Getting Started
===============

Python Packages
---------------

This module requires the python packages: multiprocessing, matplotlib, numpy, math and numba which can be installed using `pip <https://pypi.org/project/pip/>`_ from a terminal using

.. code-block:: console

   pip install multiprocessing matplotlib numpy math numba


Importing Codes
---------------

These codes can be run using the :class:`~Anis_TTF_rays.ALI_FMM` class. The other module functions are used in this class and do not need calling directly. The class can be input using the command

.. code-block:: python

   from Anis_TTF_rays import ALI_FMM

with Anis_TTF_rays.py in the same directory.

.. note::

   The folder __pycache__ stores some compiled functions. Functions requiring compiling will be slow to run initially if not already compiled.

Creating an instance of the ALI_FMM class
-----------------------------------------

An instance of the class can be created using

.. code-block:: python
  
   Forward_Model = ALI_FMM(veln, velpn, vel_map, scx, scz)

The name of this instance can be whatever you want. See :func:`~Anis_TTF_rays.ALI_FMM.__init__` command for additional arguments.

Class functions can then be called using the instance variable. For example

.. code-block:: python

   Forward_Model.plot_phase(1)

This will plot the phase velocity of material with index 1 which is stored within ``Forward_Model``. All functions must be called from an instance of the class except for :func:`~Anis_TTF_rays.ALI_FMM.generate_group_vel` and :func:`~Anis_TTF_rays.ALI_FMM.generate_phase_vel` which can also be called using

.. code-block:: python

   ALI_FMM.generate_group_vel(None, c_22, c_23, c_33, c_44, density)

.. code-block:: python

   ALI_FMM.generate_phase_vel(None, c_22, c_23, c_33, c_44, density)

Defining Materials
------------------

There are two ways in which materials can be defined for these codes. The first way is by using a table of velocity curves which can be input when the class instance is initialised. These variables can be accessed using ``Forward_Model.group_vel`` and ``Forward_Model.phase_vel``. The group and phase velocities must both be input as a table with the first column being the angle from 0 to 360 in 1 deg increments, and other columns being the velocities for each material. If no argument is given, an isotropic material with a velocity of 1m/s is used. This can be used for all isotropic materials as the variable vel_map can be used to scale all velocities.

