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VGPM

Eppley-VGPM

original CbPM

updated CbPM

right-click here and use "save as" to download the code


#include <math.h>

/*--------------------------------------------------------------------------*/

double opp_cbpm2( double chl,
		  double bbp,
		  double irr,
		  double k490,
		  double mld,
		  double zno3,
		  double daylength) {
/*

   !Description:     opp_cbpm2 - computes daily primary productivity using a chl:Carbon ratio.  
                     This is a spectrally resolved version of the cbpm, using nine separate
                     wavelengths.  It is also depth resolved, integrating the effects from
                     the surface down to a fixed depth of 200 m.

                     The cbpm2 algorithm estimates productivity using chl (m-1), bbp (m-1), 
		     surface irradiance (Einsteins m-2 d-1), k490 (m-1), mld (m), zno3 (m) 
                     and day length (hours).  

Net primary productivity is carbon * growth rate, where carbon is proportional to particulate
backscatter

	carbon = 13000 * (bbp - 0.00035)

and growth rate is a function of nutrient and temperature stress (f(nut,T) and photoacclimation 
(f(Ig))

	growth rate (u) = umax * f(nut,T) * f(Ig)

where:

	umax = 2

	f(nut,T) = ((Chl/C)sat - y0) / ((Chl/C)max - y0)

	f(Ig) = 1 - exp (-5 * Ig)


and: 

	(Chl/C)sat = ratio of satellite observed chl and carbon (carbon from bbp)

	(Chl/C)max = 0.022 + (0.045-0.022) * exp (-3 * Ig)

	Ig = median mixed layer light level 
	   = surface irradiance * exp (-k(lambda) * MLD/2)

The above items are analyzed for nine separate wavelengths, and is vertically resolved to a depth
of 200 m.

For more details, please see the paper by Westberry, et al (2008)


   !Input Parameters:  
      chl            chlorophyll concentration
      bbp            backscatter
      irr            Photosynthetically available radiation in Einsteins per
                     day per square meter
      k490           diffuse attenuation coefficient at 490 nm (units of 1 / m )
      mld            mixing layer depth in meters
      zno3           depth of the nitrocline
      daylength      length of the day in decimal hours.

   !Output Parameters: 
             Primary productivity in milligrams Carbon per square meter
                     per day

   !Dependencies:
      function austinPetzold_1986 ( double lambda, double K490 )

         given a reference k490 vlaue, determine k(lambda) for a specified lambda

         ref:
            Austin, R. W., and T. J. Petzold (1986), Spectral dependence of the diffuse
            attenuation coefficient of light in ocean waters, Opt. Eng., 25, 473 – 479

   !Revision History:  

   08-16-2010 first release version (Robert O'Malley)
      [original code written in matlab by T. Westberry]

   01-05-2011   O'Malley
      add uMax trap on mu[m]
      correct z_eu determination

   !References and Credits
   
      Westberry, T. Behrenfeld, M.J., Siegel, D.A., and Boss, E.; 2008.  Carbon-based
      primary productivity modeling with vertically resolved photoacclimation.  Global
      Biogeochemical Cycles, Vol. 22, GB2024, doi:10.1029/2007GB003078
      
*/

  double austinPetzold_1986( double, double );

  double uMax;			/* max growth rate */
  double chlCarbonMax;		/* max chl:carbon ration */
  double nutTempFunc;		/* f(nut,T) */
  double chlCarbonSat;		/* satalite chl:carbon ratio */
  double carbon;		/* bbp converted to carbon */
  double IgFunc;		/* f(Ig) */
  double IgFuncz;               /* f(Ig) below the mixed layer depth */
  double z_eu;			/* euphotic depth at 1% light level */
  double npp;                   /* net primary production */

/* --------------------- */
/*   spectral variables  */
/* --------------------- */

  double lambda[] = { 400, 412, 443, 490, 510, 555, 625, 670, 700 };
  double parFraction[] = { 0.0029, 0.0032, 0.0035, 0.0037, 0.0037, 0.0036, 0.0032, 0.0030, 0.0024 };
  double X[] = { .11748, .122858, .107212, .07242, .05943, .03996, .04000, .05150, .03000 };
  double e[] = { .64358, .653270, .673358, .68955, .68567, .64204, .64700, .69500, .60000 }; 
  double Kw[]= { .01042, .007932, .009480, .01660, .03385, .06053, .28400, .43946, .62438 };
  double Kd[9];
  double Kbio;
  double Kdif[9];

  double Klambda[9];
  double Eo[9];
  double Ez_mld[9];
  double par_mld;
  double delChlC;

  double y0;

