path: root/deproject_sbp.py

#!/usr/bin/env python3
#
# Weitian LI
# Created: 2016-06-10
# Updated: 2016-06-24
#
# Change logs:
# 2016-06-24:
#   * Split class 'Projection' to a separate module 'projection.py'
#   * Move class 'DensityProfile' to tool 'calc_mass_potential.py'
#   * Split class 'AstroParams' to separate module 'astro_params.py'
# 2016-06-23:
#   * Add configuration parameter 'sbpexp_rcut'
#   * Allow extrapolate the cooling function profile
#   * Add plot function to class 'BrightnessProfile'
#   * Update sample configuration file
#   * Remove obsolete class 'SurfaceBrightnessProfile'
# 2016-06-22:
#   * Add class 'DensityProfile', the inversion to 'BrightnessProfile'
#   * Add classes 'AstroParams' and 'BrightnessProfile'
#   * Add class 'ChandraPixel'
#   * Update documentation
# 2016-06-21:
#   * Add document about the gas density derivation
# 2016-06-20:
#   * Use configuration file instead of the tedious command line arguments
# 2016-06-16:
#   * Add methods 'save()', 'report()' and 'plot()' to class 'SBP'
# 2016-06-15:
#   * Add command line arguments
#   * Add class 'SBP' for SBP background subtraction and extrapolation
# 2016-06-14:
#   * Add class 'PLCModel' based on 'FitModel'
#   * Split class 'FitModel' from 'ABModel'
# 2016-06-13:
#   * Add class 'ABModel' to support data scaling
#   * Implement primitive SBP deprojection approach for class 'DeprojectSBP'
#

"""
Deproject the 2D surface brightness profile (SBP) into ???

The SBP deprojection is performed using a non-parametric approach with
regularization which add the constraint that the 3D gas density profile
should be smooth.

=======================================================================

Surface brightness (`SUR_FLUX` column of the dmextract'ed radial profile):
    Brightness: [ photon s^-1 cm^-2 pixel^-2 ]
where the 'cm^-2' is due to the instrumental effective area, and the
'pixel^-2' is corresponding to the solid angle with respect to the source
(i.e., [ arcsec^-2 ]).

The flux has dimension:
    Flux: [ photon s^-1 cm^-2 ]
therefore, the dimension of brightness can also be expressed as:
    Brightness: [ Flux pixel^-2 ] = [ Flux sr^-1 ]

The instrument and (time-normalized) exposure map has dimension:
    [ count photon^-1 cm^2 ]
which is used to convert the instrument-specific counts image into physical-
meaningful flux unit.

Emission measure:
    EM = \int n_e n_H dV ~= (n_e^2 / ratio_eH) V  [ cm^-3 ]
where 'ratio_eH' is the ratio of electron density to proton density (n_H).

APEC normalization returned by XSPEC is simply the *emission measure* of
the gas scaled by the distance:
    eta = (\int n_e n_H dV) / (4 pi (D_A (1+z))^2)
assuming (ref. [4]):
    n_H ~= 0.826 n_e
then the gas density (n_H or n_e) can be calculated.

The flux calculated with the XSPEC `flux` command has dimension:
    Flux: [ photon s^-1 cm^-2 ] or [ erg s^-2 cm^-2 ]

When use XSPEC APEC model to calculate the cooling function (Lambda),
its normalization is calculated with EM = 1, therefore:
    norm = 1e-14 / (4 pi (D_A (1+z))^2) * EM
         = 1e-14 / (4 pi (D_A (1+z))^2)  [ cm^-5 ]
where the 'D_A' is the angular diameter distance which can be simply
calculated from its redshift.

With the Galactic absorption (nH), temperature (varies with radius), and
abundance (assumed constant) been set, the cooling function is derived by
using the XSPEC `flux` command.
Therefore, cooling function has dimension:
    Lambda: [ Flux EM^-1 ]

By deprojecting the surface brightness, the flux per volume can be derived,
and EM can be further obtained by incorporating the cooling function,
and finally the (3D) gas density can be determined.
    (projection): EM * Lambda / A  ->  Brightness
where 'A' is the solid angle (i.e., area covered by the source).
=======================================================================

References:
[1] Croston et al. 2006, A&A, 459, 1007-1019
[2] McLaughlin, 1999, ApJ, 117, 2398-2427
[3] Bouchet, 1995, A&AS, 113, 167
[4] Ettori et al, 2013, Space Science Review, 177, 119-154
[5] AtomDB / APEC model:
    * http://www.atomdb.org/faq.php#DensityXSPECnorm
    * https://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/manual/XSmodelApec.html


Sample configuration file:
------------------------------------------------------------
## Configuration for `deproject_sbp.py`
## Date: 2016-06-23

