MATLAB (Proprietary)
MATLAB (matrix laboratory) is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, Java, and Fortran.

GNU Octave (MatLab Alternative) (GPL)
GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface (CLI) for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB. It may also be used as a batch-oriented language. As part of the GNU Project, it is free software under the terms of the GNU General Public License (GPL).

Project SEIZMO (GPL)
Project SEIZMO is a Matlab and GNU Octave based toolbox encompassing a collection of over 700 functions that provide a framework for seismic data preparation, quality control, and analysis akin to that of Seismic Analysis Code with a focus on passive source seismology at regional and global scales (where earthquakes and ambient noise are the sources). There are numerous functions for reading/writing standard seismic data formats, displaying and editing metadata, plotting seismograms, creating animations, data processing, and interactive analysis. Data processing capabilities include correlation, convolution, deconvolution, detrending, differentiation, integration, interpolation, resampling, filtering, merging, response transferring, rotation, stacking, spectral analysis, tapering, and windowing. The toolbox contains collections of functions for arrival time determination and quality control with cross correlation and cluster analysis, Rayleigh wave two plane-wave analysis, seismic ambient noise processing, and frequency-wavenumber analysis. SEIZMO utilizes direct access to the TauP toolkit to administer predicted arrival times, raypaths, pierce points, and travel time curves for several widely recognized 1D seismic earth models. Mapping in SEIZMO draws on the M_Map toolbox. The seismology toolbox also incorporates several 3D mantle models, a catalog of moment tensors from the Global CMT project, and a database of instrument responses available through IRIS. There are functions to aid in rapid prototyping and customization for new functions and documentation for every function is accessible through the inline help system.

EDT: ElastoDynamics Toolbox for MATLAB (Proprietary)
The ElastoDynamics Toolbox (EDT) provides an extensive set of MATLAB functions to model wave propagation in layered media. It is based on the direct stiffness method and the thin layer method, presented by Kausel and Ro√ęsset. The toolbox is developed at KU Leuven, where it is used both for educational purposes and in research projects. The current version 2.2 (build 20) of the ElastoDynamics Toolbox has been released in August 2010.

The GISMO Suite (GPL)
The GISMO suite is a collection of Matlab toolboxes for seismic waveform analysis built on a common platform. GISMO stands for GI Seismology Matlab Objects. The GISMO suite consist of two types of tools: (1) a core codebase of stable toolboxes; and (2) an ever-evolving set of contributed codes developed from these core products. The core products include the Waveform Suite and Correlation Toolbox.

The Waveform Suite: A Robust Platform for Manipulating Waveforms in MATLAB

Read and Write miniSEED File

List of Free of mostly Numerical Geophysics Software