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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 Tel +39-040-5582258 fax +39-040-575519. Topics. Different aspects of isostasy:

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gravity inversion and isostasy an integrated system

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

Tel +39-040-5582258 fax +39-040-575519

  • Different aspects of isostasy:
    • Local compensation
    • Regional compensation
    • Isostasy and sea level change
    • Examples:
    • Glacio-isostasy and Hydro-isostasy
  • Gravity inversion
    • Spectral properties of the gravity field
    • Downward/upward continuation
    • Iterative inversion procedure
    • Examples:
      • Synthetic root
      • Eastern Alps
  • Integration of isostasy and gravity inversion:
    • Eastern Alps
    • Tibet plateau
    • Parana’ Basin
    • South China Sea
description of geographic differences
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
description of geographic differences1
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.
geographic differences description
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
isostatic models locales equilibrium airy und pratt and regional equilibrium flexural isostasy
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
regional equilibrium flexural isostasy
Regional equilibrium (Flexural Isostasy)
  • Flexural rigidity:

Typical values:

E = 1011 N/m2

 = 0.25

regional equilibrium flexural isostasy1
Regional equilibrium (Flexural Isostasy)
  • Insert expressions for p and q:
  • Solution of equation:

k= wave-number

We put:

We obtain:

regional equilibrium flexural isostasy2
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 :

regional equilibrium flexural isostasy3
Regional equilibrium (Flexural Isostasy)
  • k= wave-number
  • W(k)= FT(w(x)) H(k)= FT (h(x))
  • To the same result you obtain by applying the Fourier Transformation (FT) of the equation:
transition to local compensation
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.

properties of the plate flexure
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
  • 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

  • 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.

i ce equivalent sea level change from the french mediterranean area compared to global curves
Ice-equivalent sea-level change,from the French Mediterranean area compared to global curves
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)

Elastic flexure model:Thin 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

convolution method
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.

total load topographic and buried load
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


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 10-25 500 19 42

2 2.38 10-25 841 33 68

5 1.20 10-210 836 65 135

10 7.11 10-315 938 105 230

20 4.23 10-320 1183 180 400

30 3.12 10-320 1604 245 520

40 2.51 10-330 1326 310 650

50 2.13 10-330 1568 360 760

60 1.85 10-330 1798 400 870

application of the convolution method in the flexure calculation
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

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.

On square overlapping windows: for Te=0,...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


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


  • 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