# config file for SBP fitting (e.g., sbpfit.conf)
sbpfit_config = sbpfit.conf

# input cooling function profile
coolfunc_profile = coolfunc_profile.txt

# redshift of the object (for pixel to distance conversion)
redshift = <REDSHIFT>

## SBP extrapolation
# cut radius from which the SBP is fitted for extrapolation,
# specified by the ratio w.r.t sbpfit rc (default: 1.2 * rc)
sbpexp_rcut_ratio = 1.2
# or directly specify the cut radius (override above rcut ratio)
sbpexp_rcut = <RCUT>
# output of the extrapolated SBP
sbpexp_outfile = sbpexp.csv
# extrapolation model information
sbpexp_json = sbpexp.json
# plot of the SBP extrapolation
sbpexp_image = sbpexp.png

## Density profiles
# deprojected 3D electron number density profile
ne_profile = ne_profile.txt
# deprojected 3D gas mass density profile
rho_gas_profile = rho_gas_profile.txt
# image of the density profiles (electron density and/or gas density)
density_profile_image = density_profile.png
------------------------------------------------------------
"""

import argparse
import json
from collections import OrderedDict

import astropy.units as au
from astropy.cosmology import FlatLambdaCDM
import numpy as np
import pandas as pd
import scipy.optimize
import scipy.interpolate
import lmfit
import matplotlib.pyplot as plt
from matplotlib.backends.backend_agg import FigureCanvasAgg as FigureCanvas
from matplotlib.figure import Figure
from configobj import ConfigObj

from astro_params import AstroParams
from projection import Projection

plt.style.use("ggplot")


class FitModel:
    """
    Base/Meta class for model fitting, with data and parameters scaling.
    """
    name = ""
    params = lmfit.Parameters()
    # optimization method
    fit_method = "lbfgsb"
    # whether the 'ydata' and 'yerr' to be scaled in order to reduce
    # the dynamical range for a more stable fitting
    scale = False
    scale_factor = 1.0

    def __init__(self, fit_method="lbfgsb", params=None, scale=True):
        self.fit_method = fit_method
        if params is not None:
            self.load_params(params)
        self.scale = scale

    @staticmethod
    def model(x, params):
        pass

    def f(self, x):
        return self.model(x, self.params) * self.scale_factor

    def load_data(self, xdata, ydata=None, xerr=None, yerr=None,
                  update_params=False):
        if xdata.ndim == 2 and xdata.shape[1] == 4:
            # 4-column data
            self.xdata = xdata[:, 0].copy()
            self.xerr = xdata[:, 1].copy()
            self.ydata = xdata[:, 2].copy()
            self.yerr = xdata[:, 3].copy()
        else:
            self.xdata = np.array(xdata)
            self.ydata = np.array(ydata)
            self.xerr = np.array(xerr)
            self.yerr = np.array(yerr)
        self.scale_data(update_params=update_params)

    def scale_data(self, update_params=False):
        """
        Scale the ydata and yerr to reduce their dynamical ranges,
        for a more stable model fitting.
        """
        if self.scale:
            y_min = np.min(self.ydata)
            y_max = np.max(self.ydata)
            self.scale_factor = np.exp(np.log(y_min*y_max) / 2)
            self.ydata /= self.scale_factor
            self.yerr /= self.scale_factor
            if update_params:
                self.scale_params()

    def scale_params(self):
        """
        Scale the paramters' min/max values accordingly.
        """
        pass

    def f_residual(self, params):
        if self.yerr is None:
            return self.model(self.xdata, params) - self.ydata
        else:
            return (self.model(self.xdata, params) - self.ydata) / self.yerr

    def fit(self, method=None):
        if method is None:
            method = self.fit_method
        self.fitter = lmfit.Minimizer(self.f_residual, self.params)
        self.fitted = self.fitter.minimize(method=method)
        self.load_params(self.fitted.params)

    def get_param(self, name=None):
        """
        Return the requested 'Parameter' object or the whole
        'Parameters' object of no name supplied.
        """
        try:
            return self.params[name]
        except KeyError:
            return self.params

    def set_param(self, name, *args, **kwargs):
        """
        Set the properties of the specified parameter.
        """
        param = self.params[name]
        param.set(*args, **kwargs)

    def dump_params(self, serialize=True):
        """
        Dump the current values/settings for all model parameters,
        and these dumped results can be later loaded by 'load_params()'.
        """
        if serialize:
            return self.params.dumps()
        else:
            return self.params.copy()

    def load_params(self, params):
        """
        Load the provided parameters values/settings.
        """
        if isinstance(params, lmfit.parameter.Parameters):
            self.params = params.copy()
        else:
            p = lmfit.parameter.Parameters()
            p.loads(params)
            self.params = p


class ABModel(FitModel):
    """
    AB model is a modified beta model, which can roughly fit both
    centrally peaked and cored models, e.g., central excess emission.

