Gravity inversion and isostasy- an integrated system

1
Gravity inversion and isostasy- an integrated
system

• Trieste, 17-20. February 2003
• Carla Braitenberg
• Dipartimento Scienze della Terra,
• Università di Trieste, Via Weiss 1, 34100 Trieste
• Berg_at_units.it
• Tel [phone] fax [phone]

2
Topics

• Different aspects of isostasy
• Local compensation
• Regional compensation
• Isostasy and sea level change
• Examples
• Glacio-isostasy and Hydro-isostasy

3
Topics

• Gravity inversion
• Spectral properties of the gravity field
• Downward/upward continuation
• Iterative inversion procedure
• Examples
• Synthetic root
• Eastern Alps

4
Topics

• Integration of isostasy and gravity inversion
• Eastern Alps
• Tibet plateau
• Parana Basin
• South China Sea

5
Geographic differences in the MSL
variation(Lambeck, Chappell 2001)
6
Description of geographic differences

• Ångermann- river sediments now in 200 m r.p.s.l.
• Transgression of sea change from fresh water to
  marine sediments
• Regression inverse
• Time scale from dating of sediments or counting
  seasonal Varves
• S-England transition from fresh-water to
  estuarine deposition
• In situ tree-stumps give upper margin to MSL

7
Description of geographic differences

• Sunda Shelf flooding of shelf
• Barbados Fossil corals age-height relation
• Dating Carbon or Uranium series methods
• North Queensland Micro-atoll-formation of
  corals. Today same corals live in 10 cm depth
  relative to minimum sea level.

8
Geographic differences- Description

• Classification of observed areas
• Central area of former ice-sheet Ångermann,
  Hudson Bay
• Marginal areas of ice-sheet or area of small
  ice-sheets Åndoya
• Medium latitudes, broad area that confined to
  ice-sheet South England. The same
  Mediterranean, Atlantic coast of SA, Gulf of
  Mexico.
• Areas far from ice-sheet-margin Barbados, Sunda
  Shelf
• Most observations regard time after LGM older
  traces were cancelled by
• A) rising MSL after LGM
• B) advancing ice-sheet before LGM

9
Isostatic Models locales equilibrium (Airy und
Pratt) and regional equilibrium (Flexural
Isostasy)

• Airy Variation of crustal thickness as function
  of topography
• Pratt Variation of crustal density as function
  of topography

10
Airy and Pratt Isostatic models
11
Regional equilibrium (Flexural Isostasy)Model of
flexure of a thin plate
12
Regional equilibrium (Flexural Isostasy)

• Flexural rigidity

Typical values E 1011 N/m2 ? 0.25
13
Regional equilibrium (Flexural Isostasy)

• Insert expressions for p and q

• Solution of equation

k wave-number
We put
We obtain
14
Regional equilibrium (Flexural Isostasy)

• An arbitrary topography can be built as the sum
  of sine-functions (Fourier-Transormation)

The flexure of the plate is then
15
Regional equilibrium (Flexural Isostasy)

• To the same result you obtain by applying the
  Fourier Transformation (FT) of the equation

• k wave-number
• W(k) FT(w(x)) H(k) FT (h(x))

16
Transition to local compensation
With very low flexural rigidity or for small
wave-numbers (great wave-lengths) the regional
isostasy goes into the Airy Isostasy
With very high rigidity or for small wave-numbers
(small wave-lengths) the load does not deform the
plate.
17
Properties of the plate-flexure

• Below the load greatest downward flexure
• In the marginal areas flexural bulge
• The smaller the elastic thickness of the plate
• The greater is the amplitude
• The smaller is the wave-length of the flexure
• At great distances of the load no effect

18
Deformation due to the ice-load
19
Examples

• In the simplified Airy case we calculate the
  subsidence (r) of the crust due to an ice load of
  thickness (h)

Maximum ice-thickness during LGM in scandinavia
and North-America estimated to max 2000-2500 m
(Lambeck and Chappell, 2001). Gives r of 600-760
m
20
Differential sea level change at stable coast and
in ocean basin
21
Examples

