GCTAPsf2D.cpp

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/***************************************************************************
 *           GCTAPsf2D.cpp - CTA 2D point spread function class            *
 * ----------------------------------------------------------------------- *
 *  copyright (C) 2012-2021 by Juergen Knoedlseder                         *
 * ----------------------------------------------------------------------- *
 *                                                                         *
 *  This program is free software: you can redistribute it and/or modify   *
 *  it under the terms of the GNU General Public License as published by   *
 *  the Free Software Foundation, either version 3 of the License, or      *
 *  (at your option) any later version.                                    *
 *                                                                         *
 *  This program is distributed in the hope that it will be useful,        *
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of         *
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the          *
 *  GNU General Public License for more details.                           *
 *                                                                         *
 *  You should have received a copy of the GNU General Public License      *
 *  along with this program.  If not, see <http://www.gnu.org/licenses/>.  *
 *                                                                         *
 ***************************************************************************/
/**
 * @file GCTAPsf2D.hpp
 * @brief CTA 2D point spread function class implementation
 * @author Juergen Knoedlseder
 */

/* __ Includes ___________________________________________________________ */
#ifdef HAVE_CONFIG_H
#include <config.h>
#endif
#include <cmath>
#include "GTools.hpp"
#include "GMath.hpp"
#include "GException.hpp"
#include "GFilename.hpp"
#include "GRan.hpp"
#include "GFits.hpp"
#include "GFitsBinTable.hpp"
#include "GCTAPsf2D.hpp"
#include "GCTASupport.hpp"

/* __ Method name definitions ____________________________________________ */
#define G_READ                                 "GCTAPsf2D::read(GFitsTable&)"
#define G_CONTAINMENT_RADIUS         "GCTAPsf2D::containment_radius(double&,"\
                       " double&, double&, double&, double&, double&, bool&)"

/* __ Macros _____________________________________________________________ */

/* __ Coding definitions _________________________________________________ */
#define G_SMOOTH_PSF

/* __ Debug definitions __________________________________________________ */

/* __ Constants __________________________________________________________ */


/*==========================================================================
 =                                                                         =
 =                        Constructors/destructors                         =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Void constructor
 *
 * Constructs empty point spread function.
 ***************************************************************************/
GCTAPsf2D::GCTAPsf2D(void) : GCTAPsf()
{
    // Initialise class members
    init_members();

    // Return
    return;
}


/***********************************************************************//**
 * @brief File constructor
 *
 * @param[in] filename FITS file name.
 *
 * Constructs point spread function from a FITS file.
 ***************************************************************************/
GCTAPsf2D::GCTAPsf2D(const GFilename& filename) : GCTAPsf()
{
    // Initialise class members
    init_members();

    // Load point spread function from file
    load(filename);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Copy constructor
 *
 * @param[in] psf Point spread function.
 *
 * Constructs point spread function by copying from another point spread
 * function.
 ***************************************************************************/
GCTAPsf2D::GCTAPsf2D(const GCTAPsf2D& psf) : GCTAPsf(psf)
{
    // Initialise class members
    init_members();

    // Copy members
    copy_members(psf);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Destructor
 *
 * Destructs point spread function.
 ***************************************************************************/
GCTAPsf2D::~GCTAPsf2D(void)
{
    // Free members
    free_members();

    // Return
    return;
}


/*==========================================================================
 =                                                                         =
 =                               Operators                                 =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Assignment operator
 *
 * @param[in] psf Point spread function.
 * @return Point spread function.
 *
 * Assigns point spread function.
 ***************************************************************************/
GCTAPsf2D& GCTAPsf2D::operator=(const GCTAPsf2D& psf)
{
    // Execute only if object is not identical
    if (this != &psf) {

        // Copy base class members
        this->GCTAPsf::operator=(psf);

        // Free members
        free_members();

        // Initialise private members
        init_members();

        // Copy members
        copy_members(psf);