    This model is used here to constrain the deprojected 3D gas density
    profile, in order to require it is smooth enough.

    References:
    [1] Pratt & Arnaud, 2002, A&A, 394, 375; eq.(2)
    [2] ref.[1], eq.(10)
    """
    name = "AB model"
    # model parameters
    params = lmfit.Parameters()
    params.add_many(  # (name, value, vary, min, max, expr)
                    ("A",     1.0e-9, True, 0.0, 1.0e-5, None),
                    ("alpha", 0.7,    True, 0.1, 1.1,    None),
                    ("rc",    30.0,   True, 1.0, 1.0e4,  None),
                    ("beta",  0.7,    True, 0.3, 1.1,    None))

    def scale_params(self):
        A_min = 1.0
        A_max = np.max(self.ydata)
        self.set_param("A", value=(A_min+A_max)*0.5,
                       min=A_min, max=A_max)

    @staticmethod
    def model(x, params):
        parvals = params.valuesdict()
        A = parvals["A"]
        alpha = parvals["alpha"]
        rc = parvals["rc"]
        beta = parvals["beta"]
        return (A * np.power(x/rc, -alpha) *
                np.power((1 + (x/rc)**2), -1.5*beta + 0.5*alpha))


class PLCModel(FitModel):
    """
    PLC model consists of a powerlaw and an constant, that is used
    to fit/approximate the outer SBP.
    Therefore, the fitted constant is used to subtract the uniform
    background from the SBP, and the fitted powerlaw index is used
    to extrapolate the SBP in order to mitigate the deprojection
    errors due to FoV limit.

    NOTE:
    I think the uniform background (i.e., by fitting the whole or
    core-excluded SBP) should be subtracted from the SBP first, then
    adopt this PLCModel to fit the outer part of SBP, with the 'bkg'
    parameter fixed at zero.
    """
    name = "PLC model"
    # model parameters
    params = lmfit.Parameters()
    params.add_many(  # (name, value, vary, min, max, expr)
                    ("A",     1.0e-9, True,  0.0, 1.0e-5, None),
                    ("rmin",  30.0,   False, 1.0, 1.0e4,  None),
                    ("alpha", 1.6,    True,  0.4, 2.8,    None),
                    ("bkg",   0.0,    False, 0.0, 1.0e-5, None))

    def load_data(self, xdata, ydata=None, xerr=None, yerr=None,
                  update_params=False):
        super().load_data(xdata=xdata, ydata=ydata, xerr=xerr, yerr=yerr,
                          update_params=update_params)
        self.set_param("rmin", value=np.min(xdata), vary=False)

    def scale_params(self):
        ymin = np.min(self.ydata)
        ymax = np.max(self.ydata)
        self.set_param("A", value=ymax, min=ymax/10.0, max=ymax*10.0)
        self.set_param("bkg", value=ymin, min=0.0, max=ymin)

    @staticmethod
    def model(x, params):
        parvals = params.valuesdict()
        A = parvals["A"]
        rmin = parvals["rmin"]
        alpha = parvals["alpha"]
        bkg = parvals["bkg"]
        return A * np.power(x/rmin, -alpha) + bkg


class DeprojectSBP:
    """
    Deproject the observed SBP to derive the 3D emission measure (EM)
    profile, using a regularization technique.

    TODO:
    * add 'mask' support
    * implement 'optimize_lbd()'

    References: ref.[1]
    """
    # input SBP data: [r, r_err, s, s_err]
    r = None
    r_err = None
    s = None
    s_err = None
    # mask used to exclude specified data point for cross-validation
    mask = None
    # 'Projection' instance for this SBP
    projector = None
    # 'ABModel' instance to fit the deprojected EM profile for rescaling data
    abmodel = None
    # smoothing parameter to balance between fidelity (chisq) and
    # consistency with the applied regularization constraint.
    lbd = 1.0
    # optimization method for scipy minimize
    opt_method = "Powell"
    # scipy optimize results from 'self.deproject()'
    deproject_res = None

    def __init__(self, r, r_err=None, s=None, s_err=None,
                 lbd=1.0, opt_method="Powell"):
        self.load_data(r=r, r_err=r_err, s=s, s_err=s_err)
        self.projector = Projection(rout=self.r+self.r_err)
        self.abmodel = ABModel(scale=True)
        self.lbd = lbd
        self.opt_method = opt_method

    def load_data(self, r, r_err=None, s=None, s_err=None):
        if r.ndim == 2 and r.shape[1] == 4:
            # 4-column data
            self.r = r[:, 0].copy()
            self.r_err = r[:, 1].copy()
            self.s = r[:, 2].copy()
            self.s_err = r[:, 3].copy()
        else:
            self.r = np.array(r)
            self.r_err = np.array(r_err)
            self.s = np.array(s)
            self.s_err = np.array(s_err)
        self.mask = np.ones(self.r.shape, dtype=np.bool)

    def deproject(self):
        """
        Deproject the observed SBP through direct deprojection
        calculation.
        """
        self.em = self.projector.deproject(self.s)
        return self.em

    def deproject_regularize(self, lbd=None, opt_method=None):
        """
        Deproject the observed SBP to derive the 3D EM profile by
        minimizing the objective function with regularization technique.

