Statistical analysis of small structures in rotated crustal blocks near the Húsavík-Flatey fault, northern Iceland

Natalie Hummel Senior Integrative Exercise

12 March 2020

Submitted in partial fulfillment of the requirements for a Bachelor of Arts degree from Carleton College, Northfield, MN.

Table of Contents

ABSTRACT INTRODUCTION……………………………...…………………………………..…1 GEOLOGIC BACKGROUND…………………..……………………………………3

Plate Configuration in Iceland……………………..………………………….4 Common Structural Patterns in Iceland……………………………..………...6 Block Rotation Near the Húsavík-Flatey Fault………………………..…...…8

STATISTICS ON DIRECTION AND ORIENTATION DATA…....………………10 The Mean…..………………………………………………………………...12 Inference on the Mean…………………………..………………………….. 13 Regressions…………………………………..………………………………14

Regression significance and confidence intervals……………….…..15 TYPES OF STRUCTURES……………………..…………………………………..16 DATA ANALYSIS AND RESULTS………………………………………..…...…20

Patterns in Small Structures with Fault-Normal distance……………....……20 Far-Fault Structures……………………………..…………………………...20 Mid-Distance Structures………………………………………..……………24

Dike Regressions………………………………..…………………...24 Vein Zone Regression…………………………………..……………28

Predictions……………………………………..……………………………..30 Near-Fault Structures………………………………………..……………….32

MODELS OF ROTATION………………………………………………..…………34 DISCUSSION……………………………………………………..…………………38

Timing of Formation……………………………………………………..…..38 Vein Zones………………………………………………………..………….39 Strike-Slip Faulting………………………………………………………..…40 Normal Faulting………………………………………………………..…….41 Anisotropy……………………………………………………..…………….42 Limitations of Stress Inversion……………………………………..………..43

CONCLUSION………………………………………….…………………………..44 ACKNOWLEDGEMENTS……………………………….………………………...45 APPENDIX …………………………………………..……………………..………46

A. MEANS TO FAR-FAULT STRUCTURES………………………..……46 B. USE OF DIKE POLES TO ESTIMATE ROTATION GRADIENT …....48 C. COMPARISON OF DIKE AND VEIN ZONE REGRESSIONS….….....50 D. TJORNES PENINSULA………………….....…………………….……..53

REFERENCES CITED………………………………………………..……...……..55

Statistical analysis of small structures in rotated crustal blocks near the Húsavík-Flatey fault, northern Iceland

Natalie Hummel Carleton College

Senior Integrative Exercise March 12, 2020

Advisor: Sarah J. Titus, Carleton College

ABSTRACT

The right lateral Húsavík-Flatey fault, in and just off the coast of northern Iceland, provides a unique opportunity to observe an oceanic transform fault on land. Structural and paleomagnetic measurements from lavas and dikes on Flateyjarskagi peninsula indicate that rocks adjacent to the fault have rotated clockwise more than 100°. I examine the effect of this rotation on the orientations of small faults and vein zones near the transform. I also use patterns in small-scale data to build on models of the style of deformation surrounding the fault. Previous studies provide constraints on the extent of rotation across Flateyjarskagi, which I use to estimate the expected orientations of rotated structures. The orientations of some small faults are consistent with expected rotations, but many structures appear to post-date rotation. Rotation of an older subset of the small faults on Flateyjarskagi accounts for patterns that have previously been attributed to variations in the stress field near the transform.

Keywords: block rotation, transform faults, directional statistics, Tjornes Fracture Zone, Iceland

1

INTRODUCTION

The Tjörnes Fracture Zone in northern Iceland accommodates right-lateral

transcurrent motion between two segments of the mid-Atlantic ridge (Fig. 1). A mantle

plume beneath Iceland has exposed portions of the Tjörnes Fracture Zone on land on the

peninsulas of Flateyjarskagi, Tjörnes, and Tröllaskagi (Wolfe et al, 1997; Hardarson et al,

1997). This presents an opportunity to study an active oceanic transform fault on land,

keeping in mind that Iceland tectonics are atypical of mid-ocean ridges due to the plume.

The Tjörnes Fracture Zone has been studied extensively to improve understanding of

tectonics in Iceland (Young et al, 1985; Fjader et al, 1994; Steffanson et al, 2006; Karson,

2017) and of transform faults in general (Bergerat et al. 2000, Angelier et al, 2000;

Garcia and Dhont, 2005; Garcia et al, 2002; Horst et al, 2018; Titus et al, 2018).

The Húsavík-Flatey fault is commonly identified as the best analogue in the

Tjörnes Fracture Zone for a mature oceanic transform and has likely accommodated tens

of kilometers of displacement (Young et al, 1985; Horst et al, 2018). Though most of the

Húsavík-Flatey fault itself is submerged, the damage zone south of the fault spans several

kilometers of northern Flateyjarskagi. Studies on stresses near the transform report

unusually complex patterns in the small faults in this damage zone (Angelier et al, 2000).

Structural and paleomagnetic measurements from lavas and dikes indicate that blocks of

crust within a couple of kilometers of the fault have rotated clockwise more than 100°

(Young et al, 1985; Horst et al, 2018; Titus et al. 2018; Young et al, 2018). Previous

workers have speculated about the complications that this rotation presents for

paleostress analysis (Young et al. 1985, Bergerat et al, 2000; Horst et al, 2018; Young et

al, 2018), but none have interpreted small-scale data with rotation in mind.

ÞH

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STA

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IR

KR

AF

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H Ú S AV Í K - F L AT E Y FA U LT

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D A LV Í K L I N E A M E N T

G R Í M S E Y L I N E A M E N T

TJORNES FRACTURE ZONE

Akureyri

Húsavík

Dalvík

T R OL LASK

AG

I

FLATEYJARSKAG

I

T J Ö R NES

WVZ

TFZ

EVZ

200 km

N

SISZ

KR

Legend:

Extrusives and seds >11 Ma

Extrusives and seds 5.3-11 Ma

Extrusives and seds 2.6-5.3 Ma

Extrusives and seds 0.8-2.6 Ma

Extrusives < 0.8 Ma and hyaloclastites <1.2 Ma

Fissure swarms

Active central volcano

Inactive central volcano

Anticlinal bedding axis

Unconformity

Fault

Fault with normal displacement

City

Figure 1. Map if the Tjornes Fracture Zone . Rocks in blue predate 5.3 Ma, and rocks in brown were deposited from 5.3 to 0.8 Ma, after the ridge jump c. 8 Ma. The inset shows the location of the Tjornes Fracture Zone (TFZ) in Iceland, and the locations of the Northern Volcanic Zone (NVZ), the left lateral South Icelandic Seismic Zone (SISZ), the Western Volcanic Zone (WVZ), and the Eastern Volcanic Zone (EVZ). Modified from Titus et al (2018).

NV

Z

2

3

I present hundreds of new measurements of structures including small faults,

dikes and veins from around the Tjornes Fracture Zone. I analyze these data, in

combination with data from previous studies (Bergerat, 2000; Garcia and Dhont, 2005;

Fjader et al, 1994), to assess the effects of regional kinematics, crustal anisotropy, and

block rotation on small structures in highly deformed areas on Flateyjarskagi and

Tjörnes. I use spatial patterns in dikes to quantify rotation across Flateyjarskagi and to

predict the orientations of rotated faults and vein zones. Populations of faults broadly

match the expected orientations in several locations, but a significant amount of

deformation appears to post-date rotation. I also use the orientations of rotated and

unrotated small faults to build on models of the timing and geometry of rotation near the

Húsavík-Flatey fault.

GEOLOGIC BACKGROUND

Plate Configuration in Iceland

A segment of the mid-Atlantic ridge--the Northern Volcanic Zone--strikes N-S

through east-central Iceland (Fig. 1), accommodating divergence between the North

American plate (western Iceland) and the European plate (eastern Iceland) (Palmason and

Saemundsson, 1974; Bodvarson and Walker, 1963). This ridge segment connects to

offshore ridges via partially-exposed zones of transcurrent motion, including the Tjörnes

Fracture Zone, a system of ESE-striking transform faults in and just off the coast of

northern Iceland. The Tjörnes Fracture Zone is composed of three sub-parallel transform

faults, as well as systems of cross-faults and zones of complex distributed deformation.

Relative plate motion in northern Iceland has an azimuth of ~102°, slightly oblique to the

4

three transforms, suggesting transtensional rather than pure transform motion (Árnadóttir

et al, 2008).

The plate boundary in Iceland is evolving due to the westward migration of the

mid-Atlantic ridge relative to the mantle plume beneath Iceland. In the last several

million years, the mantle plume passed east of the mid-Atlantic ridge. Between 9 and 7

Ma (Garcia et al, 2003), the ridge segment passing through Iceland jumped east to remain

above the hotspot, forming the Northern Volcanic Zone. Spreading was accommodated

on two parallel ridge segments for several million years as the Northern Volcanic Zone

developed (Garcia, 2003). The Tjörnes Fracture Zone, in the north, and the Mid Icelandic

Belt, in the south, formed to accommodate transform motion between the Northern

Volcanic Zone and the older ridge segments to the west. In northern Iceland, the ridge

jump created an unconformity between 12-8 Ma lavas produced at the spreading ridge to

the west and lavas younger than 5.3 Ma produced after the ridge jump (Fig. 1).

The Northern Volcanic Zone propagated to the north overtime, transferring

transform motion in the Tjörnes Fracture Zone onto more northerly structures. The

northernmost structure, the Grimsey lineament, is likely the youngest, and is currently the

most seismically active (Stefansson et al. 2008, Rögnvaldsson et al. 1998). The central

structure, the Húsavík-Flatey fault, shows evidence of significant displacement and

remains a source of seismic activity. There is little modern seismic activity along the

southernmost fault, the Dalvik lineament. However, seismicity diminishes significantly to

the south of the Dalvik lineament, indicating that the lineament might form a boundary

along which smaller faults terminate (Stefansson et al, 2008).

