Replace trapezoid rule for energy integration of background template by numerica…

…l integration in ctskymap (#1975)

Replace trapezoid rule for energy integration of background template by numerica…
…l integration in ctskymap (#1975)
Jürgen Knödlseder
1 parent 8321269e
Showing 4 changed files with 69 additions and 32 deletions
ChangeLog
NEWS
src/ctskymap/ctskymap.cpp
src/ctskymap/ctskymap.hpp
-2017-03-18
+2017-03-21
  
         * Version 1.3.0 released
         ========================
  
-        Take into account RoI in ctskymap (#1975)
+        Take into account RoI in ctskymap and use numerical integration for IRF (#1975)
         Set local language to english (#1370)
         Enhance logfile output of csroot2caldb script
         Set default deadc value to 0.98 (CTA has <2% deadtime)
 New Features and Important Changes in ctools 1.3.0.dev1
  
-18 March 2017
+21 March 2017
  
  
 Introduction
@@ -53,7 +53,10 @@ None
 ctskymap - CTA sky mapping
 --------------------------
 Any RoI is now taken into account in ctskymap so that if IRF background
-subtraction is requested the counts and background maps are coherent
+subtraction is requested the counts and background maps are coherent. In
+addition, the energy integration over the background template is now done
+numerically so that strong variations in the background rate, such as
+those occuring close to the threshold, are correctly taken into account
 (#1975).
  
  
@@ -30,7 +30,6 @@
 #endif
 #include <cstdio>
 #include "ctskymap.hpp"
-#include "GTools.hpp"
  
 /* __ Method name definitions ____________________________________________ */
 #define G_INIT_MAP                 "ctskymap::init_map(GCTAObservation* obs)"
@@ -584,7 +583,8 @@ void ctskymap::map_background(GCTAObservation* obs)
  *            No background template available in instrument response function.
  *
  * Estimates the background in the sky map using the IRF template for a given
- * observation and adds this estimate to the background sky map.
+ * observation and adds this estimate to the background sky map. The IRF
+ * template is integrated numerically in energy.
  ***************************************************************************/
 void ctskymap::map_background_irf(GCTAObservation* obs)
 {
@@ -612,24 +612,16 @@ void ctskymap::map_background_irf(GCTAObservation* obs)
         throw GException::invalid_value(G_BKG_SUBTRACT_IRF, msg);
     }
  
-    // Setup energy range covered by data
-    GEnergy  emin;
-    GEnergy  emax;
-    GEbounds ebds;
-    emin.TeV(m_emin);
-    emax.TeV(m_emax);
-
-    // Determine number of energy nodes, 20 per decade, at least 10 bins
-    int nebin = int((emax.log10TeV() - emin.log10TeV()) * 20.0);
-    if (nebin < 10) {
-        nebin = 10;
-    }
+    // Compute natural logarithm of energy range in MeV
+    double lnEmin = std::log(m_emin * 1.0e6);
+    double lnEmax = std::log(m_emax * 1.0e6);
  
     // Extract region of interest from observation
     GCTARoi roi = obs->roi();
  
-    // Setup energy nodes for computation
-    GEnergies energies(nebin, emin, emax); // More complicated nbins later
+    // Initialise statistics
+    int    calls = 0;
+    double total = 0.0;
  
     // Loop over all background map pixels
     for (int i = 0; i < m_bkgmap.npix(); ++i) {
@@ -646,28 +638,57 @@ void ctskymap::map_background_irf(GCTAObservation* obs)
             continue;
         }
  
-        // Compute background value for each energy node
-        std::vector<double> values(energies.size(), 0.0);
-        for (int k = 0; k < energies.size(); ++k) {
-            values[k] = (*bkg)(energies[k].log10TeV(), instdir.detx(),
-                                                       instdir.dety());
-        }
+        // Setup integration function
+        ctskymap::irf_kern integrand(bkg, &instdir);
+        GIntegral          integral(&integrand);
  
-        // Integrate background values according to Trapezoid rule
-        double value = 0;
-        for (int k = 1; k < energies.size(); ++k) {
-            value += 0.5 * (values[k] + values[k-1]) *
-                           (energies[k].MeV() - energies[k-1].MeV());
-        }
+        // Set precision (has been carefully adjusted using a test simulation
+        // over the energy range 20 GeV - 120 TeV)
+        integral.eps(1.0e-7);
  
+        // Do Romberg integration
+        double value = integral.romberg(lnEmin, lnEmax);
+
+        // Update number of background function calls
+        calls += integral.calls();
+        
         // Multiply background rate with livetime and solid angle
         value *= obs->livetime() * m_bkgmap.solidangle(i);
  
         // Add number of background events to background map
         m_bkgmap(i,0) += value;
  
+        // Update total number of background events
+        total += value;
+
     } // endfor: looped over background map pixels
+  
+    // Log background subtraction results
+    log_value(NORMAL, "Events in background", int(total+0.5));
+    log_value(NORMAL, "Background evaluations", calls);
  
     // Return
     return;
 }
+
+
+/***********************************************************************//**
+ * @brief Estimates the background in sky map based on IRF template
+ *
+ * @param[in] lnE Natural logarithm of energy in MeV.
+ ***************************************************************************/
+double ctskymap::irf_kern::eval(const double& lnE)
+{
+    // Get log10 of energy in TeV
+    double logE = lnE * gammalib::inv_ln10 - 6;
+
+    // Get function value
+    double value = (*m_bgd)(logE, m_dir->detx(), m_dir->dety());
+
+    // Correct for variable substitution
+    value *= std::exp(lnE);
+
+    // Return value
+    return value;
+}
+
@@ -73,6 +73,19 @@ protected:
     void map_background(GCTAObservation* obs);
     void map_background_irf(GCTAObservation* obs);
  
+    // Background integration kernel
+    class irf_kern : public GFunction {
+    public:
+        irf_kern(const GCTABackground* bgd,
+                 const GCTAInstDir*    dir) :
+                 m_bgd(bgd),
+                 m_dir(dir) { }
+        double eval(const double& lnE);
+    protected:
+        const GCTABackground* m_bgd;   //!< Pointer to background
+        const GCTAInstDir*    m_dir;   //!< Pointer to instrument direction
+    };
+
     // User parameters
     GFilename     m_outmap;      //!< Output file name
     double        m_emin;        //!< Minimum energy (TeV)