        XXX:
        Since we just deproject the observed SBP without considering
        the PSF convolution effect, the 3D EM profile can be just solved,
        therefore, there is no need (also no way) to optimize the
        smoothing parameter 'lambda'.
        """
        def fobj(x):
            return self.f_objective(x, scaled=True)

        def callback(x):
            # NOTE: 'x' here is the scaled EM solution
            x_unscaled = self.unscale_data(x)
            self.update_abmodel(x_unscaled)

        if lbd is not None:
            self.lbd = lbd
        if opt_method is None:
            opt_method = self.opt_method
        # initial guess
        em0 = self.projector.deproject(self.s)
        # scale the EM data to reduce the dynamical range
        em0_scaled = self.scale_data(em0, update_params=True)
        res = scipy.optimize.minimize(fun=fobj, x0=em0_scaled,
                                      method=opt_method,
                                      callback=callback,
                                      options={"disp": True})
        self.deproject_res = res
        self.em = self.unscale_data(res.x)
        return self.em

    def update_abmodel(self, x, xerr=None, update_params=False):
        """
        Load the supplied data into self.abmodel, and perform fitting.

        If the errors/uncertainties is not specified, it is assumed
        to have the same relative errors as the observed SBP.
        """
        if xerr is None:
            x_err = x * self.s_err / self.s
        self.abmodel.load_data(xdata=self.r, xerr=self.r_err,
                               ydata=x, yerr=x_err,
                               update_params=update_params)
        self.abmodel.fit()

    def scale_data(self, x, xerr=None, update_params=False):
        """
        Scale the data (i.e., 3D EM profile) by dividing the fitted
        AB model.

        If the errors/uncertainties is not specified, it is assumed
        to have the same relative errors as the observed SBP.
        """
        self.update_abmodel(x=x, xerr=xerr, update_params=update_params)
        x_fitted = self.abmodel.f(self.r)
        x_scaled = x / x_fitted
        return x_scaled

    def unscale_data(self, x):
        """
        Undo the data scaling by multiplying the same fitted model
        previously used to scale the data.
        """
        x_fitted = self.abmodel.f(self.r)
        x_unscaled = x * x_fitted
        return x_unscaled

    def f_objective(self, x, scaled=False):
        """
        The objective function to be minimized, in order to derive the
        best solution (i.e., deprojected SBP) for the observed SBP.

        This objective function is a combination of plain chi-squared
        and a regularization constraint.

        'lbd' is the parameter to balance the goodness-of-fit and the
        regularization constraint.

        References: ref.[1], eq.(2)
        """
        return (self.f_chisq(x, scaled=scaled) +
                self.lbd * self.f_constraint(x, scaled=scaled))

    def f_residual(self, x, scaled=False):
        """
        Calculate the residuals of each data point for the solution.

        The current solution (i.e., 3D EM profile) is first projected
        into the 2D SBP, then compared to the observed SBP.
        """
        if scaled:
            x = self.unscale_data(x)
        x_2d = self.projector.project(x)
        residuals = (x_2d - self.s) / self.s_err
        return residuals

    def f_chisq(self, x, scaled=False):
        """
        Function to calculate the chi-squared value of the current
        solution with respect to the data.
        """
        chisq = np.sum(self.f_residual(x, scaled=scaled) ** 2)
        return chisq

    def f_constraint(self, x, scaled=False):
        """
        Function to calculate the value of regularization constraint.

        References:
        [1] ref.[1], eq.(3)
        [2] ref.[3], eq.(18)
        """
        if not scaled:
            x = self.scale_data(x)
        # constraint = np.sum((x[:-1] + x[1:]) ** 2)
        constraint = np.sum((x[:-2] - 2*x[1:-1] + x[2:]) ** 2)
        return constraint

    def optimize_lbd(self, lbd0=None):
        """
        Find the optimal smoothing parameter 'lbd' by using the
        cross-validation method.