5

Common Structural Patterns in Iceland

Throughout Iceland, lavas are intruded by dikes and cut by both fissures and

normal faults that strike N-S, approximately perpendicular to plate motion. Normal

faulting is the dominant mode of faulting in most areas (Forslund and Gudmundsson,

1992). Normal faults commonly dip steeply--between 70° and 80°--indicating that they

may form by linking lava-perpendicular joints (Forslund and Gudmundsson, 1992).

Normal faults also commonly follow dike margins (Karson, 2017).

Iceland deviates from a typical mid-ocean ridge in several ways due to the

underlying mantle plume and the active evolution of the plate boundary. The mantle

beneath Iceland is hotter and less dense than average mantle beneath a spreading center,

and the crust is several kilometers thicker than most oceanic crust (Saemundsson, 1979).

Complicated structural patterns arise in regions between overlapping and propagating

ridge segments (Green et al, 2014; Karson, 2017). Motion also often takes place on

multiple structures, rather than on a single ridge or transform. For instance, spreading

takes place on the parallel Western and Eastern Volcanic Zones, and transform motion in

the Tjörnes Fracture Zone is accommodated on several major faults (Sheiber-Enslen et al,

2011).

Crustal block rotation is common in Iceland in bookshelf fault systems. Slip in the

left-lateral transform zone in southern Iceland, the South Icelandic Seismic Zone, takes

place almost exclusively along N-S-striking right-lateral faults, resulting in

counterclockwise block rotations (Fig. 2). Green et al. (2014) document smaller-scale

bookshelf faulting between overlapping ridge segments within the volcanic rift zone

Figure 2. Model of bookshelf faulting in the left lateral South Icelandic Seismic Zone from Sigmundsson et al. (1995). Overlapping ridge segments--the Reykjanes ridge to the west and Eastern Volcanic Zone to the east--are shown in dark grey. Slip occurs primarily along N-S-strik-ing right lateral faults, resulting in counterclockwise block rotations between the ridge segments.

6

7

southeast of the Tjörnes Fracture Zone. Karson (2017) also documents ridge-parallel

strike-slip faults throughout Iceland, independent of bookshelf systems (ie. without

regular sets of parallel faults). In most cases, ridge-parallel strike slip motion is attributed

to the anisotropy of oceanic crust (Karson, 2017; Green et al, 2014). Slip occurs on pre-

existing planes of weakness such as normal faults and dike margins, rather than on Riedel

shears or shear-parallel faults.

Block Rotation Near the Húsavík Flatey Fault

There is strong evidence to support significant clockwise block rotation near the

Húsavík-Flatey fault on the peninsula of Flateyjarskagi. Dikes deviate from the N-S

strikes observed elsewhere in northern Iceland and paleomagnetic data indicate that lavas

and dikes have rotated clockwise around a steep but non-vertical axis (Fig. 3). The extent

of clockwise rotation increases progressively approaching the fault, likely exceeding 100°

in the nearest on-land areas (Horst et al, 2017; Titus et al, 2018; Young et al, 2018).

Young et al. (1985) first documented a gradual change in dike strikes approaching

the Húsavík-Flatey fault in a detailed paper on the geology of the Tjörnes Fracture Zone.

They attributed this pattern to a rotation gradient caused by heterogeneous simple shear

(Fig. 4B). The deformation of exposed rocks is brittle but appears to approximate simple

shear on a large scale. This shear is heterogeneous due to the variation in rotation with

proximity to the fault. The authors propose that rotation took place as a result of broadly

distributed transcurrent deformation prior to the development of the fault, and that

rotation ceased after a major fault was established. The pattern in dike and lava rotation is

complicated by an anticline in east-central Flateyjarskagi (Fig. 1), which likely also

formed during the initiation of the Northern Volcanic Zone (Young et al, 1985).

Dik

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Figu

re 3

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s of

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how

ing

the

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tion

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(le

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cum

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(201

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ear

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fined

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g et

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how

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ts fr

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et a

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9

10

Several recent studies (Horst et al, 2018, Titus et al, 2018, Young et al, 2018)

present paleomagnetic data confirming large-magnitude block rotations post-dating dike

emplacement. Paleomagnetic remanence directions preserved in lavas and dikes deviate

from the expected direction in Iceland. These deviations increase toward the fault and

suggest clockwise rotation about a steeply plunging axis (Fig. 3). For the most part, the

magnitudes of dike and lava rotations are statistically indistinguishable, suggesting that

the majority of dikes pre-date rotation (Titus et al; 2018).

Horst et al. (2018) interpret the variability in their paleomagnetic data as

evidence for rotation of irregular crustal blocks (Fig. 4C) with potential variation along

Húsavík-Flatey fault-strike, whereas Young et al (1985) and Titus et al (2018) assume

that a gradient with fault-normal distance is sufficient to characterize the pattern of

rotation in the field area. Titus et al. (2018) use regressions of structural and

paleomagnetic measurements from dikes and lavas to quantify rotation as a function of

fault-proximity, assuming no variability parallel to the fault. This assumption agrees with

maps of dikes published by Young et al. (Fig. 3) and the heterogeneous simple shear

rotation model for rocks on Flateyjarskagi west of the anticline and central volcano.

Though current paleomagnetic data are spatially variable, they do not provide

opportunities to compare between areas with the same fault-normal distance and different

locations along strike (excluding tilted rocks near the Flateyjarskagi anticline axis).

Therefore, there is not currently good evidence for variation in rotation along strike.

STATISTICS ON DIRECTIONS AND ORIENTATIONS

11

Statistics provides tools to rigorously compare populations and describe patterns

in scattered data sets. For this study, I used the geologyGeometry library in R studio for

statistical computations (Davis and Titus, 2017). This library contains statistical tools

developed specifically for the analysis of direction and orientation data (e.g.

measurements of strike and dip).

The data used in this study consist of planes (veins, faults, dikes) and plane-ray pairs

(faults with slickenside striae), which I will describe as directions and orientations,

respectively. An orientation is fully constrained in 3-D, whereas a plane (a direction) can

rotate around its pole and appear the same. An orientation therefore has one more degree

of freedom than a direction, and must be described with three numbers (strike, dip, rake)

instead of two (strike, dip).

The geologyGeometry library represents orientations (faults with slickensides) as

rotation matrices. A rotation matrix for a plane-ray pair has the form:

,

where is the pole to the fault plane and is a ray on the fault plane that is

perpendicular to the slip vector, called the fault “vorticity vector”. If slip on the fault

described by R is approximated as simple shear deformation of a block, this deformation

would rotate the boundary of the block about . Note that since and are three

dimensional vectors, R is a 3 by 3 matrix. This matrix rotates to the y-z plane and

to the y-axis. Rotation matrices of this form have a one-to-one correspondence with

orientations and can be averaged, regressed, and manipulated (Davis and Titus, 2017).

12

Most statistical techniques applied to scalar data sets--such as the mean, variance,

linear regression, etc.--are not directly applicable to directional data. The mean of a set of

n scalars is their sum divided by n. There is no directly analogous process for averaging n

planes because it is not clear what it means to add or divide planes. Planar geologic data

are often described by two scalars: a strike and a dip, which can be individually averaged.

However, averaging strikes and dips separately is not a reliable method for averaging a

group of planes. In general, the components of a multidimensional datum, like a plane,

should be treated together (Davis and Titus, 2017). We therefore use statistical techniques

developed specifically for direction and orientation data, such as the Bingham

distribution, the Frechet and projected arithmetic means, and small circle regressions.

The Mean

The projected arithmetic mean and the Frechet mean both describe the central

tendency of a set of directions or orientations, and are usually similar but not identical.

The arithmetic mean is the sum of the elements in a dataset divided by the number of

elements. For rotations, the resulting matrix is not always a rotation (special orthogonal)

matrix. The projected arithmetic mean is the rotation matrix closest to the arithmetic

mean (Davis and Titus, 2017).

The Frechet variance:

,

describes the variance of a set of n rotations, , around a mean, . The function

computes the magnitude of the rotation that takes to (Davis and Titus, 2017). The

13

Frechet mean is computed iteratively by finding the that minimizes . A similar

method is used to calculate the Frechet mean of a set of directions.

Inference on the Mean

The mean of a dataset reflects the true mean of the sampled population with some

uncertainty. Therefore, the mean of a set of measurements can deviate to some extent

from an expected mean, even if the two values are not statistically distinct. Confidence

regions for the mean of a set of directions can be computed using bootstrapping or

probability distributions, and compared to hypothesized means. If a hypothesized mean

plots outside of the 95% confidence region for the mean of a set of data, then the true

mean of the data is distinct from the predicted mean at the 95% confidence level (p <

0.05) (Tauxe, 2003).

The Bingham distribution is a probability distribution for directional data

presented by Bingham and Mardia (1978), which describes data that form an elliptical or

girdle-like cluster around a mean (Borradaile, 2003). The geologyGeometry library uses

the Bingham distribution to construct confidence regions for means, using the method

described by Tauxe (2003).

Bootstrapping can also be used to construct a confidence region for the mean of

directional data. This approach is computationally more intensive, but does not assume

that data are distributed according to the Bingham distribution. Means are computed for

multiple bootstrapped datasets of the same size as the original data set, each formed by

randomly sampling the original dataset with replacement. The Malhalanobis distance

14

describes the distance of each bootstrapped mean from the center of the resulting multi-

dimensional cloud of means. An ellipsoid that encompasses the 95th percentile of the

Malhalanobis distances serves as a 95% confidence region for the population mean

(Davis and Titus, 2017). Both methods of inference work best on reasonably large

datasets, but the minimum number of data points for which inference is effective is a

function of the scatter in the data (Davis and Titus, 2017).

Regressions

I use regressions to quantify the average change in the direction of geologic

structures with some scalar variable (e.g. northing, width, etc.). Direction regressions can

be considered functions that take a scalar as input and predict a direction, answering

questions of the form: “What is the average expected dike direction in areas with an

easting of 5000 m?” Regressions also provide information about the rate of change in a

direction with a scalar variable. A small-circle regression assumes rotation about a fixed

axis, which results in predicted directions that follow a small-circle path.

