Formulation for the rupture distance

R: Reference point

Strike fs measured from R to L

Dip d measured from R to W

Length: l=AB=RL=UW

Width: w=CD=LU=RW

Distance from the reference point to the observation site: Drs=RS

Azimuth from the reference point to the observation site: qs

Coordinate of the vertices:

R: (xr,yr,zr): Reference point of fault

L: (xr+l sinfs, yr+l cosfs , zr )

W: (xr+w cosdcosfs, yr-wcosdsinfs , zr+wsind)

U: (xr+w cosdcosfs+lsinfs, yr-wcosdsinfs +lcosfs, zr+wsind)

S: (xr+Drssinqs, yr+Drs cosqs , 0.0 ): Observation site

Length of perpendicular SO on the fault plane: SO=x

Parameters (s,t): CO=sCD, BO=tAB

Right triangle DWAS: WS2= WA2+AS2=RW2(1-s)2+AS2

Right triangle DRAS: RS2= RA2+AS2=RW2s2+AS2

WS2- RS2= RW2(1-2s)

Then, s=1/2 – (WS2- RS2)/ 2RW2=1/2 – (WS2- RS2)/ 2w2

Right triangle DRCS: RS2= RC2+CS2=RL2(1-t)2+CS2

Right triangle DLCS: LS2= LC2+CS2=RL2t2+CS2

RS2-LS2= RL2(1-2t)

Then, t=1/2 – (RS2-LS2)/ 2RL2=1/2 – (RS2-LS2)/ 2l2

s<0, t>1 : distance from S to the vertex R

Drup=RS

s<0, 0<t<1 : distance from S to the side RL

Right triangle DLCS: CS2= LS2-LC2

Drup=CS=sqrt(LS2-LC2)=sqrt(LS2-LR2t2)= sqrt(LS2-l2t2)

s<0, t<0 : distance from S to the vertex L

Drup=LS

0<s<1, t>1: distance from S to the side RW

Right triangle DRAS: AS2= RS2-RA2

Drup= AS=sqrt(RS2-RA2) =sqrt(RS2-RW2s2) =sqrt(RS2-w2s2)

0<s<1, 0<t<1: distance from S to the fault plane

Right triangle DCOS: OS2= CS2-CO2

Right triangle DLCS: CS2= LS2-LC2

Drup=OS=sqrt(CS2-CO2)=sqrt(CS2-CD2s2)=sqrt(LS2-LC2-CD2s2)
=sqrt(LS2-LR2t2-CD2s2) = sqrt(LS2-l2t2-w2s2)

0<s<1, t<0 : distance from S to the side LU

Right triangle DLBS: BS2= LS2-LB2

Drup= BS=sqrt(LS2-LB2) =sqrt(LS2-LU2s2) =sqrt(LS2-w2s2)

s>1, t>1: distance from S to the vertex R

Drup=WS

s>1, 0< t<1: distance from S to the side WU

Right triangle DUDS: DS2= US2-UD2

Drup= DS=sqrt(US2-UD2) =sqrt(US2-WU2t2) =sqrt(US2-l2t2)

s>1, t<0: distance from S to the vertex U

Drup=US