.. collapse:: Example velocity table

   |
   **Velocity table with a single material**

   +------+-----------+
   |  0.0 | 5085.0026 |
   +------+-----------+
   |  1.0 | 5088.2812 |
   +------+-----------+
   |  2.0 | 5097.9530 |
   +------+-----------+
   |  3.0 | 5113.5586 |
   +------+-----------+
   |  4.0 | 5134.4274 |
   +------+-----------+
   |  5.0 | 5159.7804 |
   +------+-----------+
   |  6.0 | 5188.8210 |
   +------+-----------+
   |  7.0 | 5220.7981 |
   +------+-----------+
   |  8.0 | 5255.0399 |
   +------+-----------+
   |  9.0 | 5290.9659 |
   +------+-----------+
   | 10.0 | 5328.0849 |
   +------+-----------+
   | 11.0 | 5365.9866 |
   +------+-----------+
   | 12.0 | 5404.3296 |
   +------+-----------+
   | 13.0 | 5442.8306 |
   +------+-----------+
   | 14.0 | 5481.2538 |
   +------+-----------+
   | 15.0 | 5519.4023 |
   +------+-----------+
   | 16.0 | 5557.1106 |
   +------+-----------+
   | 17.0 | 5594.2389 |
   +------+-----------+
   | 18.0 | 5630.6681 |
   +------+-----------+
   | 19.0 | 5666.2964 |
   +------+-----------+
   | 20.0 | 5701.0359 |
   +------+-----------+
   | 21.0 | 5734.8104 |
   +------+-----------+
   | 22.0 | 5767.5531 |
   +------+-----------+
   | 23.0 | 5799.2058 |
   +------+-----------+
   | 24.0 | 5829.7167 |
   +------+-----------+
   | 25.0 | 5859.0401 |
   +------+-----------+
   | 26.0 | 5887.1352 |
   +------+-----------+
   | 27.0 | 5913.9657 |
   +------+-----------+
   | 28.0 | 5939.4989 |
   +------+-----------+
   | 29.0 | 5963.7055 |
   +------+-----------+
   | 30.0 | 5986.5592 |
   +------+-----------+
   | 31.0 | 6008.0363 |
   +------+-----------+
   | 32.0 | 6028.1154 |
   +------+-----------+
   | 33.0 | 6046.7773 |
   +------+-----------+
   | 34.0 | 6064.0048 |
   +------+-----------+
   | 35.0 | 6079.7826 |
   +------+-----------+
   | 36.0 | 6094.0969 |
   +------+-----------+
   | 37.0 | 6106.9356 |
   +------+-----------+
   | 38.0 | 6118.2881 |
   +------+-----------+
   | 39.0 | 6128.1450 |
   +------+-----------+
   | 40.0 | 6136.4986 |
   +------+-----------+
   | 41.0 | 6143.3422 |
   +------+-----------+
   | 42.0 | 6148.6704 |
   +------+-----------+
   | 43.0 | 6152.4793 |
   +------+-----------+
   | 44.0 | 6154.7657 |
   +------+-----------+
   | 45.0 | 6155.5281 |
   +------+-----------+
   | 46.0 | 6154.7657 |
   +------+-----------+
   | 47.0 | 6152.4793 |
   +------+-----------+
   | 48.0 | 6148.6704 |
   +------+-----------+
   | 49.0 | 6143.3422 |
   +------+-----------+
   | 50.0 | 6136.4986 |
   +------+-----------+
   | 51.0 | 6128.1450 |
   +------+-----------+
   | 52.0 | 6118.2881 |
   +------+-----------+
   | 53.0 | 6106.9356 |
   +------+-----------+
   | 54.0 | 6094.0969 |
   +------+-----------+
   | 55.0 | 6079.7826 |
   +------+-----------+
   | 56.0 | 6064.0048 |
   +------+-----------+
   | 57.0 | 6046.7773 |
   +------+-----------+
   | 58.0 | 6028.1154 |
   +------+-----------+
   | 59.0 | 6008.0363 |
   +------+-----------+
   | 60.0 | 5986.5592 |
   +------+-----------+
   | 61.0 | 5963.7055 |
   +------+-----------+
   | 62.0 | 5939.4989 |
   +------+-----------+
   | 63.0 | 5913.9657 |
   +------+-----------+
   | 64.0 | 5887.1352 |
   +------+-----------+
   | 65.0 | 5859.0401 |
   +------+-----------+
   | 66.0 | 5829.7167 |
   +------+-----------+
   | 67.0 | 5799.2058 |
   +------+-----------+
   | 68.0 | 5767.5531 |
   +------+-----------+
   | 69.0 | 5734.8104 |
   +------+-----------+
   | 70.0 | 5701.0359 |