Regressions are useful when the average direction of a population changes

gradually with some property of the population. Regressions are best applied to unimodal

data sets. The regressions described here predict a single orientation for a given value of

the independent variable. Multiple modes will effectively be averaged, obscuring any

patterns in individual populations of the data. It is also preferable to have an even

distribution of observations across the range of the independent variable. A regression is

computed by minimizing the difference between predicted and observed data points. For

orientations, this minimized quantity can be written as:

15

,

where Ri is an observed orientation paired with scalar si, R(si) is the orientation

predicted given si (Davis and Titus, 2017). The square of this difference is summed over

all data points. Functions that compute regressions attempt to find the function for R(si)

that minimizes this sum.

The proportion of variability in the data that can be explained by the regression is

defined as:

,

where is the Frechet variance of the data: a measure of the total variability. An R2

value describes how closely the predictions from a regression match the observed data,

but does not provide a measure of statistical significance.

Regression significance and confidence intervals

The statistical significance of a regression is described by a p value, which can be

determined by running regressions on permutations of the data. I create permuted data

sets by reassigning each direction to a scalar from the data set randomly, with

replacement. I then run regressions on each of these permuted data sets. The p value is

the proportion of permuted regressions with a higher than the original regression. Low

p values indicate a low probability that the data are randomly distributed.

I use bootstrapping to create confidence intervals for the parameters output by the

regressions. I compute regressions on a sample of data selected randomly, with

replacement from the original data set. These regressions give a sense for the range of

regression outputs expected for a random sample of the total population of faults or veins.

16

A small circle regression produces a starting direction, an axis of rotation, and a rate of

rotation, which can be described by a rotation matrix with 5 degrees of freedom. The

95% confidence region for a regression is the 5-dimensional ellipsoid that encompasses

the 95th percentile of Mahalanobis distances of a cloud of bootstrapped regressions. This

5-dimensional ellipsoid can then be used to solve for a 95% confidence region for a

regression axis or rotation rate (Titus et al, 2018).

TYPES OF STRUCTURES

During the summer of 2018, I worked with Sarah Titus, Maxwell Brown, Paul

Ashwell, and Seth Waag-Swift to measure hundreds of brittle structures around the

subaerial regions of the Tjörnes Fracture Zone. These measurements supplement data

collected by Fjader et al. (1994), Bergerat et al. (2000), Titus et al. (2018), and

unpublished measurements taken by Dr. Titus, students, and collaborators in previous

years. Most of our data come from river valleys, road cuts, shorelines, and quarries due to

their relative accessibility. We classify structures into several categories: fractures, veins,

small faults with slickenside striae, vein zones, dikes, and large faults with gouge or

cataclasite (Fig. 5). The smallest veins and fractures represent millimeters of relative

motion, while the largest faults observed have likely accommodated tens to hundreds of

meters of slip.

Lava flows are identified by red paleosols between flows, or by bands of

amygdules aligned parallel to bedding. Most dikes are basaltic and stand out from the

host-rock due to baked margins or sets of margin-parallel fractures. Dikes in pre-

unconformity lavas tend to be several decimeters to several meters in width.

Figure 5. Images of structures in the field area. (A) a vein with red staining and multiple layers. (B) A planar vein zone. (C) A red-stained fault surface with slickenside striae and mineralizedsteps down in the direction of motion. (D) A large fault on Tjornes with bright red and orangegouge. (E) A dike margin. (F) A faulted dike margin in a quarry in Eyjafjordur.

..

17

18

Veins in the study area range in size from ~1 mm to several centimeters, are filled

with quartz, calcite, and/or zeolites, and are sometimes stained by iron oxides or other

colorful minerals. Veins frequently follow faults, fractures, and dike margins. Many large

veins contain multiple layers, indicating several episodes of opening (Fig 5A). Mineral

fibers in some veins suggest opening perpendicular to the plane of the vein, while

displacement across other veins suggests oblique opening. For the purposes of this study,

I do not distinguish between these categories because I do not use veins for kinematic

analyses, and because in most cases, it is not possible to determine the direction of

opening. An individual vein measurement does not carry much weight because veins

frequently bend, pinch, and swell, but there is often a dominant orientation in a large set

of vein measurements. Vein zones are cm-to-m scale planar regions of sub-parallel or

anastomosing veins. These structures can only be measured by sighting, introducing

some uncertainty. Even so, vein zone measurements tend to be less scattered than vein

measurements because the zones average out some of the variability in individual veins.

Sometimes the rock on one side of a fault plane is removed, exposing a plane with

slickenside striae indicative of the fault slip direction. Observations of fault surfaces with

slickenside striae are particularly valuable, both as kinematic indicators and as fully-

constrained orientations. The sense of slip on a fault is indicated by parallel score marks

or mineral fibers pointing in the direction of slip (Fig. 5C). A variety of features on fault

surfaces can be used to determine sense of motion (Petit, 1986). We most commonly

observed steps moving down in the direction of motion. These steps can cause fault

surfaces to feel smooth in the direction of motion of the missing block, and rough in the

opposite direction. We also occasionally interpreted features as Riedel shears or lunate

19

structures (Fig. 6). It is important to note that slip indicators can be subtle and subject to

misinterpretation. For instance, whether a feature is identified as a step or a Riedel shear

changes the inferred direction of motion on the fault. Whenever possible, multiple

indicators should be identified. Since this is not always an option, there is considerable

uncertainty associated with fault-slip direction.

Fractures are planar breaks that lack evidence of slip or opening. Some features

recorded as fractures, particularly the large fractures in hyaloclastites south of the

Húsavík-Flatey fault on Tjörnes, may have accommodated some slip, but lack

displacement markers. Larger faults in the study area contain up to several decimeters of

red, green, or tan gouge, and up to several meters of intense damage. Many such faults

create topographic lows. Some faults have damage zones with several slickenside

surfaces, which occasionally have inconsistent senses of slip.

RESULTS AND DATA ANALYSIS

Patterns in small structures with fault-normal distance

When viewed together, the orientations of structures on Flateyjarskagi are

extremely scattered. Poles to dikes, veins, and small faults are distributed along

horizontal to NW-dipping girdles (Fig. 7). Local variations account for some of this

scatter. In order to parse spatial patterns in each type of structure, I divide data from

Flateyjarskagi west of the anticline into three regions based on distance from the

Húsavík-Flatey fault: far-fault (> 20 km), mid-distance (2-20 km), and near-fault (< 2

km). Structures from areas more than 20 km from the fault are well outside the rotation

gradient documented on Flateyjarskagi. I consider these far-fault structures to form

A B

Figure 6. Two examples of fault slip indicators with similar appearances that suggest opposite senses of motion. In A, reidel shears cause abrupt steps up and gradual steps down in the direction of slip, and in B, mineral fibers form an abrupt step down in the direction of slip. Figure adapted from Petit et al, 1986.

R shear

20

Approaching the HFF

Figu

re 7

. Pol

es t

o di

kes,

vein

zon

es, a

nd s

mal

l fa

ults

on

Flat

eyja

rska

gi a

nd T

rolls

kagi

wes

t of

the

Fla

teyk

arsk

agi

antic

line,

co

lore

d by

faul

t-nor

mal

dis

tanc

e, w

here

pur

ple

is n

ear-

faul

t and

yel

low

is 2

8 km

from

the

faul

t. Ye

llow

squ

ares

indi

cate

ave

rage

fa

r-fa

ult

orie

ntat

ions

and

gre

y sq

uare

s in

dica

te p

redi

cted

rot

ated

orie

ntat

ions

. B

lack

tria

ngle

s in

dica

te t

he s

trike

s of

fau

lts

pred

icte

d ba

sed

on le

ast c

ompr

essi

ve st

ress

with

an

azim

uth

of 1

02°.

Kam

b co

unto

urs a

re sh

own

for m

ultip

les o

f 3 si

gma.

21

dike

sle

ft la

tera

l fau

ltsno

rmal

faul

tsrig

ht la

tera

l fau

ltsve

in z

ones

2-5

km

10 k

m

>18

km

<2 k

m

22

“default” patterns--analogues for all locations in the study area before shearing near the

Húsavík-Flatey fault.

The orientations of most far-fault structures have unimodal distributions, which I

describe using the mean. Structures fewer than 20 km from the transform are potentially

affected by the rotation gradient. I describe patterns in mid-distance structures using

regressions against fault-normal distance and through comparisons to expected fully-

rotated orientations. Structures in the intensely deformed rocks less than 2 km from the

transform are numerous, scattered, and exhibit variable patterns along fault-strike.

Far-Fault Structures

At field stations more than 10-14 km from the Húsavík-Flatey fault, lavas dip

shallowly to the SW. Far-fault rocks contain veins, fractures, dikes, and sometimes fault

surfaces with slickenside striae. Most dikes and strike-slip faults dip steeply and are

approximately perpendicular to lava bedding, as illustrated in Figure 8. Right-lateral

faults strike mostly SE. Most left lateral faults strike SSE-NNE. Normal faults mostly

strike NNE and dip between 60° and 80°, typical of normal faults in Iceland (Forslund

and Gudmundsson, 1992). Several faults recorded as right-lateral are approximately

parallel to the primary population of left lateral faults and vice versa, indicating that the

slip senses of some faults could have been misinterpreted (Fig. 7). Excluding potentially

misidentified faults and dike-parallel faults, most far-fault slickenside surfaces are

broadly kinematically consistent with the ENE-WSW directed extension in Iceland.