    } // endif: object was not identical

    // Return this object
    return *this;
}


/***********************************************************************//**
 * @brief Return point spread function (in units of sr^-1)
 *
 * @param[in] delta Angular separation between true and measured photon
 *            directions (rad).
 * @param[in] logE Log10 of the true photon energy (TeV).
 * @param[in] theta Offset angle in camera system (rad). Defaults to 0.0.
 * @param[in] phi Azimuth angle in camera system (rad). Not used.
 * @param[in] zenith Zenith angle in Earth system (rad). Not used.
 * @param[in] azimuth Azimuth angle in Earth system (rad). Not used.
 * @param[in] etrue Use true energy (true/false). Not used.
 *
 * Returns the point spread function for a given angular separation in units
 * of sr^-1 for a given energy and offset angle.
 ***************************************************************************/
double GCTAPsf2D::operator()(const double& delta,
                             const double& logE, 
                             const double& theta, 
                             const double& phi,
                             const double& zenith,
                             const double& azimuth,
                             const bool&   etrue) const
{
    #if defined(G_SMOOTH_PSF)
    // Compute offset so that PSF goes to 0 at 5 times the sigma value. This
    // is a kluge to get a PSF that smoothly goes to zero at the edge, which
    // prevents steps or kinks in the log-likelihood function.
    static const double offset = std::exp(-0.5*5.0*5.0);
    #endif

    // Initialise PSF value
    double psf = 0.0;

    // Update the parameter cache
    update(logE, theta);

    // Continue only if normalization is positive
    if (m_norm > 0.0) {

        // Compute distance squared
        double delta2 = delta * delta;

        // Compute Psf value
        #if defined(G_SMOOTH_PSF)
        psf = std::exp(m_width1 * delta2) - offset;
        if (m_norm2 > 0.0) {
            psf += (std::exp(m_width2 * delta2)-offset) * m_norm2;
        }
        if (m_norm3 > 0.0) {
            psf += (std::exp(m_width3 * delta2)-offset) * m_norm3;
        }
        #else
        psf = std::exp(m_width1 * delta2);
        if (m_norm2 > 0.0) {
            psf += std::exp(m_width2 * delta2) * m_norm2;
        }
        if (m_norm3 > 0.0) {
            psf += std::exp(m_width3 * delta2) * m_norm3;
        }
        #endif
        psf *= m_norm;

        #if defined(G_SMOOTH_PSF)
        // Make sure that PSF is non-negative
        if (psf < 0.0) {
            psf = 0.0;
        }
        #endif

    } // endif: normalization was positive
    
    // Return PSF
    return psf;
}


/*==========================================================================
 =                                                                         =
 =                             Public methods                              =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Clear point spread function
 *
 * Clears point spread function.
 ***************************************************************************/
void GCTAPsf2D::clear(void)
{
    // Free class members
    free_members();
    this->GCTAPsf::free_members();

    // Initialise members
    this->GCTAPsf::init_members();
    init_members();

    // Return
    return;
}


/***********************************************************************//**
 * @brief Clone point spread functions
 *
 * @return Deep copy of point spread function.
 *
 * Returns a pointer to a deep copy of the point spread function.
 ***************************************************************************/
GCTAPsf2D* GCTAPsf2D::clone(void) const
{
    return new GCTAPsf2D(*this);
}


/***********************************************************************//**
 * @brief Read point spread function from FITS table
 *
 * @param[in] table FITS table.
 *
 * @exception GException::invalid_value
 *            Response table is not two-dimensional.
 *
 * Reads the point spread function form the FITS @p table. The following
 * column names are mandatory:
 *
 *     ENERG_LO - Energy lower bin boundaries
 *     ENERG_HI - Energy upper bin boundaries
 *     THETA_LO - Offset angle lower bin boundaries
 *     THETA_HI - Offset angle upper bin boundaries
 *     SIGMA_1  - 1st Gaussian sigma
 *     AMPL_2   - 2nd Gaussian relative amplitude
 *     SIGMA_2  - 2nd Gaussian sigma
 *     AMPL_3   - 3rd Gaussian relative amplitude
 *     SIGMA_3  - 3rd Gaussian sigma
 *
 * The data are stored in the m_psf member. The energy axis will be set to
 * log10, the offset angle axis to radians.
 ***************************************************************************/
void GCTAPsf2D::read(const GFitsTable& table)
{
    // Clear response table
    m_psf.clear();

    // Read PSF table
    m_psf.read(table);

    // Get mandatory indices (throw exception if not found)
    m_inx_energy = m_psf.axis("ENERG");
    m_inx_theta  = m_psf.axis("THETA");
    m_inx_sigma1 = m_psf.table("SIGMA_1");
    m_inx_ampl2  = m_psf.table("AMPL_2");
    m_inx_sigma2 = m_psf.table("SIGMA_2");
    m_inx_ampl3  = m_psf.table("AMPL_3");
    m_inx_sigma3 = m_psf.table("SIGMA_3");

