Using Semblance based Time Delays to Locate Event Directions
Definition of the Time Delay Matrix
Property 1 of the Time Delay Matrix
Example: A Vertically Incident Plane Wave
The Effect of Noise in a Vertically Incident Plane Wave
Example: Two Horizontally Incident Plane Waves.
We develop here a method that will use the relative time delays of an event between multiple satellites to find the direction and phase velocity of that event. Since we only dealing with the relative time delays of the event, we do not directly address the type or magnitude of the event. We are most interested, and hence concentrate almost exclusively on the finding the direction of the event.
In the first couple of sections, we use the tensor analysis of Harvey recast to fit our needs. Starting in the section “Use of the Slowness” vector, we develop new material that fit our needs.
In this development, we use the techniques of the taup method of seismic processing. For taup processing, one adjusts the slowness of a plane wave using , the slowness in a plot called a slantstack. In out case, we are using the concept of the slowness vector of a plane wave, i.e. , where n is the phase direction of the plane wave, and V is the phase velocity. This slowness vector casts the derivation equations into a simple form.
We hope to eventually use these techniques in a case of the superposition of plane waves.
You will find in equation (3.9) a way to calculate the slowness vector given the relative time delays. We also have in equation (4.1) through (4.3) a way to use the slowness vector to construct the time delay matrix that we have been calculating. Finally, in equations (5.5), (7.3), and Error! Reference source not found. we show (hopefully) familiar patterns of the time delay matrix.
As a bonus, we include the start of a noise analysis, and in the summary, a todo list.
Consider N satellites moving in an orbit, and suppose a plane wave impinges on these satellites. The phase velocity of that plane wave is defined as V and the direction is given by the unit vector . Then the difference of the arrival times at two satellites is given by the expression:
(1.1) .
Assume the coordinate system defined by placing the origin at the centroid of the satellites:
(1.2) .
Suppose there is some uncertainty in the locations of the spacecraft and/or the calculation of the phase of the plane wave. In that case, one should consider the quantity S, the error between the estimated and actual values of the event’s temporal location
(1.3) ,
where is simply an arbitrary fixed reference time. In the above expression, we want to minimize S. Define the slowness vector as
(1.4) .
In this case, we find that equation (1.3) may be rewritten as
where we use the tensor summation convention , i.e. a repeated index implies a summation over all possible values of that repeated index.
In general we will be minimizing S in equation (1.5) while accommodating various levels of noise in the quantities.
We look here at the question of uncertainty in the phase of the plane wave. This uncertainty is expressed in the values of and , the origin of time, and the phase slowness of the plane wave. In order to minimize (1.5) for these quantities, we need to take the partial derivatives with respect to and k=1,2,3. Setting those derivatives to 0, we find:
(2.1) .
The first equation of (2.1) can be simplified by isolating the term, and then using equation (1.2):
(2.2) .
(Remember the implied summation over k since k is a repeated index.) This last equation simply means that the best choice of the time origin is the average of the time delays.
Next consider the (volumetric) tensor defined by
(2.3) ,
the last expression using the tensor summation convention. We find that is a second order tensor, which may be represented as a 3x3 matrix. Further, we can use to simplify equation 2 of (2.1):
and so if we adopt the convention that our timing is such that , we find:
(2.4) ,
where the index j=1,2, and/or 3, and hence not summed. The tensor is a volumetric, second order tensor, and if the determinant of it is not 0, it can be inverted in (2.4) to yield:
(2.5) ,
where again the summation over k is implied.
In summary, equation (2.5) relates the unknown slowness vector to the known time delays of the event, and the known location of the satellites, given by , and the volumetric tensor therein derived using equation (2.3).
In the next tow sections, we introduce uncertainty in the variables . This uncertainty is broken up into two parts: (1) The assumption we only know the relative delays between pairs of satellites, and (2) The assumption that even those relative delays have noise.
In this section we deal with the reality that we only know the delay in the plane wave’s phase as calculated between satellites:
. 