I use bootstrapping and the Bingham distribution to compute 95% confidence

intervals for the mean poles to right-lateral, left lateral, and normal faults farther than 20

km from the Húsavík-Flatey fault. These datasets are mostly unimodal (Fig. 7). I

0

5

10

15

20

25

0 25 50 75

Degrees from Lava

coun

t

0

100

200

0 25 50 75Degrees from Lava

coun

t

0

25

50

75

100

0 25 50 75

Degrees from Lava

coun

t

0

5

10

0 25 50 75

Degrees from Lava

coun

t

0

5

10

15

20

0 25 50 75

Degrees from Lava

coun

t

Figure 8. Histograms showing the number of degrees between structures and average local lava bedding, where 90° is perpendicular and 0° is parallel. Note that plots have different y-axes. Equal area plots show poles to each type of structure after being rotated such that local lava bedding is taken to horizontal. Kamb contours represent regions containing 3, 6, 9, and 12 standard deviations of the distribution. Points on ste-reonets are colored by fault-normal distance. Unrotated structures are nearly vertical at all distances from the HFF.

Dikes Veins Vein Zones

Right Lateral Faults Normal Faults

0

10

20

30

0 25 50 75

Degrees from Lava

coun

t

Left Lateral Faults

n=302n=1008n=97

n=123n=71 n=55

23

24

removed 2 outliers from the right-lateral fault data, out of a total of 33 data points.

I analyzed all of the 47 observed left lateral faults, and 25 observed normal faults. The

relatively small number of data points in each category puts bootstrapping on shaky

ground. Davis and Titus (2017) demonstrate that with fewer than approximately 100

observations, the 95% confidence region for a mean predicted by bootstrapping contains

the true population mean less than 95% of the time. This implies that the confidence

regions produced using bootstrapping are probably too small. The Bingham distribution

analysis produced wider confidence regions, likely more reflective of the true certainty of

the analysis. For this reason, I only discuss the results of the Bingham analysis, but the

results of the bootstrapping analysis are presented in Table 1.

I compare the means of observed fault poles to expectations based on local plate

motion. GPS data indicate that relative plate motion in northern Iceland has an azimuth

close to 102° (Árnadóttir et al, 2008). Laboratory experiments on isotropic rocks

demonstrate that faults form at angles close to 60° to the least compressive stress,

consistent with Andersonian fault mechanics applied to rocks with typical internal

friction coefficients (Scholtz, 1989). Therefore, extension towards 102° might be

accommodated on normal faults striking ~012° and/or on conjugate sets of right- and left-

lateral faults, striking ~162° and ~222°, respectively (Fig. 9). The right-lateral and normal

fault poles we observed in unrotated far-fault lavas do not deviate from this prediction

with statistical significance (p > 0.05). However, a vertical fault plane striking 222°--the

predicted left-lateral fault direction--plots outside of the 95% confidence region for the

mean of the observed left-lateral fault poles (Fig. 9).

Comparison of expected fault poles to the projected arithmetic means of small fault poles farther than 20 km from the Husavik-Flatey fault. Expected strike slip fault poles assume failure occurs on planes oriented 60° to a least compressive stress with an azimuth of 102°. Expected normal faults strike perpendicular to 102° and dip 70° in either direction. Normal faults are separated into two unimodal populations according to dip. The number of normal faults dipping in either direction is very small, so I also report a mean for the entire population of normal faults. Bootstrapping rejects the null hypothe-sis at a higher rate than Bingham inference.

Figure 9. General patterns in faults with slickenside striae from far-fault areas. (A) Equal area plots showing 95% confidence regions (computed using Bingham inference) for the projected arithmetic means of poles to right lateral, normal, and left lateral faults. The normal fault data are small and somewhat multi-modal, resulting in a large confience region. Data are presented in Figure 8. Colored squares indicate poles to faults predicted based on the direction of plate motion in Iceland. (B) A schematic representation of faulting, showing the discrepency between expected left lateral faults and observed left lateral faults, which are often parallel to dikes.

9

Table 1. Results of bootstrapping and Bingham inference comparing expected and observed fault poles.

26

Mid-distance Structures

North of the Gil Lautur line on Flateyjarskagi, lavas are visibly more deformed

and many structures do not follow patterns observed in far-fault areas. Previous studies

have established a rotation gradient in this area (Young et al, 1985, Young et al, 2018,

Titus et al, 2018). It is likely that some small structures have rotated with the lavas and

dikes. In this section, I explore the extent to which rotation on Flateyjarskagi has affected

patterns in structures between 2 and 20 km from the Húsavík-Flatey fault.

Dike Regressions

I use a logarithmic small-circle regression of dike data from Flateyjarskagi and

Tröllaskagi to quantitatively describe the rotation gradient near the Húsavík-Flatey fault

(Appendix B). Figure 10 shows the results of two small circle regressions of dike poles

between 2 and 20 km from the Húsavík-Flatey fault. Regression A assumes a constant

rate of rotation approaching the fault. Regression B fits a rotation rate that is proportional

to the natural logarithm of distance, predicting a steeper rotation gradient near the fault.

The logarithmic fit has a slightly better R2 (0.68 vs. 0.64) and is used in the rest of my

analyses. However, the logarithmic regression cannot be extrapolated to areas very near

the fault because rotation increases very rapidly at small distances. Both regressions

produce poles of rotation plunging steeply to the southwest.

These regressions are similar to regressions published by Titus et al. (2018),

except that they are computed from data from a more limited area. Titus et al. run

regressions on dikes from all of Flateyjarskagi west of the anticline, up to 28 km from the

fault. Only dikes within 10-15 km of the Húsavík-Flatey fault appear to have rotated due

5 km

Gil-

Látu

r lin

ean

ticlin

e ax

is

5 km

Gil-

Látu

r lin

ean

ticlin

e ax

is

Dik

esLa

vas

NN

distance from fault (km)

40

0

30

0

distance from fault (km)

180

30

Dik

e Re

gres

sion

sno

t inc

lude

d in

t

he re

gres

sion

s

Loga

rithm

ic

R =

0.6

78p-

valu

e <

0.01

2

Cons

tant

rota

tion

R =

0.6

36p-

valu

e <

0.01

2

HFF

20 k

m

Figu

re 1

0. (A

and

B) E

qual

are

a pl

ots

show

ing

smal

l circ

le re

gres

sion

s of

the

pole

s to

dik

es o

bser

ved

betw

een

2 an

d 20

km

from

the

Hus

avik

-Fla

tey

faul

t. Th

e ax

es o

f rot

atio

n ar

e pl

otte

d in

bla

ck. P

lane

s and

pol

es to

pre

dict

ed d

ikes

are

plo

tted

at 1

km

inte

rval

s bet

wee

n 2

and

20 k

m fr

om th

e fa

ult.

Pole

s ar

e co

lore

d by

faul

t nor

mal

dis

tanc

e, w

ith fa

r-fau

lt po

les

in g

reen

and

nea

r-fau

lt po

les

in p

urpl

e, a

s sh

own

on th

e m

ap o

f dik

e da

ta in

C. E

qual

are

a pl

ots i

n C

show

dat

a us

ed in

the

regr

essi

ons.

Dik

e st

rike

data

from

You

ng e

t al.

(198

5) a

re sh

own

on th

e m

ap.

The

map

in C

is a

dapt

ed fr

om T

itus e

t al.(

2018

).

N

27

28

to off-fault deformation, so the regressions presented here only use data up to 20 km from

the fault. This increases both regression R2 values and the apparent rate of rotation. Dikes

on eastern Flateyjarskagi do not follow the pattern well and were therefore excluded from

the regression. This is consistent with the map of dike strikes from Young et al. (1985)

shown in Figure 10. I also include several measurements from Tröllaskagi that seem

consistent with the pattern on Flateyjarskagi.

Dikes nearer than 2 km from the fault are also not included in the regression.

Dikes from this interval have extremely variable directions and make up a large

proportion of our observations (66 out of 108 dikes measured east of the anticline), so

including these data has a strong effect on the regressions. Regressions including dikes

from 0-2 km predict counterclockwise rotation, despite a clear clockwise pattern in the 2-

20 km distance interval. Excluding the 0-2 km range there are only 42 data points, but the

data are more evenly distributed in space, and R2 values for the regressions are improved.

Vein Zone Regression

Small circle regressions confirm a systematic change in the poles to vein zones

with proximity to the Húsavík-Flatey fault (p value < 0.01). The regression shown in

Figure 11 predicts a clockwise increase in vein zone strike with northeasting, similar to

the pattern observed in the dikes. In contrast to the dike regression, the axis of rotation

plunges shallowly to the southeast. The vein zone regression predicts 31° of rotation each

time distance to the Húsavík-Flatey fault is halved--similar to the 30.5° of rotation

predicted by the dike regression. However, the vein zone regression predicts an axis of

rotation closer to the small circle of predicted poles, so the predicted change in vein zone

direction is smaller overall. The vein zone regression predicts a 56° difference, measured

Freq

uenc

y

0 60 120

010

0020

00

Freq

uenc

y

20 50 80

010

0020

00

n=108

2-20 km13 sites

n= 42

2-20 km13 sites

R = 0.678p-value < 0.01

2R = 0.125p-value < 0.01

2

Vein Zones Dikes

Boot

stra

pped

Rot

atio

n Ra

tes

Rate of rotation (degrees) Rate of rotation (degrees)

Figure 6. (Top) Equal area plots showing vein zones and dikes observed east of the Flateyjarskagi anticline, colored by fault-normal distance. (Middle) Equal area plots showing the results of small circle regressions on these directional data against the natural logarithm of fault-normal distance. Colored plotted points indicate poles predicted at 1 km intervals between 2 and 20 km from the fault. Grey points are axes of rotation from 500 bootstrapped regressions, and grey regions indicate 95% confidence regions for the bootstrapped axes, derived from ellipsoids fit to higher-dimension-al bootstrapped regression outputs. Histograms show rates of rotation (degrees of rotation predicted each time distance to the Husavik-Flatey fault decreases by 1/e) from the bootstrapped regressions, with black lines indicating rates from the original regressions.

Mea

sure

men

tsRe

gres

sion

s

29

30

along a great circle, between vein zones 2 km from the Húsavík-Flatey fault and vein

zones 20 km from the fault, compared to an 80° difference between the predictions for

dikes at 2 and 20 km.

