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%function sigmabar.m
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%Compute the individual effective mean Backscattering Cross Section
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% and the global Scattering Cross Section (for the scattering on a particle integrated in all directions)
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% !!! here, independent of the vertical axis and written for ONE mean size of particles a_s !!!
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%choicePart= 1 : mineral
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% 2 : organic (biological o detritical)
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% 3 : user defined
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%choiceMod= 1 : model of Thorne et al. 2002 (calibrated for sand particles)
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% 2 : model Tessier 2006, constructed from Stanton 1998 (fluid sphere, non rigid particles)
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% f [kHz] : nominal frecuency of ADCP
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% a_s [m] : radius of particle
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% rho_s [kg/m3]: density of particle
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% c_s [m/s] : sound velocity in particle
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%------------------------------------------------------------------------------
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function [sigbar,sigbartot]=sigmabar(f,a_s,rho_s,c_s,choiceMod)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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c0=1505;rho0=1024; %(T=15deg,S=34psu)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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k=2*pi/(c0/f/1000);
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ka=k.*a_s;
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ka2=ka.^2;
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ka4=ka.^4;
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%%
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% *
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% switch choicePart
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% case 1 %mineral---------------------------
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% %c1=3800;%e=16.1551;
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% c1=4500;
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% rho1=2650;
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% % c1=1680;
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% % rho1=1130;
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% g=rho1./rho0;
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% h=c1/c0;
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% e=g*h^2;
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% case 2 %zooplankton, Stanton 1998)---------
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% % c1= ;
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% rho1=rho0*1.04;
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% e=1.12;
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% g=rho1/rho0;
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% h=sqrt(e/g); %h=1.0377;
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%user defined ----------------------
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c1=c_s;
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rho1=rho_s;
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g=rho1./rho0;
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h=c1/c0;
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e=g*h^2;
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% end
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A=(e-1)/(3*e)+(g-1)./(2*g+1); %A=0.57
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R=(g*h-1)/(g*h+1); %R=0.73
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B=((e-1)/(3*e))^2+1/3*((g-1)./(2*g+1))^2; %B=0.12
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%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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switch (choiceMod)
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case 1 %Thorne et al., 1993, 2002
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%model calibrated for sand particles
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Kalpha43=2*B; %(Kalpha=0.18 sables; Kalpha=0.002 zoo)
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Kf=2*A; %(Kf=1.14 sables ; Kf=0.0779 zoo)
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C0=1.1*( 1-0.25*exp(-( (ka-1.4)./0.5 ).^2) ).*( 1+0.37*exp(-( (ka-2.8)./2.2 ).^2) ); %for sand mixture
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%C0=1.1; %for zoo?
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%to compute the SCATTER INDEX :
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%Fm=FORM function
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Fm=C0.*(Kf*ka2)./(1+Kf*ka2);
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%L=Scatter LENGHT
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L=Fm.*a_s/2;
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%Backscattering cross section
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sigbar=L^2;
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%to compute sediment attenuation :
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%XHI=NORMALISED TOTAL scattering cross section
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XHI=(Kalpha43.*ka4)./(1+1.3*ka2+Kalpha43.*ka4);
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%sig=TOTAL scattering cross section
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sigbartot= (2*pi*a_s.^2).*XHI;
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case 2 %Tessier 2006, constructed from Stanton, 1998
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%for non rigid (fluid) spheres (mineral o organic)
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xi=2*sqrt(2)*A/R;
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sigbar=(A^2).*(a_s.^2).*(ka4./(1+xi.*ka2+(2*(A^2)/R^2)*ka4));
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%asymptotes:
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sigR=A^2*(a_s.^2).*ka4; %Rayleigh Backscattering
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sigG=R^2*(a_s.^2)/2; %Geometric Scattering
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xitot=2*sqrt(2*B)/R;
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sigbartot=4*pi*B*a_s.^2.*(ka4./(1+xitot.*ka2+(2*B/R^2)*ka4));
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end
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