    // Throw an exception if the table is not two-dimensional
    if (m_psf.axes() != 2) {
        std::string msg = "Expected two-dimensional point spread function "
                          "response table but found "+
                          gammalib::str(m_psf.axes())+
                          " dimensions. Please specify a two-dimensional "
                          "point spread function.";
        throw GException::invalid_value(G_READ, msg);
    }

    // Set energy axis to logarithmic scale
    m_psf.axis_log10(m_inx_energy);

    // Set offset angle axis to radians
    m_psf.axis_radians(m_inx_theta);

    // Convert sigma parameters to radians
    m_psf.scale(m_inx_sigma1, gammalib::deg2rad);
    m_psf.scale(m_inx_sigma2, gammalib::deg2rad);
    m_psf.scale(m_inx_sigma3, gammalib::deg2rad);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Write point spread function into FITS binary table
 *
 * @param[in] table FITS binary table.
 *
 * Writes point spread function into FITS binary @p table.
 *
 * @todo Add necessary keywords.
 ***************************************************************************/
void GCTAPsf2D::write(GFitsBinTable& table) const
{
    // Create a copy of the response table
    GCTAResponseTable psf(m_psf);

    // Convert sigma parameters back to degrees (only if there are tables)
    if (psf.tables() > 0) {
        psf.scale(m_inx_sigma1, gammalib::rad2deg);
        psf.scale(m_inx_sigma2, gammalib::rad2deg);
        psf.scale(m_inx_sigma3, gammalib::rad2deg);
    }

    // Write response table
    psf.write(table);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Load point spread function from FITS file
 *
 * @param[in] filename FITS file name.
 *
 * Loads the point spread function from a FITS file.
 *
 * If no extension name is given the method scans the `HDUCLASS` keywords
 * of all extensions and loads the background from the first extension
 * for which `HDUCLAS4=PSF_3GAUSS`.
 *
 * Otherwise, the background will be loaded from the `POINT SPREAD FUNCTION`
 * extension.
 ***************************************************************************/
void GCTAPsf2D::load(const GFilename& filename)
{
    // Open FITS file
    GFits fits(filename);

    // Get the default extension name. If no GADF compliant name was found
    // then set the default extension name to "POINT SPREAD FUNCTION".
    std::string extname = gammalib::gadf_hduclas4(fits, "PSF_3GAUSS");
    if (extname.empty()) {
        extname = gammalib::extname_cta_psf2d;
    }

    // Get PSF table
    const GFitsTable& table = *fits.table(filename.extname(extname));

    // Read PSF from table
    read(table);

    // Close FITS file
    fits.close();

    // Store filename
    m_filename = filename;

    // Return
    return;
}


/***********************************************************************//**
 * @brief Save point spread function table into FITS file
 *
 * @param[in] filename FITS file name.
 * @param[in] clobber Overwrite existing file?
 *
 * Save the point spread function into a FITS file.
 *
 * If no extension name is provided, the point spread function will be saved
 * into the `POINT SPREAD FUNCTION` extension.
 ***************************************************************************/
void GCTAPsf2D::save(const GFilename& filename, const bool& clobber) const
{
    // Get extension name
    std::string extname = filename.extname(gammalib::extname_cta_psf2d);

    // Open or create FITS file (without extension name since the requested
    // extension may not yet exist in the file)
    GFits fits(filename.url(), true);

    // Remove extension if it exists already
    if (fits.contains(extname)) {
        fits.remove(extname);
    }

    // Create binary table
    GFitsBinTable table;

    // Write the background table
    write(table);

    // Set binary table extension name
    table.extname(extname);

    // Append table to FITS file
    fits.append(table);

    // Save to file
    fits.save(clobber);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Simulate PSF offset (radians)
 *
 * @param[in] ran Random number generator.
 * @param[in] logE Log10 of the true photon energy (TeV).
 * @param[in] theta Offset angle in camera system (rad).
 * @param[in] phi Azimuth angle in camera system (rad). Not used.
 * @param[in] zenith Zenith angle in Earth system (rad). Not used.
 * @param[in] azimuth Azimuth angle in Earth system (rad). Not used.
 * @param[in] etrue Use true energy (true/false). Not used.
 *
 * Draws a random offset for a three component Gaussian PSF.
 ***************************************************************************/
double GCTAPsf2D::mc(GRan&         ran,
                     const double& logE,
                     const double& theta,
                     const double& phi,
                     const double& zenith,
                     const double& azimuth,
                     const bool&   etrue) const
{
    // Update the parameter cache
    update(logE, theta);

