On the Origin of Hot Spots:A Study of Method in

the Statistical Analysis Thereof

Oliver HatfieldSupervisor: Dr Jochen Einbeck

23rd May 2014

Oliver Hatfield

On the Origin of Hot Spots

Introduction - Volcanoes and Hot Spots

Roughly 90% of Earth’s surface volcanism caused by tectonicplate movement.

Hot spots are regions of abnormally high volcanic activity.Significant subjectivity.

Three main lists used inthis project:

1. Sleep, 1990.2. Morgan, 2007.3. Courtillot et al. 2003.

Geologists want to investigate their origins.

Oliver Hatfield

On the Origin of Hot Spots

The Plume Hypothesis

Claims that plumes of buoyantmaterial rise through mantle toform hot spots.

Shear velocity data fromtomographic studies of themantle shows areas ofhigher/lower density.

Usable contours range fromapprox −1.5% to +1.5%.

Claims that hot spots lie abovelow shear velocity provinces:particularly the -1% contour.

Oliver Hatfield

On the Origin of Hot Spots

The Plate Hypothesis

Claims hot spots are caused by side effects of tectonic platemovement - particularly where crust is stretching.

Predicts that hot spots correlate with spreading plateboundaries.

3 types of plate boundary:

Spreading Boundaries (Ridges).

Convergent Boundaries(Trenches).

Transform Boundaries.

Oliver Hatfield

On the Origin of Hot Spots

The Problem

Begin with a set of basepoints, (eg. PlateBoundaries)B = {bj}j=1...m1 .

Add a set of auxiliary points,(Hotspots) A = {ai}i=1...m0 .

How best to summarisewhether or not the two arelinked?

Oliver Hatfield

On the Origin of Hot Spots

The Problem

Begin with a set of basepoints, (eg. PlateBoundaries)B = {bj}j=1...m1 .

Add a set of auxiliary points,(Hotspots) A = {ai}i=1...m0 .

How best to summarisewhether or not the two arelinked?

Oliver Hatfield

On the Origin of Hot Spots

The Problem

Begin with a set of basepoints, (eg. PlateBoundaries)B = {bj}j=1...m1 .

Add a set of auxiliary points,(Hotspots) A = {ai}i=1...m0 .

How best to summarisewhether or not the two arelinked?

Oliver Hatfield

On the Origin of Hot Spots

Mean Minimum Distance (MMD)

Definition

MMD from A to B, denoted by δmmd(A,B), is defined as

δmmd(A,B) =1

m0

m0∑i=1

minbj∈B

d(ai , bj), (1)

where d(ai , bj) is the distance between ai and bj , and m0 = ‖A‖.

MMD= 1.08

MMD= 1.81

Oliver Hatfield

On the Origin of Hot Spots

Assessing Correlation Strength

Estimate P(X ≤ xA), where xA is the observed MMD from A toB, and X is a random variable for δmmd(C,B), for a random set C.

Method 1: Monte Carlo method.Take n samples of new points, Ck , and let Xk be the MMD ofset Ck to B. Let pA = 1

n

∑nk=1 1Xk≤xA .

E (pA) =1

n

n∑k=1

E (1Xk≤xA)

= P(X ≤ xA)

Oliver Hatfield

On the Origin of Hot Spots

Spherical Sampling

Need to randomise pointsuniformly over the surface of asphere.

Spherical co-ordinates θ and φ.

Sampling θ and φ uniformly leaves points bunched near the poles.

Oliver Hatfield

On the Origin of Hot Spots

Spherical Sampling

Area element of transformation gives distortion factor of area. Forf (θ, φ) = (cosθ sinθ, sinφ sinθ, cosφ), it is dA = sinφ dθ dφ.

To get this independent of φ, let v = −cosφ, so dv = sinφ dφ.

Area element is now dA = dθ dv .

Points are uniformly distributed over the sphere.

Oliver Hatfield

On the Origin of Hot Spots

Assessing Correlation Strength

Method 2 - Central Limit Theorem:Let minbj d(ai , bj) = di .

δmmd(A,B) =1

m0

m0∑j=1

minbj∈B

d(ai , bj) = d . (2)

Therefore d−µs/√m0∼ N (0, 1), by the Central Limit Theorem.

Thus, estimate P(X ≤ xA) using a normal distribution.Estimate µ and s from sample output.

Oliver Hatfield

On the Origin of Hot Spots

Results - Plates

Monte Carlo Method Plate boundaryHot Spot List Ridges Trenches Transforms

Sleep 0.031 0.950 0.949

Morgan 0.051 0.999 0.975

Courtillot 0.026 0.992 0.959

Normal Distribution Plate boundaryHot Spot List Ridges Trenches Transforms

Sleep 0.036 0.943 0.950

Morgan 0.054 0.998 0.972

Courtillot 0.029 0.991 0.959

Oliver Hatfield

On the Origin of Hot Spots

Plates - Randomisation

To see if hot spot lists were subjectively chosen to minimise MMD,randomise sets of hot spots from across all lists, find pA for each.

Oliver Hatfield

On the Origin of Hot Spots

Results - Plumes

Monte Carlo Shear Velocity Contour at 2800km depthHot Spot List -1.5% -1% -0.5% 0% 0.5% 1% 1.25%

Sleep 0.000 0.002 0.000 0.364 0.968 0.988 0.982Morgan 0.000 0.002 0.000 0.958 0.999 1.000 0.993

Courtillot 0.000 0.002 0.000 0.259 0.977 0.987 0.990

Normal Distr. Shear Velocity Contour at 2800km depthHot Spot List -1.5% -1% -0.5% 0% 0.5% 1% 1.25%

Sleep 0.001 0.005 0.000 0.355 0.971 0.985 0.983Morgan 0.000 0.001 0.000 0.965 0.999 0.999 0.993

Courtillot 0.001 0.004 0.000 0.259 0.976 0.990 0.990

Oliver Hatfield

On the Origin of Hot Spots

Plumes - Results

Hotspots mostly lie inside 0% contour on negative side, hencecorrelation for negative contours but none of the others.

Figure: Left hand image shows 0% contour, along with hot spots’locations, rescaled to match right hand graph, which shows continuousnature of wave velocities.

Oliver Hatfield

On the Origin of Hot Spots

Plumes - Randomisation

To see if hot spot lists were subjectively chosen to minimise MMD,randomise sets of hot spots from across all lists, find pA for each.

Oliver Hatfield

On the Origin of Hot Spots

Conclusion

MMD is a characteristic which summarises the closeness oftwo sets of points.

Plate Hypothesis predicts that hot spots correlate withspreading plate boundaries, but not the others. MMD agreeswith this.

Plume Hypothesis observes (but does not appear to predict)that hotposts correlate with −1% shear wave velocity contour.MMD confirms this, but nothing unique about that contour.

Results are marginally stronger for plumes than plates.

Correlation does not imply causation!

Oliver Hatfield

On the Origin of Hot Spots

References

Foulger, Gillian R

Plates vs Plumes.

Wiley-Blackwell, (2010).

Sleep, N. H.

Hotspots and Mantle Plumes: Some Phenomenology

Journal of Geophysical Research, 95: 6715-6736 (1990).

Morgan, W.J. and Phipps Morgan, W.

Plate velocities in the hotspot reference frame

Geological Society of America, 430: 65 - 78 (2007).

Courtillot, V. , Davaillie, A. , Besse, J. et al.

Three distinct types of hotspots in the Earth’s mantle .

Earth and Planetary Science Letters, 205 : 295-308 (2003).

Oliver Hatfield

On the Origin of Hot Spots