In order to compare regressions of dikes and vein zones, I use bootstrapping to

construct confidence regions for the regression parameters (Appendix C). I compute

logarithmic small circle regressions on 500 bootstrapped datasets for each type of

structure. The p value for the null hypothesis that the dike and vein zone regressions are

identical is 0.07, indicating that the regressions are almost distinct to the 95% confidence

level.

If the regressions are in fact distinct, this may be due to differences in the rates of

rotation, axes of rotation, and/or the initial (> 20 km) directions of each structure. In order

to compare rates and axes of rotation, I fit a 95% confidence ellipsoid to the bootstrapped

regressions of each structure, then examined the ranges of each parameter within each

ellipsoid. Figure 11 shows projections of these ellipsoids on equal area plots, representing

95% confidence regions for the regression axes. Although the axes of rotation appear to

differ between vein zones and dikes, the confidence regions overlap significantly. This

implies that there is not a statistically significant difference between the axes of rotation

for the two regressions. The overlap between rotation rates shown in the histograms in

Figure 11 also do not support a significant difference between the rotation rates of the

two regression.

Predictions

I do not have enough measurements of small faults between 2 and 20 km from the

Húsavík-Flatey fault to run regressions of fault orientation against fault-normal distance.

31

Our dataset contains a total of 90 left-lateral faults in the 2-20 km range, but they are all

either 2-3.5 km, 10 km, or 19 km from the transform (Fig. 8). The right-lateral and

normal fault data are similarly sparse between 3 and 10 km. There has been little rotation

beyond 10 km, so without data in the 3-10 km range, I was unable to describe patterns in

the small faults with regressions. Additionally, at several stations, the left-lateral and

normal faults data have multiple modes, which are not well-treated by regressions.

Instead of using regressions, I look for evidence of rotated small faults by

estimating the expected orientation of each type of fault (left-lateral, normal, or right-

lateral) in each location, and comparing these expected orientations to the observed

faults. Predicted orientations assume that each type of fault initiates with an orientation

identical to the Frechet mean of the orientations observed in unrotated areas (> 20 km

from the transform). I then rotate these far-fault average orientations according to the

rotation predicted by the logarithmic small circle dike regression shown in Figure 10. The

resulting predictions are shown in Figure 8. Left-lateral faults greater than 20 km from

the Húsavík-Flatey fault have a bimodal distribution, which is not well described by the

Frechet mean. Far-fault left-lateral fault planes have a similar distribution to dike planes,

so I use the dike prediction for the left-lateral fault planes, instead of the Frechet mean.

In most areas, some but not all of the observed small faults are consistent with the

predicted orientations. Left-lateral faults mimic dikes at all distances between 2 and 20

km, with the exception of several SSE-striking faults in each area, which might be

misidentified right-lateral faults. The regression slightly overpredicts the amount of

rotation in a set of apparently unrotated right-lateral faults from a station 10 km from the

Húsavík-Flatey fault. Between 2 and 3 km, a population of right-lateral faults is well

32

predicted, and a population matches the far-fault right-lateral orientation. Normal faults

are not well predicted at any distance. Most normal faults throughout the study area strike

NNE and dip between 50 and 80 degrees, regardless of distance from the Húsavík-Flatey

fault. There is an anomalous SE-striking population of normal faults between 2 and 5 km

from the fault, but these faults do not match the predicted rotated orientations.

Near-fault Structures

Patterns in the orientations of structures nearer than 2 km from the Húsavík-Flatey

fault are more complex than patterns between 2 and 20 km from the fault, possibly due to

larger post-rotation populations. I analyze more than one thousand measurements from

numerous stations along the northern coast of Flateyjarskagi, very near the fault. There is

considerable variation along strike in this area, so I divide the coastal data into an eastern

bin, a central bin, and a western bin, and consider patterns in each bin separately.

Structures from within 2 km of the Húsavík-Flatey fault are presented on equal

area plots in Figure 12. For simplicity, only poles are plotted in Figure 12, rather than

fault orientations. Faults are binned by type (left-lateral, etc.), so there are some

constraints on the rakes of the faults on each equal area plot. Clockwise rotation about a

steep axis should not, for the most part, rotate left-lateral faults such that they appear to

be normal or right-lateral. I therefore categorize faults as right-lateral, left-lateral, or

normal based on rakes measured in the field, and assume these descriptions applied to the

faults when they initiated. Rakes are reported as degrees to the slip vector measured

clockwise from strike, along the fault plane. Rakes between 315° and 45° are classified as

B

HFF

5 km

Left

Lat

eral

Fau

ltsN

orm

al F

aults

Righ

t Lat

eral

Fau

ltsVe

in Z

ones

Dik

es

far-fault average pole

predicted rotated pole

Figure 12. (A) Map of the northern coast of central Flateyjarskagi very near the Husavik-Flatey Fault with field stations, binned and colored by distance along fault strike. (B) Equal area plots showing poles tostructures from each bin, colored as in A. Grey squared indicate poles to Frechet means of structures greater than 20 km from the HFF: the expected unrotated directions. Black squares indicate poles to structures predicted by rotating far-fault Frechet means to a fault-normal distance of 2 km according to the logarithmic small circle dike regression.

N = 18 N = 22 N = 21

N = 14 N = 78 N = 80

N = 60N = 18

N = 12 N = 62 N = 75

N = 76N = 67N = 61

A

N

33

34

left-lateral, between 45° and 135° are normal, and between 135° and 225° are right-

lateral. Thrust faults are not plotted in Figure 12. There are only 25 total recorded thrust

faults out of a total of 631 total faults with slickenside striae, and it is likely that some of

them are misidentified normal faults.

Poles to structures of all kinds largely follow girdles defined by NW-dipping

lavas. Dikes do not appear to vary systematically along strike. Most near-fault dikes are

well predicted by the small-circle dike regression, but all three bins appear to have a

second minor population striking NE rather than ESE. Most left-lateral faults in the west

and east strike NE, similar to left-lateral fault surfaces far from the Húsavík-Flatey fault.

A population of left-lateral faults is parallel to the dikes in all three bins. About half of

the left-lateral faults in the central region are slightly clockwise of a typical unrotated

left-lateral fault. Right-lateral faults vary dramatically along strike. In the east, they

match right-lateral faults in unrotated areas. In the central bin, all faults broadly match the

predicted rotated orientation. In the west, right-lateral faults appear partially rotated, with

poles similar to post-rotation normal faults. Normal faults strike NNE all along the

northern coast of Flateyjarskagi, though normal faults in the central bin are on average

slightly clockwise of faults in other bins. Most vein zones strike NE-ENE, slightly

counterclockwise of dikes on average. In the west, some vein zones strike NS, and in the

east, a minority of vein zones strike ESE, similar to the prediction for a fully-rotated vein

zone.

35

MODELS OF ROTATION

The style of rotation observed south of the Húsavík-Flatey fault is not consistent

with bookshelf fault systems documented in the South Iceland Seismic Zone and

elsewhere in Iceland. Similar to a bookshelf system, blocks likely rotate around left-

lateral faults that initiate approximately perpendicular to the transform, along N-S

striking weaknesses in the Icelandic crust. However, the >100° rotations observed within

several kilometers of the fault cannot be explained by a single set of left-lateral faults,

necessitating at least 2-3 generations of left-lateral faults, as illustrated in Figure 4B (Nur

et al, 1986), or a more irregular block geometry (Fig. 4C). Additionally, unlike a typical

bookshelf system, areas very near the Húsavík-Flatey fault have experienced more shear

than areas at mid-distances.

Previous workers have proposed several modified bookshelf-like models to

explain the block rotations on northern Flateyjarskagi. Here, I distinguish between three

models, illustrated in Figure 13. Young et al. (1985, 2018) believe that rotation took place

during the development of the Húsavík-Flatey fault, before strain concentrated along a

single transform fault (Figure 13A). They use sandbox models to demonstrate

development of a curved fabric in a transform zone prior to the development of discrete

shears at the center of the zone.

Karson (2017) attributes the rotation gradient to migration of the transform

through northern Flateyjarskagi. His rift propagation model is based on models used by

Hey et al. (1980) to show migration of a discrete transform along the Cocos-Nazca

spreading center. However, a discrete transform is not sufficient to explain rotation;

transform motion must be distributed over the rotating area. If ridge segments overlapped

A

HFF

Figu

re 1

3. T

hree

mod

els f

or th

e ge

omet

ry o

f rot

atio

n ne

ar th

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usav

ik-F

late

y fa

ult.

(A) R

otat

ion

in a

bro

ad sh

ear z

one

prio

r to

the

deve

lop-

men

t of t

he tr

ansf

orm

. Not

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t rot

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this

mod

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on th

e Hus

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tion

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cted

in y

oung

er c

rust

(dar

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ray)

. (B

) B

lock

s rot

ate

betw

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rift s

egm

ents

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ore

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ent o

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s bet

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ts a

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he ri

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8.5

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Ada

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Sig

mun

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n et

al.

(199

5)

to i

llust

rate

boo

kshe

lf fa

ultin

g in

the

Sou

th I

cela

ndic

Sei

smic

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he N

orth

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anic

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pro

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tes

north

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gion

nea

r th

e de

velo

ping

Hus

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t. Th

is

mod

el d

oes n

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ply

rota

tion

north

of t

he fa

ult.

(C) A

nti-J

styl

e ro

tatio

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g w

hen

the

trans

form

is lo

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. Thi

s mod

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etim

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the

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nger

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s.

BC

NVZ

KRKR

KR

NVZ

NVZ

HFF

HFF

36

during propagation, the resulting N-S gradient in the spreading rate along the Northern

Volcanic Zone would cause distributed transform motion between the overlapping ridges.

This distributed shear could result in clockwise rotation of the lavas between the ridges.