• In the simplified case of Airy, we calculate the
  crustal uplift (r) in case of a lowstand of MSL

In occasion of a measured sealevel fall
of about 120 m, you obtain r of 40m. The
hydro-isostatic effect of MSL-change is then
The comparison with the observations in the
Mediterranean shows that the hydro-isostatic
effect calculated in the Airy model is
over-estimated.
22
Mean Sea level at the time of the LGM.
Lambeck and Bard, 2000
23
Ice-equivalent sea-level change, from the French
Mediterranean area compared to global curves
24
Ice-equivalent sea-level change, from the French
Mediterranean area for late-Holocene time. The
continuous curve is same as (i) in (a)
(Lambeck and Bard, 2000)
25

• Elastic flexure modelThin plate approximation
• Flexure of the crust/lithosphere in frequency
  space related to topography

Problems in recovering H(k) Low spectral
energies in topography Poor spatial resolution
caused by required window size in spectral
analysis Limitations posed by rectangular window
26
Convolution method

• spectral domain

• space domain

Flexure point load response Obtained from inverse
FT of flexure transfer function
Load refers to total load, being the sum of
surface and subsurface load.
27
Total Load topographic and buried load
Buried load
Lburied inner-crustal loads, hi thickness of
the i-th layer, ?i density of the i-th layer and
?c the density of the reference
crust. Equivalent topography total load divided
by reference density
28

• section across 3D flexure response functions to
  point load for different flexure parameters

29
Maximum spatial frequency that must be covered
(f0 in 1/km) Sampling (dr in km) Number of
elements along the baselength of the square grid
(N) Minimum filter extension (Rmax in km)
required to reach percentage point 10 and 1 of
the maximum value of the impulse response

• Te f0 (1/km) dr (km) N Rmax (km) Rmax (km)
• 10 1
• 1 4.00 10-2 5 500 19 42
• 2 2.38 10-2 5 841 33 68
• 5 1.20 10-2 10 836 65 135
• 10 7.11 10-3 15 938 105 230
• 20 4.23 10-3 20 1183 180 400
• 30 3.12 10-3 20 1604 245 520
• 40 2.51 10-3 30 1326 310 650
• 50 2.13 10-3 30 1568 360 760
• 60 1.85 10-3 30 1798 400 870

30
Application of the convolution method in the
flexure calculation

• Forward model
• given load and Te spatial variation obtain
  expected flexure
• Inverse model
• given load and observed CMI variation obtain
  spatial variations of Te

31

• Inverse modeling of Te on synthetic model
• Synthetic Moho created by Paul Wyer with FD
  solution of loading a spatially varying Te-model
  with the Eastern Alps topography
• Resolution 5 km
• Size of Moho 550 km by 500 km

• Flexure Moho
• convolution of topographic load with flexure
  response functions.

32
On square overlapping windows for Te0,...30 km
(dTe 0.5 km) calculate difference between
flexure and synthetic Moho Choose Te that
minimizes rms
Moho Grid 550km by 500km. Topo enlarged grid
1200km by 1500 km Te-input model constant on
15km by 15km cells Moho with noise noise has rms
with 5 of maximal excursion Final Te resolution
in space 30-45 km
33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
(No Transcript)
38
(No Transcript)
39

• Summary isostasy
• Postglacial movements glacio and hydrostatic
  movements
• flexural isostasy- elastic thickness/flexural
  rigidity controling parameter
• Recover elastic thickness
• convolution method allows high spatial resolution
• Inner loads converted to equivalent topography

40

• References
• Braitenberg, C., Ebbing, J., Götze H.-J. (2002).
  Inverse modeling of elastic thickness by
  convolution method - The Eastern Alps as a case
  example, Earth Planet. Sci. Lett., 202, 387-404.
• Lambeck K. , Chappell J. (2001) Sea level change
  through the last glacial cycle, Science, 292,
  679-686
• Lambeck K., E. Bard (2000) Sea-level change along
  the French Mediterranean coast for the past
  30000 years , Earth Planet. Sci. Let., 175,
  203-222