   +------+-----------+
   | 71.0 | 5666.2964 |
   +------+-----------+
   | 72.0 | 5630.6681 |
   +------+-----------+
   | 73.0 | 5594.2389 |
   +------+-----------+
   | 74.0 | 5557.1106 |
   +------+-----------+
   | 75.0 | 5519.4023 |
   +------+-----------+
   | 76.0 | 5481.2538 |
   +------+-----------+
   | 77.0 | 5442.8306 |
   +------+-----------+
   | 78.0 | 5404.3296 |
   +------+-----------+
   | 79.0 | 5365.9866 |
   +------+-----------+
   | 80.0 | 5328.0849 |
   +------+-----------+
   | 81.0 | 5290.9659 |
   +------+-----------+
   | 82.0 | 5255.0399 |
   +------+-----------+
   | 83.0 | 5220.7981 |
   +------+-----------+
   | 84.0 | 5188.8210 |
   +------+-----------+
   | 85.0 | 5159.7804 |
   +------+-----------+
   | 86.0 | 5134.4274 |
   +------+-----------+
   | 87.0 | 5113.5586 |
   +------+-----------+
   | 88.0 | 5097.9530 |
   +------+-----------+
   | 89.0 | 5088.2812 |
   +------+-----------+
   | 90.0 | 5085.0026 |
   +------+-----------+
   | 91.0 | 5088.2812 |
   +------+-----------+
   | 92.0 | 5097.9530 |
   +------+-----------+
   | 93.0 | 5113.5586 |
   +------+-----------+
   | 94.0 | 5134.4274 |
   +------+-----------+
   | 95.0 | 5159.7804 |
   +------+-----------+
   | 96.0 | 5188.8210 |
   +------+-----------+
   | 97.0 | 5220.7981 |
   +------+-----------+
   | 98.0 | 5255.0399 |
   +------+-----------+
   | 99.0 | 5290.9659 |
   +------+-----------+
   |100.0 | 5328.0849 |
   +------+-----------+
   |101.0 | 5365.9866 |
   +------+-----------+
   |102.0 | 5404.3296 |
   +------+-----------+
   |103.0 | 5442.8306 |
   +------+-----------+
   |104.0 | 5481.2538 |
   +------+-----------+
   |105.0 | 5519.4023 |
   +------+-----------+
   |106.0 | 5557.1106 |
   +------+-----------+
   |107.0 | 5594.2389 |
   +------+-----------+
   |108.0 | 5630.6681 |
   +------+-----------+
   |109.0 | 5666.2964 |
   +------+-----------+
   |110.0 | 5701.0359 |
   +------+-----------+
   |111.0 | 5734.8104 |
   +------+-----------+
   |112.0 | 5767.5531 |
   +------+-----------+
   |113.0 | 5799.2058 |
   +------+-----------+
   |114.0 | 5829.7167 |
   +------+-----------+
   |115.0 | 5859.0401 |
   +------+-----------+
   |116.0 | 5887.1352 |
   +------+-----------+
   |117.0 | 5913.9657 |
   +------+-----------+
   |118.0 | 5939.4989 |
   +------+-----------+
   |119.0 | 5963.7055 |
   +------+-----------+
   |120.0 | 5986.5592 |
   +------+-----------+
   |121.0 | 6008.0363 |
   +------+-----------+
   |122.0 | 6028.1154 |
   +------+-----------+
   |123.0 | 6046.7773 |
   +------+-----------+
   |124.0 | 6064.0048 |
   +------+-----------+
   |125.0 | 6079.7826 |
   +------+-----------+
   |126.0 | 6094.0969 |
   +------+-----------+
   |127.0 | 6106.9356 |
   +------+-----------+
   |128.0 | 6118.2881 |
   +------+-----------+
   |129.0 | 6128.1450 |
   +------+-----------+
   |130.0 | 6136.4986 |
   +------+-----------+
   |131.0 | 6143.3422 |
   +------+-----------+
   |132.0 | 6148.6704 |
   +------+-----------+
   |133.0 | 6152.4793 |
   +------+-----------+
   |134.0 | 6154.7657 |
   +------+-----------+
   |135.0 | 6155.5281 |
   +------+-----------+
   |136.0 | 6154.7657 |
   +------+-----------+
   |137.0 | 6152.4793 |
   +------+-----------+
   |138.0 | 6148.6704 |
   +------+-----------+
   |139.0 | 6143.3422 |
   +------+-----------+
   |140.0 | 6136.4986 |
   +------+-----------+
   |141.0 | 6128.1450 |
   +------+-----------+
   |142.0 | 6118.2881 |
   +------+-----------+
