Plate Kinematic Reconstruction and Restoration via Fractal Error Minimization Rex H. Pilger, Jr. Highlands Ranch, Colorado
Mar 15, 2016
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 frameH
awai
ian-
Em
pero
r
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
Foun
datio
nE
aste
r
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
euni
on
Hotspotting – Tristan Reference Frame
Tasm
an
Gre
at M
eteo
r-C
anar
y
Eas
t Afri
ca
Hotspotting – Tristan Reference Frame
Eas
t Aus
tralia
Youngest dates (!)
Oldest dates
Car
ibbe
an A
rcs
to T
rista
nHotspotting – Tristan reference frame
Another approach: fractal measures
38
19
11
5
3
1
Chart Title
y = -1.0029x + 1.614R2 = 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)
Log
(cou
nt)
Count
Fractal measure: reduced by restoration
7
13
4
2
2
1
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