Sigmundsson et al. (1995) provide a similar explanation for block rotation in the South

Icelandic Seismic Zone. The greater magnitude rotations near the Húsavík-Flatey fault

could be the result of the zone between overlapping ridge segments in northern Iceland

narrowing overtime. A possible geometry is illustrated in Figure 13B. Notably, in this

case, shear would have taken place prior to the development of a mature transform fault,

consistent with the conclusions of Young et al. (1985). Unlike the model from Young et

al., the rift propagation model does not necessarily predict rotation north of the fault (i.e.

on Tjörnes).

Horst et al. (2017) and Stefansson et al. (2008) attribute the rotation to off-fault

deformation during periods of high coupling along the transform, likening the pattern to

anti-J shaped deflections of valleys near other mature oceanic transform faults (Fig 13C).

Anti-J-style rotation could occur simultaneously with slip on the Húsavík-Flatey fault and

could be on-going, unlike rotation caused by rift propagation or by heterogeneous simple

shear during development of the transform. Unlike the ridge-propagation model, anti-J

block rotation would likely affect areas north and south of the transform.

The three models are similar to one another in many respects, but differ in the

timing of rotation. Young et al. (1985) find that the amount of rotation is consistent with

shearing during a ~2 million year time interval. However, this analysis assumes that all of

the relative plate motion was accommodated in the shear zone on Flateyjarskagi during

this time, which would not have been the case if spreading was partitioned between two

37

38

parallel ridges, as suggested by Garcia (2003). Two million years is therefore an

underestimate of the duration of rotation, but it is possible that most rotation took place

during the first few million years of shearing along the Tjörnes Fracture Zone.

Paleomagnetic data from Tjörnes indicate little to no rotation of rocks younger than 5.3-

3.8 Ma (Titus et al, 2018).

My interpretation of the timing of small faults on northern Flateyjarskagi provides

weak evidence that rotation is not on-going. If the rate of rotation were constant overtime,

we might expect fault poles in rotated lavas to be evenly distributed along a girdle

representing a range from unrotated to fully-rotated, corresponding to a range in fault age

from young to old. Instead, to some extent, we observe distinct populations of fully-

rotated and unrotated faults. In fact, the majority of small faults less than 2 km from the

Húsavík-Flatey fault could easily have experienced no rotation. This might suggest that

there has been a long period of time after rotation stopped during which unrotated faults

could accumulate. This interpretation is more consistent with the heterogeneous simple

shear and rift-propagation models than the anti-J model.

The presence or absence of rotation north of the Húsavík-Flatey fault is critical to

distinguishing between the three rotation models. Unfortunately, much of this area is

submerged, with the exception of the Tjörnes Peninsula. Only portions of the exposed

lavas on Tjörnes are old enough to have recorded large-magnitude rotations, and it is

difficult to tell whether these pre-unconformity rocks have rotated like rocks on

Flateyjarskagi. We do not observe a rotation gradient in the dikes or small faults on

Tjörnes (Appendix D), but this does not rule out rotation because it is possible that initial

dike orientations were too scattered for rotation to be recorded.

39

Paleomagnetic data published by Titus et al. (2018) and Young et al. (2018) may

support clockwise rotation of rocks on Tjörnes north of the Húsavík-Flatey fault.

Paleomagnetic measurements from rocks near the fault on Tjörnes deviate from expected

unrotated directions in a similar manner to paleomagnetic measurements from northern

Flateyjarskagi. However, since paleomagnetic data are directions rather than orientations,

the rotation that caused this deviation is ambiguous. A clockwise rotation is not the

smallest rotation that could account for the observed deflections. The argument for

clockwise block rotation on Tjörnes would be much more compelling with evidence of a

rotation gradient. Data from Titus et al. (2018) span distances between approximately 2

and 6 km from the Húsavík-Flatey fault, with no apparent gradient in rotation. This

makes it unclear whether there have been large-magnitude clockwise rotations of lavas on

Tjörnes. If pre-unconformity rocks on Tjörnes are interpreted as unrotated, northward

propagation of the Northern Volcanic Zone (Fig. 13B) is the most compelling

explanation for the rotation gradient on Flateyjarskagi. It is unclear why anti-J-style

rotations or the heterogeneous simple shear model would result in asymmetrical rotations

north and south of the fault.

DISCUSSION

Timing of formation

The amount of rotation apparent in each data set may relate to the ages of the

observed structures. The rotation gradient recorded in the dike data is close enough to the

gradient in the paleomagnetic data that most dikes in the study area likely pre-date

rotation (Titus et al, 2018). This is consistent with most of the 40Ar/39Ar ages determined

40

for dikes by Garcia et al. (2003), which range from 5 to 10 Ma. Dikes therefore have

recorded the full rotation history of the host-rock. By contrast, many near-fault vein

zones and small faults likely formed during or after rotation. Structures that match

predictions shown in Figures 8 and 12 may have formed prior to rotation and rotated with

the dikes, whereas near-fault structures that match far-fault orientations likely post-date

rotation. It is possible that the girdles followed by fault and vein poles in some areas

reflect continuous formation of these structures during rotation, as shown in Figure 4B.

Ideally, inferences about the relative ages of structures would be supported with cross-

cutting relationships, but we observed too few instances of cross-cutting to inform my

interpretations.

Vein Zones

Vein zones and dikes both rotate clockwise approaching the fault around

statistically indistinguishable poles of rotation. However, bootstrapping analysis shows

that it is unlikely that the regressions of veins zones and dikes 2-20 km from the HFF

record the same pattern (p = 0.07) (Fig. 11), suggesting that vein zones and dikes have

experienced different rotation histories. In general, vein zones appear to have rotated less

than dikes, which might indicate that vein zones are younger on average, though this is

not reflected in the rotation rates of the regressions. A N-S-striking vein zone that formed

after most off-fault deformation would bring the average direction of near-fault vein

zones closer to the average far-fault direction, diminishing rotation in the vein zone

regression. Vein zones between 2 and 5 km from the fault form a continuous arc between

apparently unrotated, NS-striking structures, and apparently fully-rotated, ENE-striking

structures, implying that vein zones in this area may have formed throughout rotation.

41

Most vein zones nearer than 2 km from the Húsavík-Flatey fault appear to have

experienced most but not all of the rotation. It is possible that several degrees of rotation

took place, then a large proportion of near-fault vein zones formed striking ~N-S, after

which the majority of rotation took place. However, there is no clear reason that most

near-fault vein zones would have formed during a relatively short interval like this.

Alternatively, it is possible that these vein zones pre- or post-date rotation and that

anisotropy in the crust or variation in stress caused them to form differently from far-fault

vein zones.

Strike-Slip Faulting

Figures 8 and 12 show distinct rotated and unrotated populations of strike slip

faults in several areas. Our data are too scattered to show evidence of three or more

distinct generations of left-lateral faults, as predicted by the model in Figure 4B. Near-

fault rocks have rotated such that unrotated left-lateral fault poles appear similar to

rotated right-lateral fault poles and vice versa. Therefore, an apparently rotated left-lateral

fault might in reality be a misidentified unrotated right-lateral fault. However, it is very

unlikely that all of the apparently rotated faults have been misidentified, due to the large

number of rotated faults and the significant spatial variations in the proportions of rotated

and unrotated faults.

Our fault data capture variation in the patterns of deformation from east to west

along the north-central Flateyjarskagi coast. Left and right-lateral faults appear younger

at eastern stations than central stations, suggesting more extensive post-rotation

deformation in the east. There is a glacial valley immediately adjacent to the eastern

stations, which might follow a zone of damage. Young et al. (1985) documented a denser

42

network of large normal faults following another topographic low in the western stations.

We did not observe a higher proportion of small normal faults in this area, but right- and

left-lateral fault planes are similar to normal fault planes and may have been influenced

by extensive normal faulting.

Normal Faulting

Young et al. (1985, 2018) suggest that most normal faulting occurred after

rotation south of the fault. Our data support this assertion; normal fault planes throughout

Flateyjarskagi appear mostly unrotated. The deviation between the far-fault average and

the near-fault normal fault poles seen in Figure 12 is more likely due to the difference in

lava bedding near the fault than due to rotation. A secondary population of far-fault

normal faults also appears to have affected the west-dipping far-fault average, which

otherwise might be more similar to the near-fault observations. Though it would seem

simplest for rotation and lava tilting to occur simultaneously, this finding suggests that

much of the tilting of near-fault lavas to the NW may have occurred on normal faults

after most of the clockwise block rotation.

Normal faults very near the Húsavík-Flatey fault preferentially dip to the east. It is

likely that this preferential dip reflects the tendency for small faults to form perpendicular

to lava bedding. Since lavas on northern Flateyjarskagi dip to the northwest, an

approximately NS-striking plane perpendicular to the lavas must dip to the east. The

normal faults we observed in shallowly-dipping far-fault lavas do not preferentially dip to

the east. Therefore, it seems likely that the NW-dip of near-fault lavas imparted a

preferred dip to the normal faults.

43

Slip on mostly east-dipping normal faults could explain the NW-tilting of the lava

pile on northern Flateyjarskagi. Whereas conjugate sets of E and W-dipping normal faults

can accommodate pure shear-style extension with no lava tilting, a set of parallel faults

will result in rotation of blocks and bounding faults. Rotation about a set of ESE-dipping

normal faults would result in west-side-down, east-side-up motion of fault-bounded

blocks, consistent with observed lava dips. So, are west-dipping lavas responsible for

east-dipping normal faults or vice versa? It is possible that a small tilt in the lavas

introduced, for instance, by loading at the fossil ridge to the west of the study area, would

result in a preferred normal-fault orientation that could then exacerbate the lava tilting.

Anisotropy

Throughout the study area, structures of every kind (veins, faults, etc.) form

nearly perpendicular to lavas (Fig. 7). On equal area plots, poles to structures consistently

form girdles following the local lava bedding plane. The simplest explanation for this

pattern is that structures are influenced by inhomogeneities in the lavas. Joints form

perpendicular to lava flows as they contract during cooling. This gives lavas an inherent

anisotropy, which appears to have a strong effect on almost every type of secondary

structure. Additionally, dikes are likely to intrude vertically (even in a homogeneous host

rock). Therefore, the margins and margin-parallel fractures in dikes that intrude prior to

lava tilting serve as further lava-perpendicular planes of weakness. Forslund and

Gunsdmunsson (1992) invoke cooling joints to explain the unusually steep (70-80°) dips

of normal faults elsewhere in Iceland. We also observed steeply dipping normal faults in

areas with shallowly dipping lavas. This pattern suggests a strong effect of local

weakness on the formation of small faults and veins.