In this model, we only know , and want to apply equation (2.5) to find the slowness vector of the plane wave. First we note that
(3.2) 
, 

that is the time differences are antisymmetric. That being the case, apply equation (2.1) to and , summing over and , and using the convention :
(3.3) 
. 

Using as in equation (1.4), we find
(3.4) 
. 

As before, setting , we find
(3.5) 
. 

Isolating , we find
. 

Collecting the positive and negative terms of the left side shows:
, 

where the second term of the left hand side is 0 by equation (1.2), and the first term of the right hand side is simplified by definition (2.3). (Remember that since we used equation (1.2), we must always reorigin our vectors so that equation (1.2) works. Otherwise, the following will not work.) Using equation (3.7) in equation (3.6) yields
(3.8) 
, 

a relation between the known satellite positions, the slowness vector, and the time delays of the phase of the plane wave. We can then solve for the slowness vector as in the previous section, and find
, 

which may be written using equation (3.1) in the more computationally equivalent form
, 

where we adopt the convention that expresses the sum over the terms above the diagonal. Note that there are such entries. In the case of a tetrahedron, that is 6 terms. In this way, we have removed the obvious redundancies from equation (3.9).
Although equation (3.10) removes the redundancies of equation (3.9) by using equation (3.1), it is not preferred in the presence of noise. That is because if our estimate of is influenced by noise equation (3.1) is not always true. Thus, in our computations, we will be using equation (3.9).
Equation (3.9) gives us the slowness vector when we know the time delays of an event. Thus we can use that vector to recover the time delays. In this section we are running the process forward, i.e. assuming we know the slowness vector. We will use this technique to build a catalog of event types that we can then use to identify various superpositions of plane waves in the form of events.
Consider a slowness vector and a known array of satellites. We know from equation (1.1) that we can define an NxN time delay matrix T of the event as it propagates across the array of satellites:
. 

We also define its cousin, the space delay array
(4.2) 
. 

This is an array of vectors. (I think this is some kind of tensor, but I am not yet sure.)
This leads to the equation
. 

We will find the space delay array a computationally convenient device.
We then can use this time delay matrix to build the various patterns of events across the array.
Since the components of T are antisymmetric ( ), T itself is antisymmetric.
In practice, the actual time delays are calculated using semblance on windows. Recall the definition of Semblance:
(4.4) 


where f and g are the time series of the first and second satellite of a pair.
This formula is symmetric with respect to f and g. Thus we have:
(4.5) 
. 

This means that even in the presence of noise our estimate the elements of the time delay matrix T will be antisymmetric. If that is not so, there is an error in the software. Call this antisymmetry
The time delay matrix T is antisymmetric even in the presence of noise.
If two plane waves, with two different slowness vectors both cross the array, the resulting time delay matrix would not simple be the sum of the two events time delays, since the two events can add together in and out of phase. More on this will be presented later.
The measure of the arrival time of an event is a measure of the slowness vector by equation (3.9). We will construct several examples that demonstrate signatures of the arrival times.
We need a simple example that we can use for debugging the program that will implement this algorithm. In this simple case, the event, decomposed into plane waves all pointing in the same direction, is propagating parallel to the Zaxis on an array with four satellites. We will assume a regular tetrahedron of unit height.
Just for the record, we will derive the coordinates of the perfect tetrahedron. Assume the tetrahedron has sides of unit length. Then by symmetry arguments, we can assign the coordinates to the vertices of the tetrahedron. Since the distance between any two pairs of these points must be 1, we have the following equations:
and so we immediately have u=1/4. Next, some more algebra shows , and so using that in the first two equations, we get . Then we see that , and . The points of our tetrahedron are now . Since we wanted a unit height, we multiply all of these by , and then by setting
(5.1) 


we get
Satellite 
(x,y,z) 
1 

2 

3 

4 

This describes a tetrahedron in which satellites 2, 3, and 4 are coplanar in the z=0 plane, has a height of 1, and sides of length .
The space delay array associated with this set of satellites is:


Numerically, this is


In this case, the slowness vector is:
(5.4) 


and by (5.2) the matrix is:


where
(5.6) 
. 

