Fsa: inversion parameters

2008-04-09

Purpose

Inversion can be tuned to take into account various constraints and type of data:

fault and slip with well defined slip rake (classical inversion)
fault where only strike or dip sense of slip is defined
fault where rake of slip is defined within a range
earthquake focal mechanisms, where either nodal plane can be the fault plane

Definitions

Symbol	Definition	Comments	units
N	data set size
i	fault and slip data of index	1 ≤ i ≤ N
Ra(i)	actual rake for data i	undirectly known through Ro(i) and, eventually, Rr(i)	degrees
Ro(i)	observed rake for data i	what is actually measured	degrees	Ro(i) = Ra(i) if the measurement is perfect
Rr(i)	half rake range for data i	0 ≤ Rr(i) ≤ 180	degrees	Ro(i)-Rr(i) ≤ Ra(i) ≤ Ro(i)+Rr(i)
Rp(i)	predicted rake for data i	computed for a given stress tensor	degrees
E(i)	individual error, E(i), on the fault and slip data i	computed for a given stress tensor	degrees	0 ≤ E(i) ≤ 180
F	misfit function on the data set	computed for a given stress tensor	degrees	0 ≤ F ≤ 180

The fault plane error

The fault plane error, E(i), is the error on the fault and slip data i computed for a given stress tensor and can be either of:

the rake difference (or angular error), Ea(i), defined as
- Ea(i) = | Rp(i) - Ro(i) |
where
- | | is the absolute value
- 0 ≤ Ea(i) ≤ 180 (in degrees)
the rake range error, Er(i), defined as
- if Ea(i) ≤ Rr(i) then Er(i) = 0
- if Ea(i) ≥ Rr(i) then Er(i) = | Ea(i) - Rr(i) |
the slip sense error, Es(i), defined as
- if Ea(i) ≤ 90 then Es(i) = 0
- if Ea(i) > 90 then Es(i) = 180

For focal mechanisms, E(i) can be computed from either the first or the second nodal plane, or taken as the average of these two calculations.

Misfit function or global error

The misfit, F, of the stress tensor with respect to the global fault and slip data set can be either of:

the average error
- Fa = ( sum of E(i), i= 1, N ) / N
or the maximum error
- Fm = max of ( E(i), i= 1, N )

If only a percentage of the full data set is required to be fit, then letting P represents the corresponding number of retained data, the individual fault plane errors are sorted in ascending order:
E(1) ≤ E(2) ≤ ... ≤ E(i) ≤ ... ≤ E(P) ≤ ... ≤ E(N)
and the misfit becomes:

the average error
- Fa = ( sum of E(i), i= 1, P ) / P
or the maximum error
- Fm = max of ( E(i), i= 1, P ) = E(P)

Set up

These parameters are set within the subroutine sfsinv

 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
 @ subroutine sfsinv     
 @==========================================
 -------------------------------------------
 INVERSION PARAMETERS
 -------------------------------------------
 switch name of current set up
    1   1st  fault plane     
    1   rake difference      
    1   Average on % best    
 -------------------------------------------
 fault plane choice
 1st  fault plane     .........101
 -------------------------------------------
 fault plane error
 rake difference      .........201
 slip sense           .........202
 rake range difference.........203
 -------------------------------------------
 global error
 Average on % best    .........301
 Maximum on % best    .........302
 ---------------------------------
 % of data to explain ..........10   100.0
 -------------------------------------------
 EXIT ..........................99
 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

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