   |143.0 | 6106.9356 |
   +------+-----------+
   |144.0 | 6094.0969 |
   +------+-----------+
   |145.0 | 6079.7826 |
   +------+-----------+
   |146.0 | 6064.0048 |
   +------+-----------+
   |147.0 | 6046.7773 |
   +------+-----------+
   |148.0 | 6028.1154 |
   +------+-----------+
   |149.0 | 6008.0363 |
   +------+-----------+
   |150.0 | 5986.5592 |
   +------+-----------+
   |151.0 | 5963.7055 |
   +------+-----------+
   |152.0 | 5939.4989 |
   +------+-----------+
   |153.0 | 5913.9657 |
   +------+-----------+
   |154.0 | 5887.1352 |
   +------+-----------+
   |155.0 | 5859.0401 |
   +------+-----------+
   |156.0 | 5829.7167 |
   +------+-----------+
   |157.0 | 5799.2058 |
   +------+-----------+
   |158.0 | 5767.5531 |
   +------+-----------+
   |159.0 | 5734.8104 |
   +------+-----------+
   |160.0 | 5701.0359 |
   +------+-----------+
   |161.0 | 5666.2964 |
   +------+-----------+
   |162.0 | 5630.6681 |
   +------+-----------+
   |163.0 | 5594.2389 |
   +------+-----------+
   |164.0 | 5557.1106 |
   +------+-----------+
   |165.0 | 5519.4023 |
   +------+-----------+
   |166.0 | 5481.2538 |
   +------+-----------+
   |167.0 | 5442.8306 |
   +------+-----------+
   |168.0 | 5404.3296 |
   +------+-----------+
   |169.0 | 5365.9866 |
   +------+-----------+
   |170.0 | 5328.0849 |
   +------+-----------+
   |171.0 | 5290.9659 |
   +------+-----------+
   |172.0 | 5255.0399 |
   +------+-----------+
   |173.0 | 5220.7981 |
   +------+-----------+
   |174.0 | 5188.8210 |
   +------+-----------+
   |175.0 | 5159.7804 |
   +------+-----------+
   |176.0 | 5134.4274 |
   +------+-----------+
   |177.0 | 5113.5586 |
   +------+-----------+
   |178.0 | 5097.9530 |
   +------+-----------+
   |179.0 | 5088.2812 |
   +------+-----------+
   |180.0 | 5085.0026 |
   +------+-----------+
   |181.0 | 5088.2812 |
   +------+-----------+
   |182.0 | 5097.9530 |
   +------+-----------+
   |183.0 | 5113.5586 |
   +------+-----------+
   |184.0 | 5134.4274 |
   +------+-----------+
   |185.0 | 5159.7804 |
   +------+-----------+
   |186.0 | 5188.8210 |
   +------+-----------+
   |187.0 | 5220.7981 |
   +------+-----------+
   |188.0 | 5255.0399 |
   +------+-----------+
   |189.0 | 5290.9659 |
   +------+-----------+
   |190.0 | 5328.0849 |
   +------+-----------+
   |191.0 | 5365.9866 |
   +------+-----------+
   |192.0 | 5404.3296 |
   +------+-----------+
   |193.0 | 5442.8306 |
   +------+-----------+
   |194.0 | 5481.2538 |
   +------+-----------+
   |195.0 | 5519.4023 |
   +------+-----------+
   |196.0 | 5557.1106 |
   +------+-----------+
   |197.0 | 5594.2389 |
   +------+-----------+
   |198.0 | 5630.6681 |
   +------+-----------+
   |199.0 | 5666.2964 |
   +------+-----------+
   |200.0 | 5701.0359 |
   +------+-----------+
   |201.0 | 5734.8104 |
   +------+-----------+
   |202.0 | 5767.5531 |
   +------+-----------+
   |203.0 | 5799.2058 |
   +------+-----------+
   |204.0 | 5829.7167 |
   +------+-----------+
   |205.0 | 5859.0401 |
   +------+-----------+
   |206.0 | 5887.1352 |
   +------+-----------+
   |207.0 | 5913.9657 |
   +------+-----------+
   |208.0 | 5939.4989 |
   +------+-----------+
   |209.0 | 5963.7055 |
   +------+-----------+
   |210.0 | 5986.5592 |
   +------+-----------+
   |211.0 | 6008.0363 |
   +------+-----------+
   |212.0 | 6028.1154 |
   +------+-----------+
   |213.0 | 6046.7773 |
   +------+-----------+
   |214.0 | 6064.0048 |
   +------+-----------+
   |215.0 | 6079.7826 |
