PDEs
A partial differential equation (or briefly a PDE) is a mathematical equation that involves two or more independent variables, an unknown function (dependent on those variables), and partial derivatives of the unknown function with respect to the independent variables. The order of a partial differential equation is the order of the highest derivative involved. A solution (or a particular solution) to a partial differential equation is a function that solves the equation or, in other words, turns it into an identity when substituted into the equation. A solution is called general if it contains all particular solutions of the equation concerned.
One of my major reasons for developing PDEs is to describe certain aspects of seismic wave propagation for exploration applications and to approximate more complicated formulations. A PDE description allows for complex media treatment as it relates the local behaviour of a physical phenomenon to infinitesimal changes in one or more of the medium or experimental parameters. In the following I will start showing examples of new PDEs and what they can do:
Let me start with something I managed to derive in the late 90's, the acoustic wave equation for anisotropic media. (See details). This PDE or variations of it is currently widely used in the Industry.
The orthrohombic medium version is a sixth-order linear PDE, which has a small region of stability (See Details), presented in 2003.
Another development is the Linearized eikonal PDE for anisotropic media with official journal publication 2003 (See Details).
An approximation of anisotropy continuation is described by a simple linear PDE (See Details), this is from the late 90's.
A new wave equation PDE for anisotropic media (including isotropic) using the TAU domain instead of depth. The equations are slightly more complicated than depth counterpart and they have the potential to provide Finite difference speed ups in their implementation (Read more) from 2001.
The virtual source perturbation for the eikonal equation is given by a linear PDE (See Details), this is 2009.
The virtual source perturbation for the wave equation is given by a wave equation like PDE with a modified source function (See Details), this is from 2009.
Linear first order PDE's to calculate perturbations in traveltime as s function of changes in the anisotropy parameters for VTI, TTI, and HTI media (see Details).
The prestack exploding reflector PDE is potential fourth order in time but has an essential singularity for horizontally traveling waves (See details).
A lot more is on the way...........
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