  Considering 3D heterogeneous anisotropic media, in Part III of this study we
obtain the finite-element solution for the stationary ray path between two
fixed endpoints using the finite element approach. It involves computation of
the global (all-node) traveltime gradient vector and Hessian matrix, with
respect to the nodal location and direction components. The global traveltime
Hessian of the resolved stationary ray also plays an important role for
obtaining the dynamic characteristics along the ray. It is a band matrix that
includes the spatial, directional and mixed second derivatives of the
traveltime at all nodes and nodal pairs. In Parts V, VI and VII of this study
we explicitly use the global traveltime Hessian for solving the dynamic ray
tracing equation, to obtain the geometric spreading at each point along the
ray. Moreover, the dynamic ray tracing makes it possible to identify/classify
caustics along the ray. In this part (Part IV) we propose an original two-stage
approach for computing the relative geometric spreading of the entire ray path
without explicitly performing the dynamic ray tracing. The first stage consists
of an efficient algorithm for reducing the already computed global traveltime
Hessian into a endpoint spatial traveltime Hessian. The second stage involves
application of a known workflow for computing the geometrical spreading using
the endpoint traveltime Hessian. Note that the method proposed in this part
(Part IV) does not deliver information about caustics, nor the geometric
spreading at intermediate points of the stationary ray path. Throughout the
examples presented in Part VII we demonstrate the accuracy of the method
proposed in this part over a set of isotropic and anisotropic examples by
comparing the geometric spreading computed from this method based on the
traveltime Hessian reduction with the one computed from the dynamic ray
tracing.
