gprMax/user_libs/DebyeFit/docs/relaxation.md

Relaxation classes for multi-Debye fitting

This class is designed for modelling different relaxation functions, like Havriliak-Negami (Class HavriliakNegami), Jonscher (Class Jonscher), Complex Refractive Index Mixing (Class CRIM) models, and arbitrary dielectric data derived experimentally or calculated using some other function (Class Rawdata).

Supported relaxation classes:

• Havriliak-Negami,
• Jonscher,
• Complex Refractive Index Mixing,
• Experimental data,

Methods

1. constructor - it is called in all childern classes, creates Relaxation function object for complex material.

   It takes the following arguments:

   • sigma is a conductivity (Siemens/metre),
   • mu is a relative permeability,
   • mu_sigma is a magnetic loss (Ohms/metre),
   • material_name is definition of material name,
   • number_of_debye_poles is choosen number of Debye poles,
   • f_n is choosen number of frequences,
   • plot is a switch to turn on the plotting,
   • save is a switch to turn on the saving final material properties,
   • optimizer is a choosen optimizer to fit model to dielectric data,
   • optimizer_options is a dict for options of choosen optimizer.

   Additional parameters:

   • rl calculated real part of chosen relaxation function for given frequency points,
   • im calculated imaginary part of chosen relaxation function for given frequency points.

2. set_freq - it is inherited by all childern classes, interpolates frequency vector using n equally logarithmicaly spaced frequencies. It takes the following arguments:

   • f_min_: first bound of the frequency range used to approximate the given function (Hz),
   • f_max: second bound of the frequency range used to approximate the given function (Hz),
   • f_n: the number of frequency points in frequency grid (Default: 50).

3. run - it is inherited by all childern classes, solves the problem described by the given relaxation function (main operational method). It consists of following steps:

   1. Check the validity of the inputs using check_inputs method.
   2. Print information about chosen approximation settings using print_info method.
   3. Calculate both real and imaginary parts using calculation method, and then set self.rl and self.im properties.
   4. Calling the main optimisation module using optimize method and calculate error based on error method. a) [OPTIONAL] If number of debye poles is set to -1 optimization procedure is repeted until percentage error goes smaller than 5% or 20 Debye poles is reached.
   5. Print the results in gprMax format style using print_output method.
   6. [OPTIONAL] Save results in gprMax style using save_result method.
   7. [OPTIONAL] Plot the actual and the approximate dielectric properties using plot_result method.

4. check_inputs - it is called in all childern classes, finds an optimal set of relaxation times that minimise an objective function using appropriate optimization procedure.

5. calculation - it is inherited by all childern classes, should be definied in all new children classes, approximates the given relaxation function.

6. print_info - it is inherited by all childern classes, prints readable string of parameters for given approximation settings.

7. optimize - it is inherited by all childern classes, calls the main optimisation module with defined lower and upper boundaries of search.

8. print_output - it is inherited by all childern classes, prints out the resulting Debye parameters in a gprMax format.

9. plot_result - it is inherited by all childern classes, plots the actual and the approximated electric permittivity, along with relative error for real and imaginary parts using a semilogarithm X axes.

10. save_result - it is inherited by all childern classes, saves the resulting Debye parameters in a gprMax format.

11. error - it is inherited by all childern classes, calculates the average fractional error separately for relative permittivity (real part) and conductivity (imaginary part).

Each new class of relaxation object should:

• define constructor with appropriate arguments,
• define check_inputs method to check relaxation class specific parameters,
• overload calculation method.