A spherical Fourier approach to estimate the Moho from GOCE data
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A spherical Fourier approach to estimate the Moho from
GOCE dataMirko Reguzzoni 1, Daniele Sampietro 2
2 POLITECNICO DI MILANO, POLO REGIONALE DI COMODepartment of Hydraulic, Environmental, Infrastructure and Surveying Engineering
1 ITALIAN NATIONAL INSTITUTE OF OCEANOGRAPHY AND APPLIED GEOPHYSICSDepartment of Geophysics of the Lithosphere.
The present research has been partially funded by ASI through the GOCE ITALY project.
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AN EXAMPLE: The first digital, high-resolution map of the Moho depth for the whole European Plate, extending from the mid-Atlantic ridge in the west to the Ural Mountains in the east, and from the Mediterranean Sea in the south to the Barents Sea and Spitsbergen in the Arctic in the north.
MOTIVATION• Moho estimation is traditionally based on: - seismic data (profiles) - ground gravity data (points)
accurate information at local scale
Andrija Mohorovičić
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MOHO DEPTH OF EUROPEAN PLATE
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• Data come from early the 1970s and the 1980s to 2007.
• Older profiles were digitized by hand from published papers .
• For some areas regional Moho depth maps, compiled using deep seismic data have been used.
MOHO DEPTH OF EUROPEAN PLATE
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THE GOAL
The GOCE mission promises to estimate the Earth’s gravitational field with unprecedented accuracy and resolution. The solution of inverse gravimetric problems can benefit from GOCE.
The GOCE mission can be used to improve the existing model or to estimate the Moho in large areas from an homogeneous dataset.
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We consider a mean reference Moho (computed for example from a isostasy model).
THE HYPOTHESES
We suppose to know (and subtract from the observations) the gravitational effect of the layers from the center of the Earth to bottom of the lithosphere (e.g. using a Preliminar Reference Earth Model).
We neglect the effect of the Atmosphere.
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• Hypotheses:
- two-layer model: 1) from topography to moho2) from moho to the bottom of the lithosphere
- layers with constant density:1) ρc=2670 kg m-3
2) ρm=3300 kg m-3
Topography Moho
Lithosphere
THE HYPOTHESES
Unique solution (Barzaghi and Sansò 1988)
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THE HYPOTHESES
• Hypotheses:
- GOCE data (potential and second radial derivative) on a grid at satellite altitude with stationary noise.
- Ground gravity anomalies.
250
km
Space-wise approach
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THE METHOD (GOCE-ONLY, PLANAR APPROXIMATION)
Linearization
Fourier transform
Inverse Fourier Transform
Estimated Moho
Error cov-matrix
dx
Gdd)x(T
DD
h222
0z
DDh
h2222
2
zz dzx
Gdd
z)x(T
dDxkxTxTxT T
dDxkxTxTxTzzTzzzzzz
fˆfD̂fk̂fT̂ T0
fˆfD̂fk̂fT̂ zzT0zz zz
2D collocation
0zz0T,T T̂T̂fD̂zz
zzzz
zz
SSk̂SSk̂SS
Sk̂S
D2TD
2Tzz
TD
zzzzSSk̂SSk̂SS
Sk̂S
D2TD
2Tzz
zzTD
zzzz
zzT,T SSk̂SSk̂SS
SSS)f(S
D2TD
2Tzz
zzD
convolution error spectrum
prediction
observables
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THE METHOD (PLANAR APPROXIMATION)
Point-wise ground observations can be added to the system to improve the the estimation of the high frequency of the model.
The collocation system can be partitioned as:
z
zz
1
Tz,TzTzz,TzTzz,Tz
Tz,TzzTzz,TzzT,Tzz
Tz,TTzz,TT,T
T
Tz,D
Tzz,D
T,D
zT
zz
T
T
T
T
CCC
CCC
CCC
C
C
C
TT
TD̂
Gridded satellite observations
Ground point-wise gravity anomalies
The system can be efficiently solved
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EDGE EFFECTSConvolution kernelBording area
Δφ
Δλ
Correct convolution Edge effect
In the case of moho estimation:
Potential: Δφ =25°, Δλ=45°
First radial derivative (at ground level): Δφ=2°, Δλ=3°
Second radial derivative: Δφ=5°, Δλ=9°
MISO approach:Δφ=12°, Δλ=22°
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EDGE EFFECTS
In the case of moho estimation:
Potential: Δφ =25°, Δλ=45°
First radial derivative (at ground level): Δφ=2°, Δλ=3°
Second radial derivative: Δφ=5°, Δλ=9°
MISO approach:Δφ=12°, Δλ=22°
Convolution kernelBording area
Δφ
Δλ
Correct convolution Edge effect
We have to consider wide areas
Generalize the method to spherical approximation
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SPHERICAL APPROXIMATION
R
SRP
ER
Moho
Topography (HQ)
(MQ)
Q
PQl
R
R cos
drl
rGddPT
QE
Q
HR
MR PQ
2
)(
We start from the potential in spherical coordinates:
and introduce the coordinates system:
cos
12Rdd
dd
We approximate the distance between P and Q as:
where
2
1
22
h
SSPQ l
R
RRRl
222QPQPhl
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ddM
R
RRR
GPT Q
Q
QPQPS
S
cos
cos)(
2
1
222
The potential can be linearized with respect to the variable r around R
ddMfPT QQQQ
QPQP ,cos
cos,)(
Convolution kernel
SPHERICAL APPROXIMATION
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A SIMPLE EXAMPLE
φ=27°
φ=81°λ=-41°
λ=71°
h=10km
kmkm
Errors from 0.5 km to 1 km
Errors from 0.2km – 0.7km
Estimated model in spherical approximation
Estimated model in planar approximation
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SIMULATED MOHO IN CENTRAL EUROPE
66°
112°
The final moho will be estimated in an area of 42°x75° with a resolution of 0.25°
Bording area for Trr convolution
Bording area for T convolution
The considered region
Bording area for MISO approach
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SIMULATED MOHO IN CENTRAL EUROPE
Reference mohoEstimated model in spherical approximation
km
km
Low-medium fequencies are well estimated using GOCE only observations.
As expected details are not recovered by observations at satellite altitude.
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SIMULATED MOHO IN CENTRAL EUROPE
Differences between the starting moho model and the estimated one
Difference between reference and estimated model(Spherical approach)
Mean [km]
-0.3
r.m.s. [km]
0.5
Details are not recovered from GOCE observations
km
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SIMULATED MOHO IN CENTRAL EUROPE
Differences between planar and spherical approches
km
Reference EstimatedSpherical
Estimated Planar
Mean [km]
31.2 30.9 30.8
r.m.s [km]
1.70 2.33 3.3
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SIMULATED MOHO IN CENTRAL EUROPE
Error map obtained with potential, second derivative and adding 50 ground observations.
Adding ground observations also high frequency can be recovered.
Estimation error decrease to 0.3km (r.m.s.)
km
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CONCLUSIONS
The general problem of estimating the discontinuity surface between two layers of different constant density was investigated.
A method based on collocation and FFT has been implemented to evaluate the contribution of GOCE data in the Moho estimation. The integration between satellite and ground data has been studied.
The method has been generalized to spherical approximation. Convolution kernels have been modified in order to consider the effect of spherical approximation.
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CONCLUSIONS
FUTURE WORK
• Apply the method to real data (GOCE-ITALY project).
Open issue: how to disentangle the different gravimetric signals that are mixed up into the data?
combination with geological models and conditions
The method has been tested on simulated data for the estimation of the Moho depth in the central Europe, showing that GOCE observation can improve our knowledge of the crust structure.