As is well known, any solution of the equation (30) in case of point source can be expressed in the form
(39) |
Fundamental solution is a response of a medium to the -function as the source. It is easily understood that (in case of piecewise smooth media with restricted numbers of interfaces) consists of some singularities (discontinuities) that propagates with correspondence to eiconal's equation (31) and relations (32) and some very smooth field .
Let us consider a very simple example: a fundamental solution that corresponds to a point vertical force in a homogeneous medium (Figure ) is
(40) |
Let us perform the convolution in equation (39), bearing in mind that
f(t)*R_{q}(t) = I^{q} f(t) = f_{q}(t) | (41) |
Let be the observed field ( or ) that is given on the surface of observation and is a linear operator that transforms the function into the field in the adjacent domain
The fields and can be scalar or vectorial. The coordinates (x,y) can be curvilinear and the surface of observation can also be curved.The action of the operator depends on the choice of the function which in turn is a parameter in a fixed eikonal equation
(42) |
We don't demand that the eikonal equation (42) is a characteristic equation of some hyperbolic differential operator. It is given, that's all!
Let us suppose that the field contains a discontinuity (of some order q) with a travel time curve .It means that in a neighborhood of the surface
Generally speaking then, the wave field will contain one or several discontinuities (with orders not necessarily equal to q) and at least one of them will coincide with at .If the position of this discontinuity is described by a function , then
(43) |
We shall call the operator a kinematically-equivalent operator of wave field continuation (on short K-operator, KO) if the field necessarily contains at least one discontinuity with eikonal , that satisfies equations (42) and (43) simultaneously.
The notion of K-operator may be expanded to the case when both fields and are vectorial fields. In that case we have two equations (isotropic case) or three equations (anisotropic case):
(44) |
The question of existence of a K-operator corresponding to a given equation (42) can be simply answered (in a positive sense) if equation (42) is a characteristic equation for some equation in partial derivatives:
(45) |
Let us consider the boundary value problem for equation (45) with condition
(46) |
(47) |
It is obvious that the solution contains at least one discontinuity which at coincides with and propagates with the eikonal satisfying equation (42).
Later we shall show that KO exists for any equation (42) if there is a solution for the Cauchy problem for equation (42) with the initial condition (43). As a matter of fact there is a whole family of K-operators that corresponds to the particular eikonal equation (42).
K-equivalence is a notion which is much wider than q-equivalence or (k)-equivalence. For instance, if belong to the same family , then it is not necessary that
Fields and can have different orders and amplitudes of discontinuities (and even their number) which satisfy the equations (42) and (43).Let us consider K-operators for the classical eikonal equation
(48) |
(49) |
(50) |
In order to get a unique solution of the equation (50) it is necessary (and sufficient) to have a definite value in a starting point .We can express the vector in the form of a sum
where is a normal to at the point , and (instead of we shall write ). The value can be expressed through the derivatives of the function . If is a horizontal plane and x and y are rectangular coordinates, then where means gradient in the plane and and are unit vectors.Let us consider the case when is a curved surface and x and y are a curvilinear system of coordinates. We propose that the axis z coincides with the direction of a vector . Let us determine a tangent plane P to in the point and let's introduce the rectangular coordinates and in P, then
where and This construction we will also denote as . So, In accordance with eikonal equation (48) (), then andWe have obtained that
The sign `+' determines the solution which will be called forward eikonal continuation . The sign ``-'' determines reverse eikonal continuation , (the relationship of these continuations is shown in Figure ).It is easily understood that where is the time of propagation from to .We shall call K-operator as the K-operator of forward (reverse) wave field continuation and denote it as (similarly ) if in the neighbourhood of the field
(similarly ) contains a discontinuity with eikonal ( ) and does not contain a discontinuity with the opposite eikonal () .Each operator can be used for determination of the operator
(51) |
There are, of course, such K-operators that contain (in a neighbourhood of ) discontinuities with both eikonals and (mixed type operators). This classification of K-operators may be expanded for many other eikonal equations.
The classes of this classification are still very wide. If operators and belong to the same class (forward, reverse or mixed) of K-operators and for some k
(52) |