initial commit of likelihood fit for muons

This commit contains code to fit for the energy, position, and direction of
muons in the SNO detector. Currently, we read events from SNOMAN zebra files
and fill an event struct containing the PMT hits and fit it with the Nelder
Mead simplex algorithm from GSL.

I've also added code to read in ZEBRA title bank files to read in the DQXX
files for a specific run. Any problems with channels in the DQCH and DQCR banks
are flagged in the event struct by masking in a bit in the flags variable and
these PMT hits are not included in the likelihood calculation.

The likelihood for an event is calculated by integrating along the particle
track for each PMT and computing the expected number of PE. The charge
likelihood is then calculated by looping over all possible number of PE and
computing:

    P(q|n)*P(n|mu)

where q is the calibrated QHS charge, n is the number of PE, and mu is the
expected number of photoelectrons. The latter is calculated assuming the
distribution of PE at a given PMT follows a Poisson distribution (which I think
should be correct given the track, but is probably not perfect for tracks which
scatter a lot).

The time part of the likelihood is calculated by integrating over the track for
each PMT and calculating the average time at which the PMT is hit. We then
assume the PDF for the photons to arrive is approximately a delta function and
compute the first order statistic for a given time to compute the probability
that the first photon arrived at a given time. So far I've only tested this
with single tracks but the method was designed to be easy to use when you are
fitting for multiple particles.

1 files changed, 76 insertions, 1 deletions

diff --git a/solid_angle.c b/solid_angle.c
index d84ff22..4e44413 100644
--- a/solid_angle.c
+++ b/solid_angle.c
@@ -1,6 +1,81 @@
 #include "solid_angle.h"
 #include <gsl/gsl_sf_ellint.h>
 #include <math.h>
+#include <gsl/gsl_interp2d.h>
+#include <gsl/gsl_spline2d.h>
+
+static double hd[11] = {0.1,0.125,0.150,0.175,0.2,0.3,0.4,0.5,0.6,0.8,1.0};
+static double Rd[13] = {0.1,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4,1.5,2.0};
+
+static double lookupTable[13][11] = {
+{0.9996,0.9996,0.9996,0.9996,0.9996,0.9996,0.9996,0.9996,0.9997,1.0000,1.0003},
+{1.0340,1.0334,1.0325,1.0316,1.0306,1.0257,1.0202,1.0150,1.0108,1.0059,1.0058},
+{1.0836,1.0814,1.0788,1.0758,1.0725,1.0578,1.0429,1.0300,1.0201,1.0091,1.0072},
+{1.1789,1.1718,1.1635,1.1546,1.1452,1.1067,1.0733,1.0480,1.0303,1.0122,1.0083},
+{1.3626,1.3375,1.3107,1.2837,1.2572,1.1669,1.1045,1.0642,1.0388,1.0147,1.0092},
+{1.7390,1.6319,1.5384,1.4590,1.3923,1.2167,1.1212,1.0737,1.0437,1.0163,1.0097},
+{2.2382,1.9036,1.6924,1.5489,1.4458,1.2241,1.1262,1.0742,1.0444,1.0170,1.0101},
+{1.7336,1.5812,1.4741,1.3945,1.3331,1.1850,1.1103,1.0676,1.0418,1.0170,1.0102},
+{1.2190,1.2218,1.2151,1.2032,1.1889,1.1314,1.0876,1.0575,1.0375,1.0165,1.0102},
+{1.0806,1.0920,1.0990,1.1023,1.1026,1.0880,1.0662,1.0471,1.0325,1.0157,1.0102},
+{1.0390,1.0464,1.0523,1.0566,1.0594,1.0591,1.0494,1.0380,1.0279,1.0148,1.0100},
+{1.0222,1.0270,1.0311,1.0345,1.0371,1.0408,1.0371,1.0305,1.0237,1.0138,1.0098},
+{1.0041,1.0051,1.0061,1.0070,1.0078,1.0106,1.0120,1.0122,1.0116,1.0097,1.0084}
+};
+
+static double A(double u, double a, double h)
+{
+    return atan(((a*a-h*h)*u - 2*a*a*h*h)/(2*a*h*sqrt((u-h*h)*(u+a*a))));
+}
+
+double get_solid_angle_approx(double *pos, double *pmt, double *n, double r)
+{
+    /* Returns the approximate solid angle subtended by a circular disk of
+     * radius r at a position `pmt` with a normal vector `n` from a position
+     * `pos`.
+     *
+     * This approximation comes from calculating the solid angle subtended by a
+     * square of equal area, which has a closed form solution. For certain
+     * regimes, the approximation is not very good and so it is modified by a
+     * correction factor which comes from a lookup table.
+     *
+     * This formula comes from "The Solid Angle Subtended at a Point by a * Circular Disk." Gardener et al. 1969. */
+    double dir[3];
+    double h, r0, D, a, u1, u2, F;
+    static gsl_spline2d *spline;
+    static gsl_interp_accel *xacc;
+    static gsl_interp_accel *yacc;
+
+    dir[0] = pos[0] - pmt[0];
+    dir[1] = pos[1] - pmt[1];
+    dir[2] = pos[2] - pmt[2];
+
+    h = fabs(dir[0]*n[0] + dir[1]*n[1] + dir[2]*n[2]);
+    D = sqrt(dir[0]*dir[0] + dir[1]*dir[1] + dir[2]*dir[2]);
+    r0 = sqrt(D*D - h*h);
+
+    a = sqrt(M_PI)*r/2;
+
+    u1 = h*h + pow(r0-a,2);
+    u2 = h*h + pow(r0+a,2);
+
+    F = 1.0;
+    if (h/D > 0.1 && h/D < 1.0 && r/D > 0.1 && r/D < 2.0) {
+        if (!spline) {
+            spline = gsl_spline2d_alloc(gsl_interp2d_bilinear, sizeof(hd)/sizeof(double), sizeof(Rd)/sizeof(double));
+            gsl_spline2d_init(spline, hd, Rd, (double *) lookupTable, sizeof(hd)/sizeof(double), sizeof(Rd)/sizeof(double));
+        }
+        if (!xacc) xacc = gsl_interp_accel_alloc();
+        if (!yacc) xacc = gsl_interp_accel_alloc();
+        F = gsl_spline2d_eval(spline,h/D,r/D,xacc,yacc);
+    }
+
+    if (r0 < a) {
+        return F*(A(u1,a,h) + A(u2,a,h) + M_PI);
+    } else {
+        return F*(A(u2,a,h) - A(u1,a,h));
+    }
+}
 
 double get_solid_angle(double *pos, double *pmt, double *n, double r)
 {
@@ -23,7 +98,7 @@ double get_solid_angle(double *pos, double *pmt, double *n, double r)
     Rmax = sqrt(L*L + (r0+r)*(r0+r));
     k = sqrt(4*r0*r/(L*L + pow(r0+r,2)));
 
-    if (fabs(r0-r) < 1e-9) {
+    if (fabs(r0-r) < 1e-5) {
         /* If r0 is very close to r, the GSL routines below will occasionally
          * throw a domain error. */
         K = gsl_sf_ellint_Kcomp(k, GSL_PREC_DOUBLE);