This can be interpreted as follows: There is a time delay of between the event at satellite 1 and satellites 2, 3, and 4, but no time delays between any pair of satellites 2, 3, and 4. Thus, using a semblance as an estimator, we can determine from the various semblance peaks.
It is at this point we should now take into account the effect of noise.
Suppose the signal at satellite i is infected with a white stationary noise, that is the characteristics of the noise do not change with time. This does not mean two samples have the same noise, only that the noise two particular timespace points have similar properties. Using this assumption, we can make a statistical analysis of the various estimates. In particular, there will be a Gaussian uncertainty in the estimate of the elements of .[1] Assume the mean of the error is and the standard deviation is . Then each estimate of the events time would be a random variable of the form , where each is a different sample.
In general the errors induced by noise are symmetric, since the semblance is symmetric:
(6.1) 
. 

Thus, if the noise is temporal, and not spatial, we would have
(6.2) 


where:
(a) is the error induced by the noise at the time the event hits satellite 1,
(b) is the noise induced between the timings between satellite 1 and satellites 2, 3, and 4, and
(c) is the error induced by the noise between pairs of 2, 3, and 4.
The matrix thus has the “shape”
(6.3) 


Similarly, if the noise is only spatial, and not temporal, we would have:
(6.4) 


We now look at the case for a satellite array as in the previous section experiencing a horizontally incident plane wave. That is the wave’s vector is and so has a slowness vector of
(7.1) 
. 

In this case, using the table just prior to (5.2), we find that if the wave is from the right, and hits satellite 1 at time 0, we have, when
Satellite 
(x,y,z) 
Time Delay 
1 


2 


3 

0 
4 


Thus, setting
(7.2) 


the time delay matrix is of the form
. 

If instead we had the plane wave coming in along the yaxis, we would have:
(7.4) 
, 

Satellite 
(x,y,z) 
Time Delay 
1 


2 

0 
3 


4 


And setting
(7.5) 


we get for our delay matrix
(7.6) 
. 

This may be a recognizable pattern.
We now have a method to find the slowness vector given the relative time delays of an event. Further, we have started building a catalog of patterns of the time delay matrix which may help in recognizing events.
The next three action items are:
(1) Confirm the efficacy of equation (3.9) using my test samples, and also using John’s data. I have a program that can do this. I need to recast that program so that it will produce a time delay matrix as output.
(2) I start looking at a superposition of plane waves and/or events that propagate from more than one direction. This will be an embellishment of John’s idea.
(3) I look at the effect of noise in the calculations of the relative delays. I have started this in the section entitled “The Effect of Noise in a Vertically Incident Plane Wave.”
Examples
Horizontally Incident Plane Waves....................... 10
Vertically Incident Plane Wave.............................. 8
noise................................................................... 9, 12
plane wave pahse delay............................................. 4
semblance
definition............................................................... 6
slowness vector.................................. 2, 12. See taup
Estimation............................................................. 4
superposition of plane wav....................................... 12
taup......................................................................... 1
time delay matrix..................................................... 12
Time Delay Matrix..................................................... 6
noise..................................................................... 9
Properties............................................................. 6
volumetric tensor....................................................... 3
white noise................................................................ 9
Cases
Harvey, "Spatial Gradients and the Volumetric Tensor", Chapter 12 of the ISSI Scientific Report Analysis of MultiSpacecraft Data, SR001 of The International Space Institure, 1998.................................................................... 1
[1] We reserve for now the argument that the noise in the estimates is Gaussian. The type of statistical distribution is not important to our analysis, since we only use the first and second moments of the statistics.