    // Select in which Gaussian we are
    double sigma = m_sigma1;
    double sum1  = m_sigma1;
    double sum2  = m_sigma2 * m_norm2;
    double sum3  = m_sigma3 * m_norm3;
    double sum   = sum1 + sum2 + sum3;
    double u     = ran.uniform() * sum;
    if (sum2 > 0.0 && u >= sum2) {
        sigma = m_sigma3;
    }
    else if (sum1 > 0.0 && u >= sum1) {
        sigma = m_sigma2;
    }

    // Now draw from the selected Gaussian
    double delta = sigma * ran.chisq2();
    
    // Return PSF offset
    return delta;
}


/***********************************************************************//**
 * @brief Return maximum size of PSF (radians)
 *
 * @param[in] logE Log10 of the true photon energy (TeV).
 * @param[in] theta Offset angle in camera system (rad). Not used.
 * @param[in] phi Azimuth angle in camera system (rad). Not used.
 * @param[in] zenith Zenith angle in Earth system (rad). Not used.
 * @param[in] azimuth Azimuth angle in Earth system (rad). Not used.
 * @param[in] etrue Use true energy (true/false). Not used.
 *
 * Determine the radius beyond which the PSF becomes negligible. This radius
 * is set by this method to \f$5 \times \sigma\f$, where \f$\sigma\f$ is the
 * Gaussian width of the largest PSF component.
 ***************************************************************************/
double GCTAPsf2D::delta_max(const double& logE, 
                            const double& theta, 
                            const double& phi,
                            const double& zenith,
                            const double& azimuth,
                            const bool&   etrue) const
{
    // Update the parameter cache
    update(logE, theta);

    // Compute maximum sigma
    double sigma = m_sigma1;
    if (m_sigma2 > sigma) sigma = m_sigma2;
    if (m_sigma3 > sigma) sigma = m_sigma3;

    // Compute maximum PSF radius
    double radius = 5.0 * sigma;
    
    // Return maximum PSF radius
    return radius;
}


/***********************************************************************//**
 * @brief Return the radius that contains a fraction of the events (radians)
 *
 * @param[in] fraction of events (0.0 < fraction < 1.0)
 * @param[in] logE Log10 of the true photon energy (TeV).
 * @param[in] theta Offset angle in camera system (rad).
 * @param[in] phi Azimuth angle in camera system (rad). Not used.
 * @param[in] zenith Zenith angle in Earth system (rad). Not used.
 * @param[in] azimuth Azimuth angle in Earth system (rad). Not used.
 * @param[in] etrue Use true energy (true/false). Not used.
 * @return Containment radius (radians).
 *
 * @exception GException::invalid_argument
 *            Invalid fraction specified.
 *
 * Uses the Newton-Raphson method to find which 'a' solves the equation:
 *
 * \f[
 *
 * fraction = m\_norm * \left( e^{m\_width1 * a^2} +
 *                             m\_norm2 * e^{m\_width2 * a^2} +
 *                             m\_norm3 * e^{m\_width3 * a^2} \right)
 *
 * \f]
 *
 * Calculate the radius from the center that contains @p fraction  of the
 * events (@p fraction * 100. = Containment % ). Fraction must be
 *
 * \f[
 * 0.0 < fraction < 1.0
 * \f]
 ***************************************************************************/
double GCTAPsf2D::containment_radius(const double& fraction, 
                                     const double& logE, 
                                     const double& theta, 
                                     const double& phi,
                                     const double& zenith,
                                     const double& azimuth,
                                     const bool&   etrue) const
{
    // Set maximum number of Newton-Raphson loops before giving up and
    // required accuracy
    const int    itermax     = 20;
    const double convergence = 1.0e-6;

    // Check input argument
    if (fraction <= 0.0 || fraction >= 1.0) {
        std::string message = "Containment fraction "+
                              gammalib::str(fraction)+" must be between " +
                              "0.0 and 1.0, not inclusive.";
        throw GException::invalid_argument(G_CONTAINMENT_RADIUS, message);
    }