/* --------------------------- */
/*   depth resolved variables  */
/* --------------------------- */

  double z[200];               /* depths */
  double chl_C[200];           /* chl:c ratio */
  double chlz[200];            /* chl */
  double mu[200];              /* growth */
  double Ezlambda[9][200];     /* fraction of light at nine wavelengths */
  double parz[200];            /* total light */
  double prcnt[200];           /* percent light */
  double Cz[200];              /* carbon */
  double ppz[200];             /* npp */

  int i;
  int m;
  int mzeu;
  double r;
  double prcnt0;
  double prcnt1;
  double z0;
  double z1;
  double numerator;
  double denominator;
  double fraction;
  double deltaZ;

  if(irr <= 0.0){
    return 0.0;
  }

  /* --------------------- */
  /*   initialize values   */
  /* --------------------- */

  z_eu = -9999;     //  1.05.2011
  y0 = 0.0003;                     /* min  chl:c  when  mu = 0 */
  for (i = 0; i<200; i++){
    z[i]= (float)(i+1);
  }
  r = 0.1;
  
  uMax = 2.0;                      /* after BANSE (1991) */
  npp = 0.0;
  mzeu = 0;

  for (i=0; i<9; i++) {
    Klambda[i]= austinPetzold_1986(lambda[i],k490);
    Eo[i] = irr * parFraction[i];
    Ez_mld[i] = Eo[i] * 0.975 * exp(-Klambda[i] * mld / 2.0);
  }

  /* ----------------------------- */
  /*   reintegrate to get par at   */
  /*   depth ...                   */
  /*   do trapezoidal integration  */
  /* ----------------------------- */

  par_mld = 0.0;
  for (i = 0; i < 8; i++ ){
    par_mld += (lambda[i+1]-lambda[i])*(Ez_mld[i+1]+Ez_mld[i])/2;
  }

  par_mld /= daylength;

  IgFunc = 1 - exp(-5.0 * par_mld);

  if(bbp < 0.00035)
    bbp = 0.00036;
  carbon = 13000.0 * (bbp - 0.00035);
  
  chlCarbonSat = chl / carbon;

  if ( chlCarbonSat < y0 ) {
    chlCarbonSat = y0;
  }

  chlCarbonMax = 0.022 + (0.045-0.022) * exp(-3.0 * par_mld);
  delChlC = chlCarbonMax - chlCarbonSat;

  nutTempFunc = (chlCarbonSat - y0) / (chlCarbonMax - y0);

  /* ''''''''''''''''''''''''' */
  /*   calculate Kd offset     */
  /*   carry through to depth  */
  /*   non-chl attenuation     */
  /* ------------------------- */

  for (i=0; i<9; i++) {
    Kbio = X[i] * pow(chl,e[i]);
    Kd[i] = Kw[i] + Kbio;
    Kdif[i] = Klambda[i] - Kd[i];
  }

  /* ''''''''''''''''''''''''''''''''''' */
  /*   integrate down the water column   */
  /*   in one-meter steps                */
  /* ----------------------------------- */

  for (m=0; m<200; m++) {

    /* ---------------------------------------------- */
    /*   if you are in the mixed layer, do this way   */
    /* ---------------------------------------------- */

    if ( z[m] < mld ) {
      chl_C[m] = chlCarbonSat;
      chlz[m] = chl_C[m] * carbon;
      mu[m] = uMax * nutTempFunc * IgFunc;

      if ( mu[m] > uMax ) {     //  1.05.2011
	mu[m] = uMax;           //  1.05.2011
      }                         //  1.05.2011
  
      for ( i=0; i<9; i++) {
	Ezlambda[i][m] = Eo[i]*0.975*exp(-Klambda[i]*z[m]);
      }

      parz[m] = 0.0;
      for (i = 0; i < 8; i++ ){
	parz[m] += (lambda[i+1]-lambda[i])*(Ezlambda[i+1][m]+Ezlambda[i][m])/2;
      }

      Cz[m] = carbon;

    } else {

      /* '''''''''''''''''''''''''''''''''''''''''''''''''''''''''' */
      /*   if below mixed layer must treat properties differently   */
      /* ---------------------------------------------------------- */

      for (i=0; i<9; i++) {
	Kbio = X[i] * pow(chlz[m-1],e[i]);     /*  after Morel & Maritorena (2001)  */
	Kd[i] = Kw[i] + Kbio;
	Kd[i] += Kdif[i];
	Ezlambda[i][m] = Ezlambda[i][m-1]*exp(-Kd[i]*1.0);
      }

      parz[m] = 0.0;
      for (i = 0; i < 8; i++ ){
	parz[m] += (lambda[i+1]-lambda[i])*(Ezlambda[i+1][m]+Ezlambda[i][m])/2;
      }

      deltaZ = zno3 - z[m];
      if ( deltaZ < 0 ) {
	deltaZ = 0;
      }

      chl_C[m] = (0.022 + (0.045-0.022) * exp(-3.0 * parz[m] / daylength));
      chl_C[m] -= delChlC * (1-exp(-0.075*deltaZ));