44

Bingham distribution analyses of the small faults measured greater than 20 km

from the Húsavík-Flatey fault support that the mean right-lateral and normal fault

orientations are not significantly different from predictions based on WNW-directed

spreading in northern Iceland. However, observed left-lateral faults are on average about

30° clockwise of their predicted NE strikes. Despite the small number of far-fault data

points and the wide confidence region for the mean of the left-lateral fault poles, the

difference is statistically significant.

This discrepancy likely reflects the influence of dike margins and NS-striking

normal faults on the formation of the left-lateral faults. Figure 8 shows a strong

relationship between the directions of dikes and the directions of left-lateral fault planes

throughout the study area. Near the Húsavík-Flatey fault, dikes have rotated away from a

direction that could influence left-lateral fault planes. In these areas, most left-lateral

faults that I interpret to post-date rotation strike close to 222°, consistent with plate

motion (Fig. 8).

Limitations of Stress Inversion

Previous studies have attempted to calculate principal stresses using fault slip data

from the Tjörnes Fracture Zone (Garcia and Dhont, 2005; Bergerat et al, 2000, Garcia et

al. 2002, Angelier et al, 2000). These studies deal with multimodal datasets by grouping

faults into kinematically consistent sets and calculating a stress state for each. Bergerat et

al. (2000) and Angelier et al. (2000) identify eight distinct stress states on Flateykarskagi,

which they attribute to short term stress fluctuations caused by periodic locking on the

45

Húsavík-Flatey fault, variations in fluid pressure, and motion along block boundaries.

Garcia and Dhont (2005) identify two stress states near the fault on Tjörnes.

I do not interpret stresses from our fault slip data for a number of reasons. Rotated

faults are not useful for stress analysis. Bergerat et al. (2000) assign stress states to

populations of faults that I interpret as rotated and populations that may be misidentified

(right-lateral rather than left-lateral). In addition, previous studies calculated stresses from

kinematic indicators assuming a homogeneous and isotropic crust. However, the

orientations of small faults in our field area are significantly influenced by anisotropies in

the crust, as evidenced by faulting on dike margins, the perpendicularity of faults to

lavas, and observations of reactivated fault planes with multiple senses of slip. In addition

to systematic effects, like the apparent effect of dikes on left lateral faults, local

inhomogeneities are likely to introduce scatter that poses problems for binning faults into

many stress regimes.

Kinematic analysis is more justified, but not the most informative treatment of

data from northern Flateyjarskagi. As noted by Allmendinger (1989), kinematic axes (eg.

the direction of greatest extension) can be calculated from fault slip data without

assuming the mechanical properties of the crust. Without knowledge of the volume of

rock being deformed or the amount of slip on each fault, I could at best estimate the

relative magnitudes of kinematic axes at each field site. Many faults appear to have

rotated, so kinematic axes would not accurately reflect the direction of motion at the time

of faulting. They would merely reflect the cumulative change in the shape of the rocks in

each area. Additionally, kinematic axes vary from station to station and our spatial

46

coverage of Flateyjarskagi is limited, so it would be difficult to draw conclusions about

the kinematics of the region as a whole.

CONCLUSION

I present new measurements of small faults and vein zones from the Tjörnes

Fracture Zone in northern Iceland, which, in combination with data from previous

studies, provide insight into the style of deformation around a major oceanic transform

fault. The orientations of structures within several kilometers of the transform fault are

extremely variable. Many small faults do not match patterns observed in far-fault areas

and do not appear kinematically consistent with E-W directed extension in Iceland.

Previous workers have interpreted this variability as an indication of variable stress

regimes. However, other studies have documented a gradual change in dike strikes and

paleomagnetic directions that support large-magnitude rotations of crust near the

transform. Therefore, fault variability might instead reflect the presence of older, rotated

structures in addition to younger, unrotated structures. Some populations of small faults

near the transform appear similar to the orientations expected for structures that rotated in

the same manner as nearby dikes. Very near the Húsavík-Flatey fault, areas with a higher

proportion of apparently unrotated small faults coincide to some extent with topographic

lows, and may have experienced more deformation post-rotation. There is also a

significant change in the poles to planar vein zones with proximity to the transform fault.

Rotation of the vein zones is distinct from rotation of the dikes to the 90% confidence

level, but not the 95% confidence level, suggesting that the vein zones might post-date

the dikes, on average. The presence of apparently rotated vein zones and small faults is

47

consistent with proposed block rotations, and presents a problem for paleostress analysis

near the transform.

ACKNOWLEDGEMENTS

I thank Sarah Titus for her support, her comments on my draft, and for providing

me with the opportunity to travel to Iceland and work on this project. I thank the other

faculty, staff, and students in the Carleton Geology Department for their help. I’m

grateful to my parents for their encouragement throughout the process. This project was

made possible by NSF grant T51G-0258.

APPENDIX A. MEANS OF FAR-FAULT STRUCTURES

I used the following code to calculate means and confidence regions for small faults measured by the research group of Sarah Titus, Bergerat et al. (2000), and Garcia and Dhont (2005).

#filter small faults by rake, distance from the fault, and remove outliers.

farLL <- filter(allSlicks, slipSense=="L", distance > 20000) farRL <- filter(allSlicks, slipSense=="R", distance > 20000) farRL <- farRL[-33,] #remove one outlier with a strike of 77 farRL <- farRL[-16,] #another outlier with a dip of 42 farN <- filter(allSlicks, slipSense == "N", distance > 20000) farN_1a <- filter(allSlicks, slipSense == "N", distance > 20000, strikeDeg <90) farN_1b <- filter(allSlicks, slipSense == "N", distance > 20000, strikeDeg >270) farN_2 <- filter(allSlicks, slipSense == "N", distance > 20000, strikeDeg > 90, strikeDeg < 270) farN_1 <- rbind(farN_1a, farN_1b) #south-dipping

#I cannot take the average of the left lateral fault orientations because they are bimodal, but the #poles to the left lateral faults are unimodal, so I will compute average poles for all faults:

#Right Lateral Fault Bootstrapping RLpoleInfer <- lineBootstrapInference(farRL$pole, 10000) lineEqualAreaPlot(farRL$pole, colors="yellow")

48

par(new=TRUE) lineEqualAreaPlot(list(RLpoleInfer$center)) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(162, 90))), colors="blue") RLpoleInfer$pvalueLine(geoCartesianFromStrikeDipDeg(c(162, 90))) geoStrikeDipDegFromCartesian(RLpoleInfer$center)

#Right Lateral Fault Bingham Inference pdf(paste("farRLBingCRandPred.pdf", sep = ""), useDingbats = FALSE, width = 10, height = 6) par(pty="s") par(mar=c(1,1,1,1)) RPoleBingInfer <- lineBinghamInference(farRL$pole, numPoints = 1000) lineEqualAreaPlot(RPoleBingInfer$points) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(162, 90))), colors="red") dev.off()

#Left Lateral Fault Bootstrapping LLpoleInfer <- lineBootstrapInference(farLL$pole, 10000) lineEqualAreaPlot(farLL$pole, colors="yellow") par(new=TRUE) lineEqualAreaPlot(list(LLpoleInfer$center)) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(222, 90))), colors="blue") LLpoleInfer$pvalueLine(geoCartesianFromStrikeDipDeg(c(222, 90))) geoStrikeDipDegFromCartesian(LLpoleInfer$center)

#Left Lateral Fault Bingham Inference LPoleBingInfer <- lineBinghamInference(farLL$pole, numPoints = 1000)

#Plot confidence region: lineEqualAreaPlot(LPoleBingInfer$points) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(222, 90))), colors="red")

#Normal Fault Bootstrapping Inference for the Mean

#All normal faults analyzed together: NpoleInfer <- lineBootstrapInference(farN$pole, 10000) lineEqualAreaPlot(list(NpoleInfer$center)) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(012, 90))), colors="blue") NpoleInfer$pvalueLine(geoCartesianFromStrikeDipDeg(c(012, 90))) geoStrikeDipDegFromCartesian(NpoleInfer$center)

49

#note that, treated together, normal faults are not unimodal

#Normal faults separated by dip direction: N1poleInfer <- lineBootstrapInference(farN_1$pole, 10000) #11 observations lineEqualAreaPlot(list(N1poleInfer$center)) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(012, 70))), colors="blue") N1poleInfer$pvalueLine(geoCartesianFromStrikeDipDeg(c(012, 70))) geoStrikeDipDegFromCartesian(N1poleInfer$center)

N2poleInfer <- lineBootstrapInference(farN_2$pole, 10000) #14 observations lineEqualAreaPlot(list(N2poleInfer$center)) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(012, 90))), colors="blue") N2poleInfer$pvalueLine(geoCartesianFromStrikeDipDeg(c(192, 70))) geoStrikeDipDegFromCartesian(N2poleInfer$center)

#Note that there are not enough observations here to trust the bootstrapping confidence intervals

#Normal fault Bingham Inference:

#All normal faults treated together: NPoleBingInfer <- lineBinghamInference(farN$pole, numPoints = 1000)

#Plot confidence region: lineEqualAreaPlot(NPoleBingInfer$points) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(012, 90))), colors="red")

#normal faults treated separately: N1PoleBingInfer <- lineBinghamInference(farN_1$pole, numPoints = 1000)

#Plot confidence region: lineEqualAreaPlot(N1PoleBingInfer$points) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(012, 70))), colors="red")

N2PoleBingInfer <- lineBinghamInference(farN_2$pole, numPoints = 1000) #Plot confidence region: lineEqualAreaPlot(N2PoleBingInfer$points) par(new=TRUE) lineEqualAreaPlot(list(geoCartesianFromStrikeDipDeg(c(192, 70))), colors="red")

50

APPENDIX B. USE OF DIKE POLES TO ESTIMATE ROTATION GRADIENT

Dike data are ideal for constraining the amount of rotation in the study area

because dike poles broadly follow a great circle around the apparent axis of rotation. The

lava poles and the steeply plunging ChRMs obtained from paleomagnetic analysis follow

small circles, nearer to the axis of rotation. As noted by Young et al. (2018), a given

rotation about this axis results in a larger change in dike direction than in ChRM or lava

direction, making dikes a more sensitive measure of rotation. In Figure 3, it is clear that

the dike poles follow a larger arc than the paleomagnetic directions, despite recording

similar rotations.