        References: ref.[3], eq.(23)
        """
        if lbd0 is not None:
            self.lbd = lbd0
        pass

    def predict_obs(self):
        """
        Predict the observation data (i.e., surface brightness) by
        projecting the interpolated solved EM profile.
        """
        pass

    def plot(self, ax=None, fig=None):
        if ax is None:
            fig, ax = plt.subplots(1, 1)
        # SBP data points
        eb = ax.errorbar(self.r, self.s, xerr=self.r_err, yerr=self.s_err,
                         fmt="none", elinewidth=2, capthick=2,
                         label="Brightness profile")
        r_min = 1.0
        r_max = max(self.r + self.r_err)
        s_min = min(self.s) / 1.2
        s_max = max(self.s + self.s_err) * 1.2
        ax.set_xlim(r_min, r_max)
        ax.set_ylim(s_min, s_max)
        ax.set_xscale("log")
        ax.set_yscale("log")
        ax.set_xlabel("Radius (%s)" % "pixel")
        ax.set_ylabel(r"Surface Brightness (photons/cm$^2$/pixel$^2$/s)")
        # deprojected EM profile
        ax2 = ax.twinx()
        line, = ax2.plot(self.r, self.em, color="black",
                         linestyle="solid", linewidth=2,
                         label="EM profile")
        em_min = min(self.em) / 1.2
        em_max = max(self.em) * 1.2
        ax2.set_xlim(r_min, r_max)
        ax2.set_ylim(em_min, em_max)
        ax2.set_yscale(ax.get_yscale())
        ax2.set_ylabel(r"Deprojected Emission Measure (???)")
        # legend
        handles = [eb, line]
        labels = [h.get_label() for h in handles]
        ax.legend(handles=handles, labels=labels, loc=0)
        fig.tight_layout()
        return (fig, ax, ax2)


class SBP:
    """
    X-ray surface brightness profile class.

    This class deals with SBP background subtraction and SBP extrapolation.
    """
    # input SBP data: [r, r_err, s, s_err]
    r = None
    r_err = None
    s = None
    s_err = None
    # uniform background been subtracted
    bkg = None
    # cut/minimal radius from which the SBP is fitted to the PLCModel
    rcut = None
    # PLCModel instance used to extrapolate the SBP
    plcmodel = None

    def __init__(self, r, r_err=None, s=None, s_err=None, rcut=None):
        self.load_data(r=r, r_err=r_err, s=s, s_err=s_err, rcut=rcut)
        self.plcmodel = PLCModel(scale=True)

    def load_data(self, r, r_err=None, s=None, s_err=None, rcut=None):
        if r.ndim == 2 and r.shape[1] == 4:
            # 4-column data
            self.r = r[:, 0].copy()
            self.r_err = r[:, 1].copy()
            self.s = r[:, 2].copy()
            self.s_err = r[:, 3].copy()
        else:
            self.r = np.array(r)
            self.r_err = np.array(r_err)
            self.s = np.array(s)
            self.s_err = np.array(s_err)
        self.rcut = rcut

    def subtract_bkg(self, bkg):
        """
        Subtract the uniform background from the brightness.

        The value of background can be acquired by fitting the whole
        or core-exclude SBP with model consisting of a plain beta model
        and a constant.
        The "AB model" maybe also applicable.
        """
        self.bkg = bkg
        self.s -= bkg
        self.bkg_subtracted = True

    def extrapolate(self, rcut=None):
        """
        Extrapolate the SBP by assuming that the outer SBP follows the
        following relation:
            S_X = A * r^{-alpha},
        which can be determined by model fitting.

        The SBP is extrapolated to the region where the brightness is
        lower than the current observed minimal brightness by one order
        of magnitude, and the extrapolated SBP bins have the same width
        and relative errors as the last SBP bin observed.

        Note that the uniform background should be subtracted first.

        Return:
          * self.r_extrapolated
          * self.r_err_extrapolated
          * self.s_extrapolated
          * self.s_err_extrapolated
          * self.mask_extrapolated
        """
        if rcut is not None:
            self.rcut = rcut
        self.mask = self.r >= self.rcut
        self.plcmodel.load_data(xdata=self.r[self.mask],
                                ydata=self.s[self.mask],
                                xerr=self.r_err[self.mask],
                                yerr=self.s_err[self.mask],
                                update_params=True)
        self.plcmodel.set_param("bkg", value=0.0, vary=False)
        self.plcmodel.fit()
        last_r_err = self.r_err[-1]
        last_s = self.s[-1]
        last_s_err = self.s_err[-1]
        #
        r_exp = self.r.tolist()
        r_err_exp = self.r_err.tolist()
        s_exp = self.s.tolist()
        s_err_exp = self.s_err.tolist()
        mask_exp = [False] * len(r_exp)
        # do extrapolation
        r_tmp = r_exp[-1] + 2*r_err_exp[-1]
        s_tmp = self.plcmodel.f(r_tmp)
        while s_tmp > last_s / 10.0:
            r_exp.append(r_tmp)
            r_err_exp.append(last_r_err)
            s_exp.append(s_tmp)
            s_err_exp.append(s_tmp * last_s_err / last_s)
            mask_exp.append(True)
            r_tmp = r_exp[-1] + 2*r_err_exp[-1]
            s_tmp = self.plcmodel.f(r_tmp)
        # convert to numpy array
        self.r_extrapolated = np.array(r_exp)
        self.r_err_extrapolated = np.array(r_err_exp)
        self.s_extrapolated = np.array(s_exp)
        self.s_err_extrapolated = np.array(s_err_exp)
        self.mask_extrapolated = np.array(mask_exp)