      IgFuncz = 1 - exp(-5.0 * parz[m]/daylength);
      mu[m] = uMax * nutTempFunc * IgFuncz;

      if ( mu[m] > uMax ) {     //  1.05.2011
	mu[m] = uMax;           //  1.05.2011
      }                         //  1.05.2011
  
      if (mu[m-1] >= r ) {
	Cz[m] = carbon;
      } else {
	Cz[m] = carbon * mu[m-1] / r;
      }

      chlz[m] = chl_C[m] * Cz[m];

    }
	   
    prcnt[m] = parz[m] / (irr * 0.975);

    /*  track this to get to the euphotic depth  */

    if ( prcnt[m] >= 0.01 ) {
      mzeu = m;
    } else {

      /* ''''''''''''''''''''''''''' */
      /*   now find 1% light depth   */
      /*   in case the user wants    */
      /*   to use this information   */
      /* --------------------------- */

      if (z_eu == -9999 ) {     // 01.05.11
        prcnt0 = prcnt[mzeu];
        prcnt1 = prcnt[mzeu+1];
        z0 = z[mzeu];
        z1 = z[mzeu+1];
        numerator = prcnt0 - 0.01;
        denominator = prcnt0 - prcnt1;
        fraction = numerator / denominator;
        z_eu = z0 + (z1-z0)*fraction;
      }
    }

    ppz[m] = mu[m] * Cz[m];

  }

  /* ------------------------------- */
  /*   do trapezoidal integration    */
  /*   from m = 0 to m = 200         */
  /* ------------------------------- */

  //  note:  186 m is the euphotic depth for pure water

  if ( mzeu < 186 ) {
    npp = 0;
    for (i = 0; i < 199; i++ ){
      npp += (z[i+1]-z[i])*(ppz[i+1]+ppz[i])/2;
    }
  } else {
    npp = -9999;
  }

  return npp;
}

/* =================================================================  */

double austinPetzold_1986 ( double lambda,
                             double K490 ) {

  double wave[] = { 350, 360, 370, 380, 390, 400,
                    410, 420, 430, 440, 450, 460, 470, 480, 490, 500,
                    510, 520, 530, 540, 550, 560, 570, 580, 590, 600,
                    610, 620, 630, 640, 650, 660, 670, 680, 690, 700 };

  double M[] = { 2.1442, 2.0504, 1.9610, 1.8772, 1.8009, 1.7383,
		 1.7591, 1.6974, 1.6108, 1.5169, 1.4158, 1.3077, 1.1982, 1.0955, 1.0000, 0.9118, 
		 0.8310, 0.7578, 0.6924, 0.6350, 0.5860, 0.5457, 0.5146, 0.4935, 0.4840, 0.4903, 
		 0.5090, 0.5380, 0.6231, 0.7001, 0.7300, 0.7301, 0.7008, 0.6245, 0.4901, 0.2891 };

  double Kdw[] = { 0.0510, 0.0405, 0.0331, 0.0278, 0.0242, 0.0217, 
		   0.0200, 0.0189, 0.0182, 0.0178, 0.0176, 0.0176, 0.0179, 0.0193, 0.0224, 0.0280, 
		   0.0369, 0.0498, 0.0526, 0.0577, 0.0640, 0.0723, 0.0842, 0.1065, 0.1578, 0.2409, 
		   0.2892, 0.3124, 0.3296, 0.3290, 0.3559, 0.4105, 0.4278, 0.4521, 0.5116, 0.6514 };

  double l0;
  double l1;
  double k0;
  double k1;
  double m0;
  double m1;
  double kdiff;
  double mdiff;
  double num;
  double den;
  double frac;
  double Kdw_l;
  double M_l;
  double Kd;

  int ref;
  int i;

  // -- INTERPOLATE TO WAVELENGTH OF INTEREST --  //

  for (i = 1; i < 36; i++) {
    if ( wave[i] >= lambda ) {
      l1 = wave[i];
      k1 = Kdw[i];
      m1 = M[i];
      l0 = wave[i-1];
      k0 = Kdw[i-1];
      m0 = M[i-1];
      break;
    }
  }

  num = lambda - l0;
  den = l1 - l0;
  frac = num / den;

  kdiff = k1 - k0;
  Kdw_l = k0 + frac*kdiff;

  mdiff = m1 - m0;
  M_l = m0 + frac*mdiff;
  

  // -- GET REFERENCE WAVELENGTH (=490 FOR NOW) AND APPLY MODEL -- //

  ref = 14;

  Kd = (M_l/M[ref]) * (K490 - Kdw[ref]) + Kdw_l;

  return Kd;

}

Last modified: 13 January 2015
by: Robert O'Malley

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