   +------+-----------+
   |216.0 | 6094.0969 |
   +------+-----------+
   |217.0 | 6106.9356 |
   +------+-----------+
   |218.0 | 6118.2881 |
   +------+-----------+
   |219.0 | 6128.1450 |
   +------+-----------+
   |220.0 | 6136.4986 |
   +------+-----------+
   |221.0 | 6143.3422 |
   +------+-----------+
   |222.0 | 6148.6704 |
   +------+-----------+
   |223.0 | 6152.4793 |
   +------+-----------+
   |224.0 | 6154.7657 |
   +------+-----------+
   |225.0 | 6155.5281 |
   +------+-----------+
   |226.0 | 6154.7657 |
   +------+-----------+
   |227.0 | 6152.4793 |
   +------+-----------+
   |228.0 | 6148.6704 |
   +------+-----------+
   |229.0 | 6143.3422 |
   +------+-----------+
   |230.0 | 6136.4986 |
   +------+-----------+
   |231.0 | 6128.1450 |
   +------+-----------+
   |232.0 | 6118.2881 |
   +------+-----------+
   |233.0 | 6106.9356 |
   +------+-----------+
   |234.0 | 6094.0969 |
   +------+-----------+
   |235.0 | 6079.7826 |
   +------+-----------+
   |236.0 | 6064.0048 |
   +------+-----------+
   |237.0 | 6046.7773 |
   +------+-----------+
   |238.0 | 6028.1154 |
   +------+-----------+
   |239.0 | 6008.0363 |
   +------+-----------+
   |240.0 | 5986.5592 |
   +------+-----------+
   |241.0 | 5963.7055 |
   +------+-----------+
   |242.0 | 5939.4989 |
   +------+-----------+
   |243.0 | 5913.9657 |
   +------+-----------+
   |244.0 | 5887.1352 |
   +------+-----------+
   |245.0 | 5859.0401 |
   +------+-----------+
   |246.0 | 5829.7167 |
   +------+-----------+
   |247.0 | 5799.2058 |
   +------+-----------+
   |248.0 | 5767.5531 |
   +------+-----------+
   |249.0 | 5734.8104 |
   +------+-----------+
   |250.0 | 5701.0359 |
   +------+-----------+
   |251.0 | 5666.2964 |
   +------+-----------+
   |252.0 | 5630.6681 |
   +------+-----------+
   |253.0 | 5594.2389 |
   +------+-----------+
   |254.0 | 5557.1106 |
   +------+-----------+
   |255.0 | 5519.4023 |
   +------+-----------+
   |256.0 | 5481.2538 |
   +------+-----------+
   |257.0 | 5442.8306 |
   +------+-----------+
   |258.0 | 5404.3296 |
   +------+-----------+
   |259.0 | 5365.9866 |
   +------+-----------+
   |260.0 | 5328.0849 |
   +------+-----------+
   |261.0 | 5290.9659 |
   +------+-----------+
   |262.0 | 5255.0399 |
   +------+-----------+
   |263.0 | 5220.7981 |
   +------+-----------+
   |264.0 | 5188.8210 |
   +------+-----------+
   |265.0 | 5159.7804 |
   +------+-----------+
   |266.0 | 5134.4274 |
   +------+-----------+
   |267.0 | 5113.5586 |
   +------+-----------+
   |268.0 | 5097.9530 |
   +------+-----------+
   |269.0 | 5088.2812 |
   +------+-----------+
   |270.0 | 5085.0026 |
   +------+-----------+
   |271.0 | 5088.2812 |
   +------+-----------+
   |272.0 | 5097.9530 |
   +------+-----------+
   |273.0 | 5113.5586 |
   +------+-----------+
   |274.0 | 5134.4274 |
   +------+-----------+
   |275.0 | 5159.7804 |
   +------+-----------+
   |276.0 | 5188.8210 |
   +------+-----------+
   |277.0 | 5220.7981 |
   +------+-----------+
   |278.0 | 5255.0399 |
   +------+-----------+
   |279.0 | 5290.9659 |
   +------+-----------+
   |280.0 | 5328.0849 |
   +------+-----------+
   |281.0 | 5365.9866 |
   +------+-----------+
   |282.0 | 5404.3296 |
   +------+-----------+
   |283.0 | 5442.8306 |
   +------+-----------+
   |284.0 | 5481.2538 |
   +------+-----------+
   |285.0 | 5519.4023 |
   +------+-----------+
   |286.0 | 5557.1106 |
   +------+-----------+
   |287.0 | 5594.2389 |
   +------+-----------+
   |288.0 | 5630.6681 |
