algorithm PStomo_eq. The algorithm uses the LSQR method to solve the inverse problem. The model is regularized by smoothing of slowness perturbations, variation of Vp/Vs ratio and the cross-gradient.

The data of highest quality are selected as the basis for the tomography algorithm. It is run with 89122 recorded phases from 3850 earthquakes selected from original 19449. The resulting velocity models are well constrained from 2 to 9 km depth.

The Vp/Vs ratio within the area varies from 1.74 to 1.82. The velocities increase with depth starting from 2 km where the P-velocity is 4.6 km/s and the S-velocity is 2.7 km/s, at depth 10 km the P-velocity is 6.9 km/s and S-velocity is 4.0 km/s.

A large low-velocity anomaly is found between depths of 2 km to 4 km, stretching from 21.5° to 21.2° W. It is interpreted to be caused by high porosity within the area. In the horizontal slices a high velocity area is seen in the north-western part of model. This is interpreted to be caused by a magma intrusive lithifying as crystalline rock under very high pressure.

The depth to the base of the brittle crust increases from 7 km in west to 9 km in the middle of the model.

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8. Acknowledgements

A special thanks to my supervisors Ari Tryggvason and Ólafur Gudmundsson who provided me with great inspiration that made my interest for seismology grow stronger. Ari for all your patience concerning my little experience with programming combined with a very complex tomography algorithm. Ólafur for your great pedagogical calmness while trying to answer all my questions. I also wish to thank to Cecilia Johansson who guided me through the steps of how to perform a bachelor project and answered all questions that came up along the way.

I wish to dedicate this report to a dear relative, friend and neighbor of mine Margareta Klingberg who tragically past away during the end of this project. She had a great passion for the earth and all that was going on inside of it, she always came up with tricky questions and was eager to get an update about my proceedings in the project. She will be missed.

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