Plate Kinematic Reconstruction and Restoration via Fractal Error Minimization

Rex H. Pilger, Jr.Highlands Ranch, Colorado

What’s the problem?

Current “standard” models:• Plate-to-plate

o Great circle approximations of spreading and fracture zone segments

o Fit to chron and fracture zone crossings, stationary and rotated

• Plate-to-hotspoto Pacific (single plate): spline-

parameterized locio Atlantic/Indian (multiple plates): Great

circle approximations of trace locio Fit to average or oldest dates from

inferred hotspot traces

Current models: plate – to - plate

Advantages Disadvantages

Iterative least-squares with partial derivatives

Great circle can be poor approximation to transform fault

Uncertainties for plate-to-plate, individual chrons

“Nuisance” parameters significantly outnumber rotation parameters: two for each spreading and fracture segment

Plate circuit uncertainties for individual chrons

How to eliminate or propagate nuisance uncertainties?

Kinematic discontinuities at reconstruction ages

How to handle uncertainties in interpolated reconstructions?

Current models: plate – to - hotspot

Advantages Disadvantages

Least-squares criteria and iterative solution May not sample oldest hotspot data (with kinematic significance)

Uncertainties relative to the assumed model

Need to use “best” isotopic dates

Plate circuit reconstructions Difficult to evaluate over all uncertainties

Size/shape of hotspot (plume?) is unknown and may vary through time

Plate-to-hotspot models

Hawaii: paradigmatic hotspot

How to evaluate fits… Hotspotting”TM”

restoration

Plate-to-hotspot: “hotspotting”

How to evaluate fits… Hotspotting”TM”

restoration

“TM” Wessel and Kroenke (1997)

Hotspotting – Hawaiian reference frame

Haw

aiia

n-E

mpe

ror

Hawaii

Orange <= 48 Ma

Green > 48 Ma**47-48 Ma: Age of Hawaiian - Emperor Bend

Loci: +/- 5 my

Hotspotting – Hawaiian reference frameC

ook

Orange <= 48 Ma

Green > 48 Ma

Mac

dona

ld

Hotspotting – Hawaiian reference frameS

amoa

Orange <= 48 Ma

Green > 48 Ma

Fou

ndat

ion

Eas

ter

Orange <= 25 Ma

Blue > 25 Ma*

*25 Ma: Nazca and Cocos plates form from Farallon plate

Hotspotting – Tristan reference frame

Tris

tan-

St.

Hel

ena

Ker

guel

en-R

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on

Hotspotting – Tristan Reference Frame

Tas

man

Gre

at M

eteo

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anar

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Eas

t A

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Hotspotting – Tristan Reference Frame

Eas

t A

ustr

alia

Youngest dates (!)

Oldest dates

Car

ibbe

an A

rcs

to T

rista

nHotspotting – Tristan reference frame

Another approach: fractal measures

38

19

11

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Chart Title

y = -1.0029x + 1.614

R2 = 0.9851

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Log (spacing)

Lo

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nt)

Count

Fractal measure: reduced by restoration

7

13

4

2

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Fractal synthesis: Hawaiian frame models

Plate reconstructions: Monte Carlo

Monte Carlo “trial and error” Linearly “random” Equal area cells, equatorially-centered for each

restored trace Sum of fractal counts over range of delta-spacing

for each realization Five percent variation in total rotation

pseudovectors & asymmetry 50,000 realizations Retain minimum sum of restored hotspot date

cells

Hawaiian Hotspots

Plate Reconstructions

Australia-Antarctica Isochron crossings1

Background gravity field2

1Cande & Stock, 20042Sandwell & Smith, 1997

Plate reconstructions - Australia-Antarctica

Reconstruction parameters: Spline-interpolated pseudovectors

“Half” total rotations

If spreading was symmetric, reconstructions should produce tight “linear” clustering

Plate reconstructions - Australia-Antarctica

Assuming symmetrical spreading, divergent clusters indicate asymmetrical spreading or ridge-jumping

Plate reconstructions – cell counting

Equal area cells Sum of fractal counts over range of delta-spacing

for each realization

Plate reconstructions

Fractal Count: Coarse

Fractal Count: Fine

Plate reconstructions: cell counts

Plate reconstructions: Monte Carlo

Monte Carlo “trial and error” Linearly “random” Five percent variation in total rotation

pseudovectors & asymmetry 40,000 realizations (6 hrs on 2 Core, 2.40 GHz, 4GB RAM)

Retain minimum sum of restored chrons and fracture zones cells

Plate reconstructions – “final”

“Best fit”: Minimum summed fractals

Realization 35,261 of 40,000

Sequence of minimum interations: 0, 343, 464, 2468, 4751,

4912, 9025, 18497, 25793, 26613, 32105, 32298, 32476, 35261

Plate Reconstructions – “final”

Tighter clustering of chrons

Plate reconstructions – comparison

Initial: Yellow, orange, green

“Final”: Red, pink, blue

Plate reconstructions – comparison

Initial: Yellow, orange, green

“Final”: Red, pink, blue

Plate reconstructions – comparison

Why fractals?

A Google Search (10/25/2010) for “fractals” produces 6,780,000 results

However, very few if any of these articles recognize that: Within an iterative, scaling process fractals “maximize

information entropy” with respect to persistent information content

That is, following Jaynes’ principle:• Across a range of scales maximizing:

o = – pn log pn – 0( pn – 1) – k(Ek (p,x) – <Ik>)

Produces Mandelbrot’s fractal equation:• N = a x –d

Application: Parameters for minimum sum of fractals, producing maximum entropy scaled solution

What’s next…

Plate-to-plate• More iterations for Monte Carlo• Apply to full data sets• Introduce uncertainties• Provide pseudo-gradients for iterative solutions, instead

of Monte Carlo• Plate circuits with uncertainties

Plate-to-hotspot• Incorporate plate-to-plate results• Include uncertainties• Pseudo-gradients for iterative solutions, instead of

Monte Carlo• Hotspot & plates to paleomagnetic models

Virtual worlds

GoogleEarth, World Wind, Bing…

Three roles:• Evaluating reconstruction models with data, especially if

tied to “real-time” calculations• Presentations like this• Exchanging data (e.g., via xml)

o Rawo Interpretedo Meta (via embedded hyperlinks)

Key references

Plate reconstruction methods: • Pilger, 1978, Geophys. Res. Lett., 5, 469-472.• Hellinger, 1981, J. Geophys. Res., 86B, 9312-9318.• Wessel & Kroenke, 1997, Nature, 387, 365-369.

Maximum Entropy: • Jaynes, 1957, Phys. Rev., 106, 620-630.

Fractals: • Mandelbrot, 1967, Science, 156, 636-638.

Maximum entropy and fractals: • Pastor-Satorras & Wagensberg, 1998, Physica A, 251,

291–302. SE Indian Ocean magnetic isochrons (digitized from map):

• Cande & Stock, 2004, Geophys. J. Int., 157, 399-414.

RIP

Benoit Mandelbrot November 20, 1924 – October 14, 2010

Edwin Jaynes July 5, 1922 – April 30, 1998