mogiM: Mogi Model

In geophys: Geophysics, Continuum Mechanics, Gravity Modeling

Description

Mogi model deformation returns the deformation from a point source presurized inflation in an elastic medium.

Mogi's model (point source in elastic half-space). computes radial and vertical displacements Ur and Uz, ground tilt Dt, radial and tangential strain Er and Et on surface, at a radial distance R from the top of the source due to a hydrostatic pressure inside a sphere of radius A at depth F, in a homogeneous, semi-infinite elastic body and approximation for A << F (center of dilatation). Formula by Anderson [1936] and Mogi [1958].

Usage

mogiM(R = 1, F = 1, A = 0.1, P = 1e+05, E = 1e+10, nu = 0.25)

Arguments

R	Hoirizontal Distance frm source, m
F	Depth below surface, m, positive down
A	radius of magma chamber
P	hydrostatic pressure change in the sphere
E	elasticity (Young's modulus)
nu	Poisson's ratio

Details

Original paper by Mogi used poisson's ratio equale to 0.25, i.e. lame parameters lambda and nu were equal.

Value

list:

ur	radial displacements Ur
uz	vertical displacements Uz, Uz > 0 = UP
dt	ground tilt Dt
er	radial strain Er
et	tangential strain Et on surface

Author(s)

Jonathan M. Lees<[e-mail]>

References

Anderson, E.M., Dynamics of the formation of cone-sheets, ring-dikes, and cauldron-subsidences, Proc. R. Soc. Edinburgh, 56, 128-157, 1936.

Mogi, K., Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them, Bull. Earthquake Res. Inst. Univ. Tokyo, 36, 99-134, 1958.

Examples

data(PXY)

delV = 2.3E13/(100^3)    ##### (convert to meter^3 from cm^3)
F = 2.8E5/100      ##### (convert to meter from cm    )

EX = seq(from=0, by=100, to= 9000)


Atest = mogiM(R=EX,F=F,A=delV)



 plot(PXY, pch=6, col='purple', xlim=c(0,9), ylim=c(0, 1) )
    ###  model
    lines(EX/1000, Atest$uz/max(Atest$uz))


############ best fit   optimization

library(stats)

    fr<-function(x)
      {

        Atest = mogiM(R=PXY$x*1000 ,F=x[1],A=x[2])


        rms = sum ( (PXY$y - Atest$uz/max(Atest$uz))^2 )

        return(rms)
      }
xin = c(2600, 2.0e+07)

FOUT = stats::optim(xin , fr)

  Btest = mogiM(R=EX,F=FOUT$par[1] ,A=FOUT$par[2])

   plot(PXY, pch=6, col='purple', xlim=c(0,9), ylim=c(0, 1) )
 
 lines(EX/1000, Btest$uz/max(Btest$uz))







    

geophys documentation built on May 1, 2019, 9:26 p.m.