A spherical Fourier approach to estimate the Moho from GOCE data Mirko Reguzzoni 1, Daniele Sampietro 2 2 POLITECNICO DI MILANO, POLO REGIONALE DI COMO.

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.

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

MOHO DEPTH OF EUROPEAN PLATE

• 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

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.

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.

• 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)

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

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

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

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°

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

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

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

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

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

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.

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

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

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

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.

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.