    // Update the parameter cache
    update(logE, theta);

    // Set minimum and maximum radius (avoid starting at zero radius)
    double amax = delta_max(logE, theta, phi, zenith, azimuth, etrue);
    double amin = 0.000001 * amax;

    // Set initial radius to start Newton's method
    double a = 0.20 * amax;

    // Set normalisation constants
    double norm1 = (m_width1 != 0.0) ? -gammalib::pi/m_width1 : 0.0;
    double norm2 = (m_width2 != 0.0) ? -gammalib::pi/m_width2 * m_norm2 : 0.0;
    double norm3 = (m_width3 != 0.0) ? -gammalib::pi/m_width3 * m_norm3 : 0.0;

    // Do the Newton-Raphson loops
    int iter = 0;
    for (; iter < itermax; ++iter) {

        // Compute square of test radius
        double a2 = a*a;

        // Precompute exponentials
        double exp1 = std::exp(m_width1 * a2);
        double exp2 = std::exp(m_width2 * a2);
        double exp3 = std::exp(m_width3 * a2);

        // Calculate f(a)
        double fa = 0.0;
        fa       += norm1 * (1.0 - exp1);
        fa       += norm2 * (1.0 - exp2);
        fa       += norm3 * (1.0 - exp3);
        fa       *= m_norm;
        fa       -= fraction;

        // If we've met the desired accuracy then exit the loop
        if (std::abs(fa) < convergence) {
            break;
        }

        // Calculate f'(a)
        double fp = 0.0;
        fp       += norm1 * exp1 * m_width1;
        fp       += norm2 * exp2 * m_width2;
        fp       += norm3 * exp3 * m_width3;
        fp       *= -2.0 * a * m_norm;

        // Calculate next point via x+1 = x - f(a) / f'(a)
        if (fp != 0.0) {
            a -= fa / fp;
            if (a < amin) {
                a = amin;
            }
            if (a > amax) {
                a = amax;
            }
        }
        else {
            break;
        }

    } // endfor: Newton-Raphson loops

    // Warn the user if we didn't converge
    if (iter == itermax-1) {
        std::string msg = "Unable to converge within " +
                          gammalib::str(convergence)+" of the root in less "
                          "than "+gammalib::str(itermax)+" iterations." ;
        gammalib::warning(G_CONTAINMENT_RADIUS, msg);
    }

    // Return containment radius
    return a;
}


/***********************************************************************//**
 * @brief Print point spread function information
 *
 * @param[in] chatter Chattiness.
 * @return String containing point spread function information.
 ***************************************************************************/
std::string GCTAPsf2D::print(const GChatter& chatter) const
{
    // Initialise result string
    std::string result;

    // Continue only if chatter is not silent
    if (chatter != SILENT) {

        // Append header
        result.append("=== GCTAPsf2D ===");
        result.append("\n"+gammalib::parformat("Filename")+m_filename);

        // Initialise information
        int    nebins     = 0;
        int    nthetabins = 0;
        double emin       = 0.0;
        double emax       = 0.0;
        double omin       = 0.0;
        double omax       = 0.0;

        // Extract information if there are axes in the response table
        if (m_psf.axes() > 0) {
            nebins     = m_psf.axis_bins(m_inx_energy);
            nthetabins = m_psf.axis_bins(m_inx_theta);
            emin       = m_psf.axis_lo(m_inx_energy,0);
            emax       = m_psf.axis_hi(m_inx_energy,nebins-1);
            omin       = m_psf.axis_lo(m_inx_theta,0);
            omax       = m_psf.axis_hi(m_inx_theta,nthetabins-1);
        }

        // Append information
        result.append("\n"+gammalib::parformat("Number of energy bins") +
                      gammalib::str(nebins));
        result.append("\n"+gammalib::parformat("Number of offset bins") +
                      gammalib::str(nthetabins));
        result.append("\n"+gammalib::parformat("Energy range"));
        result.append(gammalib::str(emin)+" - "+gammalib::str(emax)+" TeV");
        result.append("\n"+gammalib::parformat("Offset angle range"));
        result.append(gammalib::str(omin)+" - "+gammalib::str(omax)+" deg");

