Title Slide + Overview:
In this talk  I am examining  problems with 9 component data as far as they relate to
source behavior.  I will give you an overview of the problem setting. How it occurs
and try to suggest a strategy on how to remedy it. This strategy will employ the
idea of symmetric sampling in component as well as spatial extent.  From this 
premise it is possible to employ reciprocity arguments to  set up a minimization problem, in order to find a relative source filter. This filter then backs out the 
source effects from the prestack data.

Slide 2 + 3:
On you left you see a basic diagram of a multi component experiment. The source
consists of 3 orthogonal vectors and so does the receiver.  To get the full 9component
data set we have to initiate the source in 3 orthogonal directions. and this can be done
in  several practical ways.  
Foreach of this orthogonal direction we  record 3 components at each receiver location.  So we end up finally with a 9 component trace.
On the right slide you see the mathematical expression on , a outer product
of the 3 component source with the 3 component receiver, giving a 3x3 tensor
quantity, related to source pos. r_1 and receiver position r_2, registered in time.
As a convention the source components are Uppercase while the receiver components are lowercase.

Slide 4+5: 
Looking at the mathematical expression on the right we can see that the source
behavior influences the elements in the trace, each source component a different 
row.   Clearly if the source varies, either locally , component by component,
or spatially, we will see be able to observe it distinctly.
Such a spatial change in behavior wit have numerous effects on our data set.
It ranges from uneven CMP stacks to biased inversion results, or misinterpreted
reflectivity.  One of the most simple effects are source and receiver rotations.

slide6+7:
One of the striking features of elastic sources as opposed to pressure point 
sources is that,  even in a homogeneous medium away from boundaries 
they have a clear radiation pattern attached to them. This pattern is in 
general non-isotropic.  These patterns can vary from location to location
and also from component to component.   If we take a typical surface land source,
as most multi-comp. sources are,  we will see some variation for each source.









slide 7 + 8
Are those effects observable ? And how do you recognize them ?
To study that an easy plot s the total energy in the prestack data trace.
For 2D shooting geometry, w see here the 9 component plot of  for
the part of the experiment. As in the mathematical description,  X source toprow
Y source midde row and Z source is bottom.
Diagonal Elements are Xx, Yy, Zz, the off-diagonal elements measure
unlike components.
The cdp axis is the big line in the middle, where zero offset was not  recorded.
In an ideal world we would see a very smooth  change with offset and 
a change in the offset direction depending on the structure.
Here the most dominant effects are clearly caused by  source or receivers
and not by the structure, which is what we are looking for.

What I just showed you was an example of a totally symmetric data acquisition.
It covers source an receiver space evenly including the component space
for sources and receivers.
This offers some unique possibilities in processing.

slide 9 + 10:
Let us consider the following data model:.....
Then if the wave field would be truly symmetric:   SGR = transpose of SGR
To measure ow asymmetric real data actually is we can compute, the following
 symmetric part:   and deviatoric part
If it is symmetric: deviatoric part is zero.


slide 11 + 12:
What I show here is  a time slice taken out of prestack data volume.
It is around the time where I expect some events to be present in the data.
Left:   added data and its transpose,  on right  I subtracted them.
Clearly the deviatoric part is not zero, it is actually on the same order of magnitude
as the symmetric part. However you notice a feature: the are those slanted
lines of coherent energy. Those are representing  energy coming from
some subsurface structure, while the deviatoric part has them not.


slide 13+ 14:

well what can we do with this information ?
We really want to invert data for the subsurface modeled the source function at
he same time. However there are a number of reasons to not do that and
take a simpler approach, like estimating source, separate wave types,
find velocity model and Invert.

slide 15+16
I am concentrating on the first part, estimating the source behavior.
If asymmetry is observable  as we saw before, we can use that info to
base an optimization problem on the lack of symmetry.
That amounts to  using  reciprocity property of the dataset.
What does reciprocity  for a vector wave field actually mean ?
Lets look at the  picture on the right,  Take a single component source
in one place, eg. Z  , it will convert in some wave type, p propagate
down to the scatterer  convert into some other wave type and reach 
surface as s wave and will get registered in one component , eg. x.
Now what is the reciprocal experiment, well exchange the source and receiver
position, but  attach the vectors to the surface. That means  source ---
now into | receiver. .  Or mathematically on the left hand side.
exchange the source and receiver position and also transpose the 3x3 matrix
of components.

slide 17:

if asymmetry is observable, as it was, we can base now optimization problem
on the reciprocity property. Namely minimize the differences in reciprocal
trace pairs. that will force symmetry.  And the way to do that is in a location
and component consistent approach. Such that the residual gets projected
into those spaces.

Slide 18 + 19:
The function we then seek to minimize s either the energy  of the differences
in individual components,  we have trace pair and to each trace there is a filter
attached, for that  source and receiver location and component.
We can do that either globally or for each component pair individually.

slide 20+21:
I did not specify in any way up to now how those filters look like,
and we have actually multitude of choices, it can be 1D or multidimensional.
I chose as a first order approach a 1D filter with relatively few filter coefficients,
that means we do an implicit angular averaging, which is a reasonable thing to do 
given an event a a limited aperture.
One way to solve it is using CG l2 algorithm

slide 22 + hold right
This a typical set of filters obtained from solving the minimization problem.
It is the filter component which acts on the X source component contribution.

slide 23+ 24:
the next on is the filter set which gets applied to the Y component


slide 24+ hold right:
and finally the Z component.
These filter sets night not show dramatic changes, however these are components
of vector filters.  Those have the effect of rotation the components around, with the
effect of rotating the source and changing the effective radiation pattern.

slide 25 + 26:
After seeing the filter we are curious to see what it did to the data. 
It effectively rotated the data  and scaled and slightly filtered the data.
Looking at the time slice again after applying the filter, we see that the continuity
of events in the structure direction is increased, we are able to follow the event 
now all the way out  with the same polarity and character unlike before.
So it did a reasonable job in equalizing the data and improving the coherency
of events that was diminished before by source and receiver effects.
It did do this by removing  relative changes within the data set, it did not remove
the source wavelet itself so it si not a deconvolution process in the traditional sense.

slide 27+28:
The algorithm I employed here has some strength and also some limitations.
First of all no subsurface velocity is necessary to perform the process.
Thus it is a convenient preprocessing tool, which removes differential
variations in the multi component data set.
By using 1D filters a I showed, we perform an implicit angular averaging,
which may or may not be desirable depending on the data, the structure and problem.It is susceptible to noise that varies the same way as the source behavior
does. It cannot  discern the two.

In conclusion I showed you a first order process to remove source variations
from multi component data. It tries to t do that using the reciprocity principle and
an a source location consistent and component consistent minimization
algorithm applied to reciprocal trace pairs. And is the basis for the determination
of absolute radiation patterns.

slide  ???
Id like to thank Chevron for  supplying me with the the 9c data set and  facilities
and I thanks the sponsors for SEP for the  continuing
 support.