    def report(self, outfile=None):
        """
        Report the extrapolation model fitting results.
        """
        results = OrderedDict([
            ("bkg", self.bkg),
            ("bkg_subtracted", self.bkg_subtracted),
            ("rcut", self.rcut),
            ("model", self.plcmodel.name),
            ("params", OrderedDict([
                (pn, [par.value, par.min, par.max, par.vary])
                for pn, par in self.plcmodel.params.items()
            ])),
        ])
        results_json = json.dumps(results, indent=2)
        if outfile is None:
            print(results_json)
        else:
            open(outfile, "w").write(results_json+"\n")

    def plot(self, ax=None, fig=None):
        """
        Make a plot of the SBP extrapolation.
        """
        if ax is None:
            fig, ax = plt.subplots(1, 1)
        # ignored data points
        mask_ignore = np.logical_not(self.mask)
        if np.sum(mask_ignore) > 0:
            ax.errorbar(self.r[mask_ignore], self.s[mask_ignore],
                        xerr=self.r_err[mask_ignore],
                        yerr=self.s_err[mask_ignore],
                        fmt="none", elinewidth=1, capthick=1)
        # data points used to fit the PLC model
        ax.errorbar(self.r[self.mask], self.s[self.mask],
                    xerr=self.r_err[self.mask],
                    yerr=self.s_err[self.mask],
                    fmt="none", elinewidth=2, capthick=2)
        # extrapolated data points
        ax.errorbar(self.r_extrapolated[self.mask_extrapolated],
                    self.s_extrapolated[self.mask_extrapolated],
                    xerr=self.r_err_extrapolated[self.mask_extrapolated],
                    yerr=self.s_err_extrapolated[self.mask_extrapolated],
                    fmt="none", elinewidth=1, capthick=1)
        # original data points without background subtraction
        eb = ax.errorbar(self.r, self.s+self.bkg,
                         xerr=self.r_err, yerr=self.s_err,
                         fmt="none", elinewidth=1, capthick=1)
        eb[-1][0].set_linestyle("dashdot")
        eb[-1][1].set_linestyle("dashdot")
        # PLC model
        mask_fit = self.mask_extrapolated.copy()
        mask_fit[:len(self.mask)] = self.mask
        r_fit = self.r_extrapolated[mask_fit]
        s_fit = self.plcmodel.f(r_fit)
        ax.plot(r_fit, s_fit, color="black", linestyle="solid")
        # adjust layout
        r_min = 1.0
        r_max = self.r_extrapolated[-1] + self.r_err_extrapolated[-1]
        s_min = min(self.s_extrapolated) / 1.2
        s_max = max(self.s_extrapolated + self.s_err_extrapolated) * 1.2
        ax.set_xscale("log")
        ax.set_yscale("log")
        ax.set_xlim(r_min, r_max)
        ax.set_ylim(s_min, s_max)
        # labels
        ax.set_xlabel("Radius (%s)" % "pixel")
        ax.set_ylabel(r"Surface Brightness (photons/cm$^2$/pixel$^2$/s)")
        ax.text(x=r_max/1.2, y=s_max/1.2,
                s=r"reduced $\chi^2$: %.2f / %.2f = %.2f" % (
                    self.plcmodel.fitted.chisqr, self.plcmodel.fitted.nfree,
                    self.plcmodel.fitted.chisqr/self.plcmodel.fitted.nfree),
                horizontalalignment="right", verticalalignment="top")
        fig.tight_layout()
        return (fig, ax)

    def get_data(self):
        """
        Get the extrapolated data, for following use.
        """
        return np.column_stack([self.r_extrapolated,
                                self.r_err_extrapolated,
                                self.s_extrapolated,
                                self.s_err_extrapolated])

    def save(self, outfile):
        """
        Save the (extrapolated) SBP to the given output file in CSV format.
        """
        df = pd.DataFrame()
        df["radius"] = self.r_extrapolated
        df["radius_err"] = self.r_err_extrapolated
        df["brightness"] = self.s_extrapolated
        df["brightness_err"] = self.s_err_extrapolated
        df["flag_extrapolation"] = self.mask_extrapolated
        flag_fit = np.zeros(self.mask_extrapolated.shape, dtype=bool)
        flag_fit[:len(self.mask)] = self.mask
        df["flag_fit"] = flag_fit
        df.to_csv(outfile, header=True, index=False)


