== MS_anisotropy

 MS_ANISOTROPY - Simple measures of anisotropy 

 // Part of MSAT - The Matlab Seismic Anisotropy Toolkit //

 Calculate the degree of anisotropy of an elasticity matrix. 

 [ uA, ... ] = MS_anisotropy( C, ... )

 Usage: 
     [ uA ] = MS_anisotropy( C )                    
         Return the Universal Elastic Anisotropy Index of Ranganathan
         and Ostoja-Starzewski (2008). Valid for any elasticity matrix,
         uA is zero for an isotropic case and increases for increasing 
         anisotropy.

     [ uA, lmA ] = MS_anisotropy( C )
         Also return the general measure of anisotropy proposed by
         Ledbetter and Miglion (2006). This the the ratio of the fastest
         and slowest squared shear wave velocity over all propogation
         and polarization directions. Equal to one in the isotropic case,
         increases with increasing anisotropy.

     [ uA, lmA, zA ] = MS_anisotropy( C )
         Also return the Zenner (1948) measure of anisotropy. This is
         only valid for cubic crystals (NaN is returned if C does not 
         represent a cubic crystal). zA is 1 for an isotropic case and
         increases or decreases with incresing anisotropy.

     [ uA, lmA, zA, cbA ] = MS_anisotropy( C )
         Also return the Chung-Buessem (1967) anisotropy index. This
         is a single valued measure of anisotropy derived from zA. Like
         uA, this is zero for an isotropic case and increases for increasing 
         anisotropy. Only valid for matricies representing cubic crystals.

     [ uA, lmA, ... ] = MS_anisotropy( C, n )
         Set the number of random directions to sample for the calculation
         of lmA. Defaults to 1000, which seems to give results accurate to
         two decimal places. Ledbetter and Miglion (2006) use 10000 which 
         gives results reproducable to three decimal places and a 
         noticable slow down.

 Notes:
     These measures of anisotropy are independent of orientation. However,
     the test for cubic symmetry assumes the matrix is in an ideal
     orention. Use MS_AXES to reorentate the imput matrix for the general 
     case. MS_NORMS can be used to provide an alternate measure of
     anisotropy. Ledbetter and Miglion (2006) claim lmA is identcal to zA
     for cubic cases but Ranganathan and Ostoja-Starzewski (2008) point 
     out cases where zA < 1 while lmA > 1 by construction. 

 References:
     Zenner, C. (1948) Elasticity and Anelasticiy of Metals. University 
     of Chicago. 

     Chung, D. H. and W. R. Buessem (1967) Journal of Applied Physics 
     vol.38 p.5 

     Ledbetter, H. and A. Miglion (2006) "A general elastic-anisotropy 
     measure" Journal of Applied Physics vol.100 art.num.063516
     http://dx.doi.org/10.1063/1.2338835

     Shivakumar, I. and M. Ostoja-Starzewski (2008) "Universal Elastic
     Anisotropy Index" Physical Review Letters vol.101 art.num.055504.
     http://dx.doi.org/10.1103/PhysRevLett.101.055504

 See also: MS_POLYAVERAGE, MS_NORMS, MS_AXES, MS_PHASEVELS
