Conservation laws for a time field (the eikonal equation solutions) in kinematic seismics (geometric optics)



A number of non-classical formulas of vector analysis are represented as differential identities that, on the one hand, relate the modulus to the direction of an arbitrary smooth vector field in a three- and two-dimensions. On the other hand, these formulas, in a sense, separate these characteristics. In particular, for an arbitrary smooth plane vector field υ (x, y) = |υ|τ with the module |υ| and a direction τ, the conservation law in two equivalent forms is presented: in terms of the direction τ and in terms of the vector field υ. It is established that each of these forms is equivalent to a conservation law for vector lines of a field υ, expressed in terms of the curvature vector of vector lines Lτ of the field υ and of the curvature vector of orthogonal to them curves. Application of these formulas to the solutions τ of the eikonal equation has allowed us to discover a number of the new identities relating the time field τ, the refractive index n(x, y) and the ray slope (direction) angle α. In particular, differential conservation laws for the time field τ (the eikonal equation solutions) in the kinematic seismics (geometrical optics) were discovered. The geometric interpretation of these conservation laws from the point of view of the differential geometry in terms of curvature vectors of rays and fronts of waves corresponding to the time field τ is obtained.