detection/fuel_left_cell.m

   1 function varargout=fuel_left_cell(t,fuel_time)
   2 % frac=fuel_left(t,fuel_time)
   3 % [frac,dfracdt]=fuel_left(t,fuel_time)
   4 % [frac,dfracdt,dfrac]=fuel_left(t,fuel_time)
   5 %
   6 % compute fuel fraction left in one cell
   7 %
   8 % in:
   9 %   t           fire arrival time (0=now) at corners, size (2,2)
  10 %   fuel_time   time the fuel takes to burn to 1/e = 0.36
  11 % out:
  12 %   frac        the fuel fraction
  13 %   dfracdt     derivative the all components of t change by the same
  14 %   dfrac       partial derivatives wrf to components of t
  15
  16 % to test:
  17 % t=[-1,0;0,1.1], s=0.1, f=1, dt=0.01
  18 % p1=fuel_left_cell(t-dt-s,f);p2=fuel_left_cell(t+dt-s,f);
  19 % [p,dp]=fuel_left_cell(t-s,f);d=(p2-p1)/(2*dt);err=d-dp
  20
  21 ps=ssum(t);
  22 aps=ssum(abs(t))+realmin;
  23 area=0.5*(1-ps/aps);
  24 t0=min(t,0);
  25 ta=0.25*ssum(t0);
  26 frac=area*exp(ta/fuel_time) + (1. - area);
  27
  28 varargout(1)={frac};
  29
  30 if nargout < 2, return, end
  31
  32 % derivative wrt same increment in all 4 values of t
  33 dpsdt=4;
  34 dapsdt=ssum(sign(t));
  35 dareadt=0.5*(dapsdt*ps-dpsdt*aps)./(aps.*aps);
  36 dtadt=0.25*ssum(0.5*(1-sign(t)));
  37 dfracdt=(dareadt+(area/fuel_time)*dtadt)*exp(ta/fuel_time)-dareadt;
  38
  39 varargout(2)={dfracdt};
  40
  41 if nargout < 3, return, end
  42
  43 % for heat flux, need derivative wrt same increment
  44 dps=ones(2);
  45 daps=sign(t);
  46 darea=0.5*(daps*ps-dps*aps)./(aps.*aps);
  47 dta=0.25*(0.5*(1-sign(t)));
  48 dfrac=((area/fuel_time)*dta+darea)*exp(ta/fuel_time)-darea;
  49
  50 varargout(3)={dfrac};
  51
  52 end