   +------+-----------+
   |289.0 | 5666.2964 |
   +------+-----------+
   |290.0 | 5701.0359 |
   +------+-----------+
   |291.0 | 5734.8104 |
   +------+-----------+
   |292.0 | 5767.5531 |
   +------+-----------+
   |293.0 | 5799.2058 |
   +------+-----------+
   |294.0 | 5829.7167 |
   +------+-----------+
   |295.0 | 5859.0401 |
   +------+-----------+
   |296.0 | 5887.1352 |
   +------+-----------+
   |297.0 | 5913.9657 |
   +------+-----------+
   |298.0 | 5939.4989 |
   +------+-----------+
   |299.0 | 5963.7055 |
   +------+-----------+
   |300.0 | 5986.5592 |
   +------+-----------+
   |301.0 | 6008.0363 |
   +------+-----------+
   |302.0 | 6028.1154 |
   +------+-----------+
   |303.0 | 6046.7773 |
   +------+-----------+
   |304.0 | 6064.0048 |
   +------+-----------+
   |305.0 | 6079.7826 |
   +------+-----------+
   |306.0 | 6094.0969 |
   +------+-----------+
   |307.0 | 6106.9356 |
   +------+-----------+
   |308.0 | 6118.2881 |
   +------+-----------+
   |309.0 | 6128.1450 |
   +------+-----------+
   |310.0 | 6136.4986 |
   +------+-----------+
   |311.0 | 6143.3422 |
   +------+-----------+
   |312.0 | 6148.6704 |
   +------+-----------+
   |313.0 | 6152.4793 |
   +------+-----------+
   |314.0 | 6154.7657 |
   +------+-----------+
   |315.0 | 6155.5281 |
   +------+-----------+
   |316.0 | 6154.7657 |
   +------+-----------+
   |317.0 | 6152.4793 |
   +------+-----------+
   |318.0 | 6148.6704 |
   +------+-----------+
   |319.0 | 6143.3422 |
   +------+-----------+
   |320.0 | 6136.4986 |
   +------+-----------+
   |321.0 | 6128.1450 |
   +------+-----------+
   |322.0 | 6118.2881 |
   +------+-----------+
   |323.0 | 6106.9356 |
   +------+-----------+
   |324.0 | 6094.0969 |
   +------+-----------+
   |325.0 | 6079.7826 |
   +------+-----------+
   |326.0 | 6064.0048 |
   +------+-----------+
   |327.0 | 6046.7773 |
   +------+-----------+
   |328.0 | 6028.1154 |
   +------+-----------+
   |329.0 | 6008.0363 |
   +------+-----------+
   |330.0 | 5986.5592 |
   +------+-----------+
   |331.0 | 5963.7055 |
   +------+-----------+
   |332.0 | 5939.4989 |
   +------+-----------+
   |333.0 | 5913.9657 |
   +------+-----------+
   |334.0 | 5887.1352 |
   +------+-----------+
   |335.0 | 5859.0401 |
   +------+-----------+
   |336.0 | 5829.7167 |
   +------+-----------+
   |337.0 | 5799.2058 |
   +------+-----------+
   |338.0 | 5767.5531 |
   +------+-----------+
   |339.0 | 5734.8104 |
   +------+-----------+
   |340.0 | 5701.0359 |
   +------+-----------+
   |341.0 | 5666.2964 |
   +------+-----------+
   |342.0 | 5630.6681 |
   +------+-----------+
   |343.0 | 5594.2389 |
   +------+-----------+
   |344.0 | 5557.1106 |
   +------+-----------+
   |345.0 | 5519.4023 |
   +------+-----------+
   |346.0 | 5481.2538 |
   +------+-----------+
   |347.0 | 5442.8306 |
   +------+-----------+
   |348.0 | 5404.3296 |
   +------+-----------+
   |349.0 | 5365.9866 |
   +------+-----------+
   |350.0 | 5328.0849 |
   +------+-----------+
   |351.0 | 5290.9659 |
   +------+-----------+
   |352.0 | 5255.0399 |
   +------+-----------+
   |353.0 | 5220.7981 |
   +------+-----------+
   |354.0 | 5188.8210 |
   +------+-----------+
   |355.0 | 5159.7804 |
   +------+-----------+
   |356.0 | 5134.4274 |
   +------+-----------+
   |357.0 | 5113.5586 |
   +------+-----------+
   |358.0 | 5097.9530 |
   +------+-----------+
   |359.0 | 5088.2812 |
   +------+-----------+
   |360.0 | 5085.0026 |
   +------+-----------+