Instead of using a regression, I could have determined how much each field

station had rotated individually by finding a rotation that takes an average far-fault dike

to the average dike direction at that station. In some ways, this method requires fewer

assumptions: the rate of rotation does not need to be constant or logarithmic, and rotation

can vary along strike. However, an unrotated dike and a rotated dike do not provide

enough information to solve for a rotation because dikes are directions, not orientations.

There are an infinite number of rotations that take one dike pole to another. Using a

regression against fault-normal distance as a proxy for rotation over time eliminates this

ambiguity. There are many rotations that take a far-fault dike pole to a near-fault dike

pole, but there is only one path that best follows the gradient in the dike directions over a

range of distances. Dikes at intermediate distances constrain the rotation.

There are drawbacks to using the dike regression to quantify the “full rotation”

history of northern Flateyjarskagi. The spatial distribution of our dike data is limited. The

regression uses measurements from stations between 2 and 3 km from the fault and from

51

stations approximately 6, 10, 13, 18 and 19 km from the fault. This leaves significant

gaps in the sampled distances, particularly between 3 and 10 km, where the rotation

gradient is likely the steepest. The dike regression assumes that there is no change in the

extent of rotation along Húsavík-Flatey fault -strike. This assumption is difficult to

validate with our data because we have very limited sampling along fault strike within

each distance interval. Data from Young et al. (1985) have more spatial coverage and are

consistent with our measurements in areas of overlap. These data show little variation

along fault strike in the area described by our dike regressions. Unfortunately, these dike

data lack dips and therefore cannot be used in our regressions.

It is possible that regressions of dike data underestimate the amount of rotation

that has taken place. The rate of rotation predicted by the dike regression is lower than

the rate predicted by regressions of dike paleomagnetic data from Titus et al. (2018). The

dike regression also systematically underpredicts rotation in the paleomagnetic data

published by Horst et al. (2018). High scatter in the near-fault dike poles suggests that

either some near-fault dikes intruded later and did not experience the full rotation or that

some near-fault dikes did not intrude striking N-S. The paleomagnetic data provide no

evidence that dikes have systematically experienced less rotation than lavas (Titus et al

2018), possibly suggesting that the initial orientations of dikes were more variable near

the HFF. Regardless of the cause, near-fault scatter is likely to cause dike regressions to

underestimate rotation. Additionally, the sparse distribution of dike data at mid-distances

(between 5 and 18 km from the HFF) necessitate that I use data points from slightly

outside the likely range of the rotation gradient (beyond 12-14 km from the fault) to

constrain the far-fault end of the regression. Including points from less- or un-rotated

52

areas likely causes the regression to further underestimate the amount of rotation per

kilometer in near-fault areas. The logarithmic fit helps to diminish this effect.

APPENDIX C. COMPARISON OF DIKE AND VEIN ZONE REGRESSIONS

In order to compare the logarithmic small circle regressions of vein zones and

dikes, I compared each of the 500 bootstrapped vein zone regressions to one of the 500

bootstrapped dike regressions by multiplying the vein zone regression rotation matrix by

the transpose of the dike regression rotation matrix. If the two regressions were identical,

this multiplication would result in the identity matrix. The 95% confidence ellipsoid for

the resulting cloud of 500 rotations contains the identity matrix, but the 90% confidence

ellipsoid does not contain the identity matrix. This implies that the difference between the

vein zone and dike regressions is significant to the 90% confidence level, but not the 95%

confidence level.

dikeDF <- filter(dikes, distance > 2000, distance<20000, easting<641500) dikeDF$distance <- lapply(dikeDF$distance, function(x) x/1000) bzDF <- filter(bzs, distance > 2000, distance<20000, easting<641500) bzDF$distance <- lapply(bzDF$distance, function(x) x/1000) dikeDF$lndist <- lapply(dikeDF$distance, function(x) log(x)) bzDF$lndist <- lapply(bzDF$distance, function(x) log(x))

logDikeSCReg <- lineSmallCircleRegression(dikeDF$lndist, dikeDF$pole, numSeeds=10, numSteps=100, numPoints=0, angleBound=2) logBZSCReg <- lineSmallCircleRegression(bzDF$lndist, bzDF$pole, numSeeds=10, numSteps=100, numPoints=0, angleBound=2)

#Plot axes of rotation for the dike and breccia zone regressions: lineEqualAreaPlotTwo(list(logBZSCReg$pole), list(logDikeSCReg$pole), colorA = "red") #Plot predictions, shown in Figures 5 and 6

xs <- c(1:20) #hues <- hues(xs, c(0, 28000)) xs <- lapply(xs, log)

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xs <- lapply(xs, function(x) x/10) poles <- lapply(xs, logDikeSCReg$prediction) curves <- lapply(poles, rayGreatCircle)

lineEqualAreaPlot(poles, curves=curves)

#Bootstrap regressions:

lineGeodesicRegressionBootstrap <- function(xs, ls, numSteps=1000, numPoints=0) { indices <- sample(1:length(ls), length(ls), replace=TRUE) lineRescaledGeodesicRegression(xs[indices], ls[indices], numSteps=1000, numPoints=0) }

lineGeodesicRegressionBootstraps <- function(xs, ls, numBoots, numSteps=1000, numPoints=0) { boots <- replicate( numBoots, lineGeodesicRegressionBootstrap(xs, ls, numSteps, numPoints), simplify=FALSE) } lineSmallCircleRegressionBootstrap <- function(xs, ls, numSteps=200, numPoints=0) { indices <- sample(1:length(ls), length(ls), replace=TRUE) lineSmallCircleRegression(xs[indices], ls[indices], numSeeds = 20, numSteps=200, angleBound=2, numPoints=0) } lineSmallCircleRegressionBootstraps <- function(xs, ls, numBoots, numSteps=200, numPoints=0) { boots <- replicate( numBoots, lineSmallCircleRegressionBootstrap(xs, ls, numSteps, numPoints), simplify=FALSE) }

f <- function(result) rotMatrixFromAxisAngle(c(result$pole, result$angle))

SCdikeBoots <- lineSmallCircleRegressionBootstraps(dikeDF$lndist, dikeDF$pole, 500, numSteps=100, numPoints=0) bzBoots <- lineSmallCircleRegressionBootstraps(bzDF$lndist, bzDF$pole, 500, numSteps=100, numPoints=0)

dikeRots <- lapply(SCdikeBoots, f) bzRots <- lapply(bzBoots, f)

#View bootstrapped regressions in rotation space:

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rotEqualVolumePlot(dikeRots) fit <- rotMahalanobisPercentiles(dikeRots, rotProjectedMean(dikeRots)) fitTris <- rotEllipsoidTriangles( fit$center, fit$leftCovarInv, level=fit$q095^2, numNonAdapt=5) fitRs <- unlist(fitTris, recursive=FALSE) rotEqualVolumePlot(fitRs, simplePoints=TRUE)

# Obtain confidence regions for axes and rotation rates from the ellipsoid fit to the bootstrapped regressions:

fitAxes <- lapply(fitRs, function (r) rotAxisAngleFromMatrix(r)[1:3]) rayEqualAreaPlot(fitAxes) fitMags <- sapply(fitRs, function (r) rotAxisAngleFromMatrix(r)[[4]]) hist(fitMags / degree)

#Compare small circle dike and vein zone regressions:

rs <- dikeRots qs <- bzRots rotEqualVolumePlot(qs) rotEqualVolumePlot(rs)

# Find differences between bootstrapped regressions, pairwise:

f <- function(r, q) {r %*% t(q)} diffs <- thread(f, rs, qs) rotEqualVolumePlot(diffs) #the differences form an ellipsoid

fit <- rotMahalanobisPercentiles(diffs, rotProjectedMean(diffs)) #fits an ellipsoid in rotation space fitTris <- rotEllipsoidTriangles(fit$center, fit$leftCovarInv, level=fit$q095^2, numNonAdapt=5) fitDiffs <- unlist(fitTris, recursive=FALSE) rotEqualVolumePlot(fitDiffs, simplePoints=TRUE) #the 95% confidence ellipsoid just intersects the origin of the plot, implying that the regressions are not distinct to the 95% confidence level. fit$pvalue(diag(c(1, 1, 1))) # Note that the process used here to compute a p value is not extremely precise. # The process yielded a p value of 0.06 for a set of synthetic data designed to have a p-#value of 0.05

APPENDIX D. TJORNES PENINSULA

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The majority of small faults on Tjörnes have similar orientations to far-fault

averages. A small number of faults at several stations are well-predicted by the

logarithmic small circle dike regression, assuming symmetrical rotation on either side of

the Husavik-Flatey fault. However, predictions could easily match these faults by chance

(Fig A1).

Normal Faults Right Lateral Faults Left Lateral Faults

3 km

Left Lateral Faults

Figure A1. Equal area plots showing small fault data from Tjornes Peninsula near the Husav-ik-Flatey fault. Fault slip vectors are colored by fault-normal distance. Predictions for fault orien-tations based on the logarithmic small circle dike regression are shown as bold, colored great circles with slip vectors. Predictions assume symmetrical rotation on either side of the transform, and are plotted for even kilometer distances (4, 2, and 1 kilometers from the fault). The fault is shown in bold. Pre-unconformity lavas are shown in blue.

N

56

57

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