CyberShake Magnitude Area Relationship

This page documents the magnitude-area relationship used in CyberShake

Original UCERF2 Modifications

Characteristic sources:

1. For each branch on UCERF2 logic tree (relevant here are the 2 M(A) branches: Ellsworth B and HB 2002):
   1. compute median magnitude from area with given scaling relationship
   2. compute total moment rate for source
   3. compute moment balanced Gaussian aleatory magnitude distribution with sigma=0.12, two sided truncation at 2 sigma.

        Note 1: this means that the resultant moment rate from this distribution will equal the moment rate from (1.2)
        Note 2: this is now discretized into 0.1 magnitude units

1. Create a single averaged source across all above branches (average the MFDs for the ruptures from (1), which could be multi-modal if the two M(A) relationships are significantly different)
2. If the CyberShake DDW fix is enabled (true for all CS studies to date), then for each source:
   1. compute implied area from Somerville 06 from UCERF2 magnitude
   2. compute DDW correction factor as ratio of implied area to original U2 area: ddwCorrFactor = som06Area / origU2Area
   3. extend DDW such that the new rupture area equals the implied Somerville 06 area: newDDW = origDDW * ddwCorrFactor

        Note: what this does, holding magnitude constant, is extend the area such that if you were to use Somerville 06 to compute magnitude it would equal the input UCERF2 magnitude.

Floating sources:

1. Compute target G-R MFD for source up to maximum magnitude (do this individually for each of the 2 M(A) branches, then average the MFD). This MFD is discretized in 0.1 magnitude bins
2. for each magnitude bin in G-R MFD:
   1. compute rupture area from scaling relationship
      1. If CyberShake DDW fix is enabled, use Somerville 06 and extended source surface
      2. Else use HB 02 and original source surface (not sure why it's only HB 02 and not an average with Ellsworth B here)
   2. build N floating ruptures with that area
   3. set rate of each rupture to G-R rate for magnitude divided by N (equal weight to each floating rupture for that magnitude bin)