|
Anisotropic materials can also be defined using a stifness tensor and density of the material. This is done using the :func:`~Anis_TTF_rays.ALI_FMM.add_materials` function which will generate a table of velocities. Alternatively for orthotropic materials, velocities can be calculated at run time using the parameter ``stif_den`` which has the stifness tensor and density at all grid points. All stifness tensors using this method must be input using MPa not Pa and have type ``np.int64``, which is used to aviod overflow errors. Any points not using this method will not be used and so can be left as zeros.

.. tip::

   Using ``stif_den`` is prefered for anisotropic materials as it gives more accurate velocities.

Assigning Materials
-------------------

There are four variables which define the materials at each grid point.

 * ``veln`` is the anisotropic orientations at each grid point i.e rotation of the velocity curve.
 * ``velpn`` is the material index of each grid point, with 0 being a grid point where the velocity is calculated at run time using ``stif_den`` and others being the column in the velocity tables for that material.
 * ``vel_map`` is the velocity scaling of each grid point. This is often set to 1 for most anisotropic materials, however for isotropic materials this parameter defines the velocity at a grid point (when used with a material with velocity of 1 in tables of velocities).
 * ``stif_den`` is the stifness tensors and density at grid points which is only used when the material index is 0.

Calculating Traveltime Fields
-----------------------------

Travel time fields can be calculated through three different functions:

 * :func:`~Anis_TTF_rays.ALI_FMM.update_i` calculates the travel time field for a single source.
 * :func:`~Anis_TTF_rays.ALI_FMM.update` calculates travel time fields for all sources unless specific ones are selected through the ``sources`` parameter.
 * :func:`~Anis_TTF_rays.ALI_FMM.update_parallel` is the same as :func:`~Anis_TTF_rays.ALI_FMM.update`, however parallelisation is used to speed up computation.

Calculating Ray Paths
---------------------

Ray paths can be calculated using :func:`~Anis_TTF_rays.ALI_FMM.find_all_TTF_rays` or in parrallel using :func:`~Anis_TTF_rays.ALI_FMM.find_all_TTF_rays_parallel`. These functions will calculate the nessessary travel time fields and perform ray tracing through them. The parameter ``trans_pairs`` defines which source reciever pairs have ray paths calculated. If ray paths are saved into the class they can be retrieved using the :func:`~Anis_TTF_rays.ALI_FMM.ray_path` function.

.. tip::

   Travel times of ray paths are identical when traveling forward or backwards along the ray. Swapping the source and receivers using the paramter ``trans_pairs`` so that less recievers are used will result in faster computation as less travel time fields are required.

   .. collapse:: Example
   
      |
   
      Both values of ``trans_pairs`` use the same source/receiver pairs, however using a single reciever with three sources is computationally cheaper than one source and three receivers

      +---+---+---+---+
      | 0 | 0 | 0 | 0 |
      +---+---+---+---+
      | 1 | 0 | 0 | 0 |
      +---+---+---+---+
      | 1 | 0 | 0 | 0 |
      +---+---+---+---+
      | 1 | 0 | 0 | 0 |
      +---+---+---+---+

      +---+---+---+---+
      | 0 | 1 | 1 | 1 |
      +---+---+---+---+
      | 0 | 0 | 0 | 0 |
      +---+---+---+---+
      | 0 | 0 | 0 | 0 |
      +---+---+---+---+
      | 0 | 0 | 0 | 0 |
      +---+---+---+---+
   
   |