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In the framework of the ray method one looks for a solution of an equation using
the following representation of a wave field:

| |
(28) |

where are complex vector amplitudes:
, *f*_{n}(*t*) are ``complex
forms'' which satisfy the relation
| |
(29) |

If we put expression (28) into the equation

| |
(30) |

(it can be, for instance, Lame's equation; see Chapter 3, equation (24)),
we derive, after some manipulating, that function satisfy
eiconal's equation
| |
(31) |

(*Q* stands for *P* or *S*), and complex amplitudes satisfy the recurrent
relations:
| |
(32) |

where is the same operator as in equation (30), is the
same matrice that is in Chapter 3 (see equation (23)), and is a
first order differential operator:
| |
(33) |

We see that ``ray waves'' propagate with the same eiconal as discontinuities.
It is not accidental because standard discontinuities satisfy the same
relation (29) as ``complex forms'' *f*_{n}(*t*).

Moreover, if a source function has a beginning at the *t*=0 (as it usually
occurs), then the solution of the equation (30) will
inevitably have discontinuities that propagate with eiconal's satisfying
equation
(31). This means that all complex forms *f*_{n}(*t*) must have a
discontinuity at the point *t*=0. Therefore, despite the choice of a complex
form *f*_{0}(*t*), any ``ray series'' descriptions of the wave field are equivalent
in the sense that was introduced earlier.

Of course, a choice of *f*_{0}(*t*) influences accuracy of wave field approximation
behind the wave front , but in any case, convergence of the
series (28) can be guaranteed only in a neighborhood of wave front
(Babich, 1961).

If *f*_{0}(*t*)=*R*_{q<<933>>0}(*t*) then series (28) represents ordinary
Tailor's expansion in a neighborhood of the front (in the case of the
``weak discontinuity'', *q*_{0}=2).

In the general case

where *q*_{0} can vary depending on the location of the point .We observe this dependence in a neighborhood of caustics and other special
domains. But in this case we have to learn how to manipulate with discontinuities
of noninteger order.
We see that usage of discontinuities is absolutely equivalent to ray series approach.
Both approaches give the same geometrical law of wave propagation and the same
behavior of wave amplitudes during propagation.

If the senior discontinuity has the order *q*_{0}, then the equivalence of the
order *q*_{0} corresponds to the zero-approximation of ray series; equivalence
of the order *q*_{0}+1 corresponds to the first-order ray-series approximation
and so on.

Usually the value of *q*_{0} does not matter. So it will be convenient to write
, if there is such an order *q*_{0}
that for all , but not for *r*<0. With this agreement,
zero-approximation corresponds to (0)-equivalence, first-order ray-series
approximation corresponds to (1)-equivalence, and so on. Usually we
shall omit symbol (0) (so ).

** Next:** Zero-approximation.
** Up:** 4: CONNECTION WITH RAY
** Previous:** 4: CONNECTION WITH RAY
Stanford Exploration Project

1/13/1998