######################################################################


class ChandraPixel:
    """
    Chandra pixel unit conversions.
    """
    angle = 0.492 * au.arcsec
    z = None
    # cosmology calculator
    cosmo = None
    # angular diameter distance
    D_A = None
    # length of one pixel at the given redshift
    length = None

    def __init__(self, z=None):
        self.z = z
        self.cosmo = FlatLambdaCDM(H0=AstroParams.H0,
                                   Om0=AstroParams.OmegaM0)
        if z is not None:
            self.D_A = self.cosmo.angular_diameter_distance(z)
            self.length = self.D_A * self.angle.to(au.radian).value

    def get_angle(self):
        return self.angle

    def get_length(self, z=None):
        if z is None:
            length = self.length
        else:
            D_A = self.cosmo.angular_diameter_distance(z)
            length = D_A * self.angle.to(au.radian).value
        return length


class BrightnessProfile:
    """
    Calculate the electron number density and/or gas mass density profile
    by deprojecting the observed X-ray surface brightness profile and
    incorporating the cooling function profile.

    NOTE:
    * The radii should have unit [ pixel ] (Chandra pixel)
    * The brightness should have unit [ photon s^-1 cm^-2 pixel^-2 ],
      i.e., [ Flux pixel^-2 ] (radial profile column `SUR_FLUX`)
    """
    # allowed density profile types
    DENSITY_TYPES = ["electron", "gas"]
    # input SBP data: [r, r_err, s, s_err]
    r = None
    r_err = None
    s = None
    s_err = None
    # redshift of the source
    z = None
    # `ChandraPixel` instance for unit conversion
    pixel = None
    # flag to indicate whether the units are converted
    units_converted = False

    def __init__(self, sbp_data, cf_data, z):
        self.load_data(data=sbp_data)
        self.load_cf_data(data=cf_data)
        self.z = z
        self.pixel = ChandraPixel(z)

    def load_data(self, data):
        # 4-column SBP: [r, r_err, brightness, brightness_err]
        self.r = data[:, 0].copy()
        self.r_err = data[:, 1].copy()
        self.s = data[:, 2].copy()
        self.s_err = data[:, 3].copy()

    def load_cf_data(self, data):
        # 2-column cooling function profile
        self.cf_radius = data[:, 0].copy()
        self.cf_value = data[:, 1].copy()

    def convert_units(self):
        """
        Convert the units of input data:
           radius: pixel -> cm
           brightness: Flux / pixel**2 -> Flux / cm**2
        """
        if not self.units_converted:
            cm_per_pixel = self.pixel.get_length().to(au.cm).value
            self.r *= cm_per_pixel
            self.r_err *= cm_per_pixel
            self.cf_radius *= cm_per_pixel
            self.s /= cm_per_pixel**2
            self.s_err /= cm_per_pixel**2
            self.units_converted = True

    def get_radius(self):
        return (self.r.copy(), self.r_err.copy())

    def calc_electron_density(self, outfile=None):
        """
        Deproject the surface brightness profile to derive the 3D
        electron number density (and then gas mass density) profile
        by incorporating the cooling function profile.

        unit: [ cm^-3 ] if the units converted for input data
        """
        projector = Projection(rout=self.r+self.r_err)
        s_deproj = projector.deproject(self.s)
        # allow extrapolation
        # XXX: cubic spline / smooth spline better ??
        cf_interp = scipy.interpolate.interp1d(x=self.cf_radius,
                                               y=self.cf_value,
                                               kind="linear",
                                               bounds_error=False,
                                               fill_value=self.cf_value[-1],
                                               assume_sorted=True)
        # emission measure per unit volume
        em_v = s_deproj / cf_interp(self.r)
        ne = np.sqrt(em_v * AstroParams.ratio_ne_np)
        self.ne = ne
        # save results to output if specified
        if outfile is not None:
            ne_profile = np.column_stack([self.r, self.r_err, ne])
            np.savetxt(outfile, ne_profile,
                       header="radius[cm]  radius_err[cm]  " +
                              "electron_number_density[cm^-3]")
        return ne

    def calc_gas_density(self, outfile=None):
        """
        Calculate the gas mass density based the calculated electron
        number density.