    } // endif: chatter was not silent

    // Return result
    return result;
}


/*==========================================================================
 =                                                                         =
 =                            Private methods                              =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Initialise class members
 ***************************************************************************/
void GCTAPsf2D::init_members(void)
{
    // Initialise members
    m_filename.clear();
    m_psf.clear();
    m_inx_energy = 0;
    m_inx_theta  = 1;
    m_inx_sigma1 = 0;
    m_inx_ampl2  = 1;
    m_inx_sigma2 = 2;
    m_inx_ampl3  = 3;
    m_inx_sigma3 = 4;
    m_par_logE   = -1.0e30;
    m_par_theta  = -1.0;
    m_norm       = 1.0;
    m_norm2      = 0.0;
    m_norm3      = 0.0;
    m_sigma1     = 0.0;
    m_sigma2     = 0.0;
    m_sigma3     = 0.0;
    m_width1     = 0.0;
    m_width2     = 0.0;
    m_width3     = 0.0;

    // Return
    return;
}


/***********************************************************************//**
 * @brief Copy class members
 *
 * @param[in] psf Point spread function.
 ***************************************************************************/
void GCTAPsf2D::copy_members(const GCTAPsf2D& psf)
{
    // Copy members
    m_filename   = psf.m_filename;
    m_psf        = psf.m_psf;
    m_inx_energy = psf.m_inx_energy;
    m_inx_theta  = psf.m_inx_theta;
    m_inx_sigma1 = psf.m_inx_sigma1;
    m_inx_ampl2  = psf.m_inx_ampl2;
    m_inx_sigma2 = psf.m_inx_sigma2;
    m_inx_ampl3  = psf.m_inx_ampl3;
    m_inx_sigma3 = psf.m_inx_sigma3;
    m_par_logE   = psf.m_par_logE;
    m_par_theta  = psf.m_par_theta;
    m_norm       = psf.m_norm;
    m_norm2      = psf.m_norm2;
    m_norm3      = psf.m_norm3;
    m_sigma1     = psf.m_sigma1;
    m_sigma2     = psf.m_sigma2;
    m_sigma3     = psf.m_sigma3;
    m_width1     = psf.m_width1;
    m_width2     = psf.m_width2;
    m_width3     = psf.m_width3;

    // Return
    return;
}


/***********************************************************************//**
 * @brief Delete class members
 ***************************************************************************/
void GCTAPsf2D::free_members(void)
{
    // Return
    return;
}


/***********************************************************************//**
 * @brief Update PSF parameter cache
 *
 * @param[in] logE Log10 of the true photon energy (TeV).
 * @param[in] theta Offset angle in camera system (rad).
 *
 * This method updates the PSF parameter cache.
 ***************************************************************************/
void GCTAPsf2D::update(const double& logE, const double& theta) const
{
    // Only compute PSF parameters if arguments have changed
    if (logE != m_par_logE || theta != m_par_theta) {

        // Save parameters
        m_par_logE  = logE;
        m_par_theta = theta;

        // Interpolate response parameters
        std::vector<double> pars = (m_inx_energy == 0) ? m_psf(logE, theta) :
                                                         m_psf(theta, logE);

        // Set Gaussian sigmas
        m_sigma1 = pars[m_inx_sigma1];
        m_sigma2 = pars[m_inx_sigma2];
        m_sigma3 = pars[m_inx_sigma3];

        // Set width parameters
        double sigma1 = m_sigma1 * m_sigma1;
        double sigma2 = m_sigma2 * m_sigma2;
        double sigma3 = m_sigma3 * m_sigma3;

        // Compute Gaussian 1
        if (sigma1 > 0.0) {
            m_width1 = -0.5 / sigma1;
        }
        else {
            m_width1 = 0.0;
        }

        // Compute Gaussian 2
        if (sigma2 > 0.0) {
            m_width2 = -0.5 / sigma2;
            m_norm2  = pars[m_inx_ampl2];
        }
        else {
            m_width2 = 0.0;
            m_norm2  = 0.0;
        }

        // Compute Gaussian 3
        if (sigma3 > 0.0) {
            m_width3 = -0.5 / sigma3;
            m_norm3  = pars[m_inx_ampl3];
        }
        else {
            m_width3 = 0.0;
            m_norm3  = 0.0;
        }

        // Compute global normalization parameter
        double integral = gammalib::twopi *
                          (sigma1 + sigma2*m_norm2 + sigma3*m_norm3);
        m_norm = (integral > 0.0) ? 1.0 / integral : 0.0;

    }

    // Return
    return;
}