1. Testing limits of applicability of Montagner, Griot-Pommera, and Lavé’s method.
We use Montagner et al.’s [2000] (M) method to strip off the splitting due to the lithosphere. On average the splitting inferred from the surface waves is ~0.2 sec and periods of the SKS waves are 6 seconds or longer. M’s criterion is that

( z z
0
)

V / 2 V
2
S
0
 2  / 6

1 or
 t / 2

1. We use T=6 seconds as the extreme case. Let
. Then
 t

0.1047 , which is less than 1, but not much, much less than 1. So we performed synthetic tests to estimate the uncertainty that is associated with the approximation. We found that only in the infinitesimal limit is the method commutative, and even then, it is inaccurate because if the incident polarization is parallel to the fast or slow direction of the lower layer that layer has no effect on splitting, yet M’s formula is independent of incident azimuth.
For the synthetic we generated east and west components of a 6 sec cosine wave modulated by a Gaussian, and rotated to fast and slow directions, changed the phase and rotated back for a series of up to 4 layers. The splitting operator is defined by
 operating on the seismogram
S 


N
E


where N and E are the components. Let

be
S s



N
E s s

 


 cos sin
  sin 
 cos 




 exp i  t /2
0 exp
0
 i  t /2




 cos
 sin
 sin 
 cos 





N
E

 (1)
  R ' HR
After multiple splits
S s
  i
 i
S
.
We determined the apparent 1-layer splitting parameters [
 a t a
] using standard numerical schemes (Silver and Chan, 1991; Davis, 2003) on the split components and calculated theoretical estimates of their values using two methods (1) the M approximation and (2) Silver and Savage’s approximation. Silver and Savage note that splitting parameters for multiple layers exhibit 4z periodicity, where z in the azimuth of incoming waves. We find that large values are obtained when the tangential energy generated by the two layers goes to near zero. So we averaged the effective splitting parameters when the tangential energy is greater than 5% of the radial energy. The results for the two-layer case and four-layer cases are shown below. The calculated splitting parameters found from minimizing tangential energy are
[  from the multi-layer expressions of Silver and Savage are [ 
SilvSav
,  t calc
,
SilvSav
 t
] calc
]
the results
and those using
M’s expressions are [ 
M
,  t
M
]
. The figures below give the fast directions and delays in each layer in the title, and in the third frame give the various estimates of the averaged parameters. The (blue) solid line gives the numerical calculated values from the waveforms, the stars are the Silver Savage estimates that agree very well with the numerical values, and the horizontal red lines are M estimates which are approximately correct. We use splitting values typical of the surface wave estimates for the lithospheric

layers in our case. We note the M estimates, that assume commutivity, are ~mean values of those obtained when the order of the layers is reversed.
2. Justification of applying an inverse splitting operator to remove the effects of the lithosphere.
Synthetic seismograms were computed numerically for the two-layered anisotropic model (Figure 7) using David Okaya’s code. We then performed the splitting analysis by first stripping off the effects of the first layer using the inverse splitting operator
  1 

 cos
 sin

2

2 sin 
2 cos 
2



 exp  i  t
2
0
/2 exp
0 i  t
2
/2



 cos sin

2

2
 sin cos

2

2

 and solving for the fast direction and splitting time. Since
  1 


1
1


I we expect that this process should be exact, a supposition that was confirmed by the above experiment.
The earth model has the two layers (100 km with 200 km layer below it). The background mantle velocity is 8 km/sec and 4.7 km/s for Vp and Vs. The top layer is
1.6% anisotropy, which would produce .35 sec splitting over 100 km. Layer 2 is 3.5% anisotropy which would produce 1.54 sec splitting over 200 km. The bottom-most layer is isotropic. Layer 1 fast axis is oriented N45E and Layer 2 is E-W.
The results obtained using our stripping method are listed below: top layer
 lower layer

2
1
= 45.0
 t
=89.85
 t
2
1
= 0.35 stripped of using

2
= 1.582 compared with model input values of [90, 1.54].
We conclude that in this case the stripping is accurate.
3. Effect of radial anisotropy
To test effects of radial anisotropy as well as the inverse splitting operator we generated synthetics using the propagator matrix synthetics using the methods of Keith and

Crampin (1977). We assume orthorhombic symmetry and use a two anisotropic layer model described below in table (1) to approximate lithosphere over asthenosphere.
C ij
(GPa) 
Lithosphere
3 3.50 100.00 km
255.86 101.23 101.23 0.00 0.00 0.00
255.86 101.23 0.00 0.00 0.00
255.86 0.00 0.00 0.00
76.31 0.00 0.00
78.31 0.00
76.31
Asthenosphere:
2 3.50 200.00 km
243.10 079.10 077.20 0.00 0.00 0.00
201.40 087.30 0.00 0.00 0.00
210.70 0.00 0.00 0.00
62.60 0.00 0.00
65.25 0.00
68.50
Table 1. Two-layer model. Elastic moduli of the anisotropic tensor are in GPa. Fast directions of the orthorhombic tensor were varied in each layer.
Synthetic seismograms were generated for different delay times and azimuths of fast direction in each layer. Then the effects of the top layer were stripped off the synthetic seismograms, and the splitting parameters for the lower layer determined for different amounts of radial anisotropy. The incidence angle in the lower layer was set at
15 degrees.
For example using the above parameters the upper and lower layers have theoretically calculated values of [ 
2
,  t
2
]  [143 o ,0.275
s ] and [ 
1
,  t
1
]  [123 o ,1.004
s ]
After stripping off the upper layer we obtain for the lower layer
[ 
1
,  t
1
]  [123.14
o ,1.119
s ]
Then changing the anisotropy in the lower layer from
C
66
 68.5
 78.5
[ 
1
,  t
1
]  [124.53
o ,1.136
s ]
i.e., a small difference.
GPa,
Davis (2003) investigated the dip of the anisotropy tensor and showed it gives rise to an azimuthal variation in splitting parameters that have 2z and 1z variation about the average.
In conclusion the results showed that the fast directions for the lower layer were determined and were largely independent of the radial anisotropy. We concluded that the

stripping works in this case also, and is not appreciably affected by the amount of radial anisotropy.
SKS Splitting values.
Table 1 gives SKS splitting values used in Figure 7. Note fast directions are measured clockwise from North.
Supplementary Figure Captions
Figures 1-6 are various estimates of the effective 1-layer spitting parameters for synthetic models in which the number of layers is two (Figures 1 and 2) or four (Figures 3-6). The layer fast directions and splitting times are listed at the top of each figure. The various estimates are listed in the third panel. Excellent agreement is observed between
and expressions given by Silver and Savage are obtained using M’s expressions are
. Those
less accurate in angle than splitting time
(Note is measured anticlockwise from east). However for reasons we do not fully comprehend both the Silver Savage expressions and the waveform analysis give large deviations when the tangential energy goes close to zero, possibly a numerical instability.
Thus in some sense the M expressions are to be preferred that avoid these regions.
For our data we used both methods and, other than at a small subset of stations when the chosen back-azimuth gave very large values of the SS values because of the abovementioned instability, the two methods agree.
Figure 7.
Synthetic seismograms were computer numerically for the two-layered anisotropic model using David Okaya’s code.