        unit: [ g cm^-3 ] if the units converted for input data
        """
        ne = self.calc_electron_density()
        rho = ne * AstroParams.mu_e * AstroParams.m_atom
        self.rho_gas = rho
        # save results to output if specified
        if outfile is not None:
            rho_profile = np.column_stack([self.r, self.r_err, rho])
            np.savetxt(outfile, rho_profile,
                       header="radius[cm]  radius_err[cm]  " +
                              "gas_mass_density[g/cm^3]")
        return rho

    def plot(self, ax=None, fig=None, density_type="electron"):
        if density_type not in self.DENSITY_TYPES:
            raise ValueError("invalid density_types: %s" % density_type)
        if density_type == "electron":
            density = self.ne
            d_name = "Deprojected electron number density"
            d_unit = "cm$^{-3}$"
        else:
            density = self.rho_gas
            d_name = "Deprojected gas mass density"
            d_unit = "g/cm$^3$"
        #
        if self.units_converted:
            # convert from [cm] to [kpc]
            r = self.r * au.cm.to(au.kpc)
            r_err = self.r_err * au.cm.to(au.kpc)
            r_unit = "kpc"
            s_unit = "flux/cm$^2$"
        else:
            r = self.r
            r_err = self.r_err
            r_unit = "pixel"
            s_unit = "flux/pixel$^2$"
        #
        if ax is None:
            fig, ax = plt.subplots(1, 1)
        # SBP data points
        eb = ax.errorbar(r, self.s, xerr=r_err, yerr=self.s_err,
                         fmt="none", elinewidth=2, capthick=2,
                         label="Brightness profile")
        r_min = 1.0
        r_max = max(r + r_err)
        s_min = min(self.s) / 1.2
        s_max = max(self.s + self.s_err) * 1.2
        ax.set_xlim(r_min, r_max)
        ax.set_ylim(s_min, s_max)
        ax.set_xscale("log")
        ax.set_yscale("log")
        ax.set_xlabel("Radius (%s)" % r_unit)
        ax.set_ylabel(r"Surface brightness (%s)" % s_unit)
        # deprojected density profile
        ax2 = ax.twinx()
        line, = ax2.plot(r, density, color="black",
                         linestyle="solid", linewidth=2,
                         label="Density profile")
        d_min = min(density) / 1.2
        d_max = max(density) * 1.2
        ax2.set_xlim(r_min, r_max)
        ax2.set_ylim(d_min, d_max)
        ax2.set_yscale(ax.get_yscale())
        ax2.set_ylabel(r"%s (%s)" % (d_name, d_unit))
        # legend
        handles = [eb, line]
        labels = [h.get_label() for h in handles]
        ax.legend(handles=handles, labels=labels, loc=0)
        fig.tight_layout()
        return (fig, ax, ax2)

######################################################################


def main():
    parser = argparse.ArgumentParser(
            description="Deproject the surface brightness profile (SBP)")
    parser.add_argument("config", nargs="?", default="sbpdeproj.conf",
                        help="config for SBP deprojection " +
                             "(default: sbpdeproj.conf)")
    args = parser.parse_args()

    config = ConfigObj(args.config)
    sbpfit_conf = ConfigObj(config["sbpfit_config"])
    try:
        sbpfit_outfile = sbpfit_conf[sbpfit_conf["model"]]["outfile"]
    except KeyError:
        sbpfit_outfile = sbpfit_conf["outfile"]
    sbpfit_results = json.load(open(sbpfit_outfile))
    sbpdata = np.loadtxt(sbpfit_conf["sbpfile"])
    rc = sbpfit_results["params"]["rc"][0]
    bkg = sbpfit_results["params"]["bkg"][0]

    print("SBP background subtraction and extrapolation ...")
    sbp = SBP(sbpdata)
    rcut = rc * config.as_float("sbpexp_rcut_ratio")
    try:
        rcut = config.as_float("sbpexp_rcut")
    except KeyError:
        pass
    sbp.subtract_bkg(bkg=bkg)
    sbp.extrapolate(rcut=rcut)
    sbp.save(outfile=config["sbpexp_outfile"])
    sbp.report(outfile=config["sbpexp_json"])
    fig = Figure(figsize=(10, 8))
    FigureCanvas(fig)
    ax = fig.add_subplot(1, 1, 1)
    sbp.plot(ax=ax, fig=fig)
    fig.savefig(config["sbpexp_image"], dpi=150)

    # TODO: smooth the extrapolated SBP

    print("SBP deprojection -> density profile ...")
    redshift = config.as_float("redshift")
    cf_data = np.loadtxt(config["coolfunc_profile"])
    sbpdata_extrapolated = sbp.get_data()
    brightness_profile = BrightnessProfile(sbp_data=sbpdata_extrapolated,
                                           cf_data=cf_data,
                                           z=redshift)
    brightness_profile.convert_units()
    brightness_profile.calc_electron_density(outfile=config["ne_profile"])
    brightness_profile.calc_gas_density(outfile=config["rho_gas_profile"])
    fig = Figure(figsize=(10, 8))
    FigureCanvas(fig)
    ax = fig.add_subplot(1, 1, 1)
    brightness_profile.plot(ax=ax, fig=fig)
    fig.savefig(config["density_profile_image"], dpi=150)


if __name__ == "__main__":
    main()