## Taking uncertainties into account

In this example we are demonstrating how to allow for uncertainties during a Reflectometry fitting job. The sample to fit consists of twenty Titanium-Nickel bilayers. Assuming that all Titanium layers have the same thickness, the goal is to find that thickness.

The reference data was generated with GENX, setting the thickness of the Ti layers equal to 3 nm.

This example follows closely the tutorial on Fitting reflectometry data. The main points to focus on here are the following:

• Added artificial uncertainties to the data being fitted
• Use of the the $RQ^4$ view for plotting
• Use of $\chi^2$ with $L_1$ normalization as the objective metric
• Setting a genetic algorithm as the minimizer
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140  """ Example demonstrates how to fit specular data. Our sample represents twenty interchanging layers of Ti and Ni. We will fit thicknesses of all Ti layers, assuming them being equal. Reference data was generated with GENX for ti layers' thicknesses equal to 3 nm This example uses exactly the same data as in FitSpecularBasics. However this time we will add artificial uncertainties and use RQ^4 view. Besides we will set Chi squared with L1-normalization as the objective metric and use genetic algorithm as the minimizer. """ import numpy as np import bornagain as ba from matplotlib import pyplot as plt from os import path def get_sample(params): """ Creates a sample and returns it :param params: a dictionary of optimization parameters :return: the sample defined """ # substrate (Si) si_sld_real = 2.0704e-06 # \AA^{-2} density_si = 0.0499 / ba.angstrom ** 3 # Si atomic number density # layers' parameters n_repetitions = 10 # Ni ni_sld_real = 9.4245e-06 # \AA^{-2} ni_thickness = 70 * ba.angstrom # Ti ti_sld_real = -1.9493e-06 # \AA^{-2} ti_thickness = params["ti_thickness"] # defining materials m_air = ba.MaterialBySLD() m_ni = ba.MaterialBySLD("Ni", ni_sld_real, 0.0) m_ti = ba.MaterialBySLD("Ti", ti_sld_real, 0.0) m_substrate = ba.MaterialBySLD("SiSubstrate", si_sld_real, 0.0) # air layer and substrate form multi layer air_layer = ba.Layer(m_air) ni_layer = ba.Layer(m_ni, ni_thickness) ti_layer = ba.Layer(m_ti, ti_thickness) substrate_layer = ba.Layer(m_substrate) multi_layer = ba.MultiLayer() multi_layer.addLayer(air_layer) for i in range(n_repetitions): multi_layer.addLayer(ti_layer) multi_layer.addLayer(ni_layer) multi_layer.addLayer(substrate_layer) return multi_layer def get_real_data(): """ Loading data from genx_interchanging_layers.dat Returns a Nx2 array (N - the number of experimental data entries) with first column being coordinates, second one being values. """ if not hasattr(get_real_data, "data"): filename = "genx_interchanging_layers.dat.gz" filepath = path.join(path.dirname(path.realpath(__file__)), filename) real_data = np.loadtxt(filepath, usecols=(0, 1), skiprows=3) # translating axis values from double incident angle (degrees) # to incident angle (radians) real_data[:, 0] *= np.pi / 360 get_real_data.data = real_data return get_real_data.data.copy() def get_real_data_axis(): """ Get axis coordinates of the experimental data :return: 1D array with axis coordinates """ return get_real_data()[:, 0] def get_real_data_values(): """ Get experimental data values as a 1D array :return: 1D array with experimental data values """ return get_real_data()[:, 1] def get_simulation(params): """ Create and return specular simulation with its instrument defined """ wavelength = 1.54 * ba.angstrom # beam wavelength simulation = ba.SpecularSimulation() scan = ba.AngularSpecScan(wavelength, get_real_data_axis()) simulation.setScan(scan) simulation.setSample(get_sample(params)) return simulation def run_fitting(): """ Setup simulation and fit """ real_data = get_real_data_values() # setting artificial uncertainties (uncertainty sigma equals a half # of experimental data value) uncertainties = real_data * 0.5 fit_objective = ba.FitObjective() fit_objective.addSimulationAndData(get_simulation, real_data, uncertainties) plot_observer = ba.PlotterSpecular(units=ba.AxesUnits.RQ4) fit_objective.initPrint(10) fit_objective.initPlot(10, plot_observer) fit_objective.setObjectiveMetric("Chi2", "L1") params = ba.Parameters() params.add("ti_thickness", 50 * ba.angstrom, min=10 * ba.angstrom, max=60 * ba.angstrom) minimizer = ba.Minimizer() minimizer.setMinimizer("Genetic", "", "MaxIterations=40;PopSize=10") result = minimizer.minimize(fit_objective.evaluate, params) fit_objective.finalize(result) if __name__ == '__main__': run_fitting() plt.show() 
FitWithUncertainties.py