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First Order Multiples - Analytics

To identify different multiples and to understand their behavior, I analytically modeled the ray paths for all possible first-order multiple events for the given synthetic model. For simplicity of referencing to these events I use the nomenclature given in Table 1.
Table 1. Nomenclature for different first order multiples.
 Type Ray-path Name 1 Both the bounces on the flat part FF 2 First bounce on the flat part and second on the dipping FD 3 First bounce on the dipping part and second on the flat DF 4 Both the bounces on the dipping part DD

shot5000
Figure 3
(a) A shot record corresponding to a shot located at a surface location of 5000. (b) Shot record with first order multiples modeled analytically overlaid on them

Equations 4 and 5 give the travel time and offset as a function of (take-off angle) for the event FF,
 (4)

 (5)
In the equations above, d is the depth of the flat reflector, V is the velocity, and h and t are offset and travel time respectively. There are different conditions and constraints on the value of depending on the shot location, which determines the range of take-off angles where these events take place and get recorded. Equations 6 and 7 give the expressions for travel time and offsets for the event FD.
 (6) (7)
In these equations, xd is the horizontal location where the dipping reflector meets the flat layer, xs is the source location and is the dip of the dipping reflector. There are two other possibilities for the first-order multiples: DF and DD. Travel time and offsets are given in equations 8 and 9 for the first category and equations 10 and 11 for the second category.
 (8)
 (9)
 (10)
 (11)
Once again, all the above equations are functions of the parameter and are valid only for a range of values corresponding to angles at which the event actually takes place. On Figure (b) are overlaid the first-order multiple events modeled using the above equations.

Next: Surface Related Multiple Prediction Up: Synthetic Data Previous: Synthetic Data
Stanford Exploration Project
5/6/2007