Costfunction

be the number of rivers taken into account and the relative contribution of a river isri with

ri ²[0 : 1] and ^X

i r= 1 (2.58)

The relative contribution of the rivers can be set for organic carbon, inorganic car-bon, and silicate seperately. According to the redeld ratios, the additional source terms in boxes near riversnr become:

q(nr;i)N;P;C;O;Si;TALK =ri ^X

columnsJ(^S)N;P;C;O;Si;TALK (2.59)

2.7.5 Comment on the constraints of model particle uxes

The amount of data available associated with a given parameter determines how well the parameter is constrained. All model particle uxes determined with the adjoint model depend on data of dissolved nutrients, no additional a priori knowl-edge is regarded. As can be seen in the denitions of biogeneous particle uxes, organic carbon is constrained by phosphate, nitrate, oxygen, total carbon, and al-kalinity. C^org particle formation and remineralization is determined using the total information about sources and sinks of these properties in the water column. The relation of data density to the number of parameters to be optimized is thus quite good for organic carbon. Silicate measurements are also quite common but the constraint for Opal uxes is vaguer compared to organic carbon. The parameter weakest constrained are export production, remineralization, and accumulation of CaCO3. Measurements of total carbon and alkalinity are rare and not evenly dis-tributed over the world ocean. This should be kept in mind when interpreting the results presented later on and when comparing these results with other models. The dynamic model of Maier-Reimer (1993) assumes deterministic ratios of shell mate-rial to soft tissue (^Opal+CaCO_Corg ³) with limitations to Opal production (availability of silica). This gives a much stronger constraint forCaCO3 uxes because data of dis-solved silicate are used to determine calcite formation. In the model presented here, all particle uxes are calculated on the basis of data of the constituents involved in particle formation and remineralization only.

2.8 Costfunction

Once all parameters are set in the advection/diusion matrix and for source terms in Equation (2.17), the system is solved for all properties of interest. This gives the model distributions of vertical velocities, temperature, salinity, oxygen, carbon dioxide, and dissolved nutrients (all dependent parameters ~p). Once the model state (the complete vector p = [p;p~]) is determined, the solution is evaluated by calculating the costfunction F(p;p~). The costfunction F(p;p~) is a scalar function of all model parameters and might contain any undesired feature formulated in terms of functions of independent and dependent parameters. These features must not be

32 The adjoint model fullled exactly but 'as good as possible' and terms in F(p;p~) are consequently named 'soft constraints'. How close a particular constraint is fullled by the model can be adjusted by multiplying the individual terms by weight factors.

2.8.1 Terms of costfunction

F(p;p~)

All terms of the costfunction used in this study are described below. Most terms of the costfunction are discussed in detail in earlier publications of Schlitzer (1993, 1999). Here, only a short review of terms used in this study is given. New and/or rened terms of the costfunction are marked ^y.

1.

Deviations from initial geostrophic shear

Outside the equatorial band (10S ^,10N), the vertical shear uz = @u=@z and vz = @v=@z of the horizontal velocities is required to be close to original geostrophic shear u_z = @u=@z and v_z = @v=@z computed from geostrophic ow calculations. p is a spatially varying weight factor depending on statis-tical information (quality indicator) and is a normalization factor.

X

2.

Deviations of mixing coecients from 'mixing coecient data'

Mixing coecients are kept close to 'mixing coecient data' from literature and/or earlier model calculations especially performed to determine best

val-ues. ^"

pk^,Pk

P^k

#2

(2.61) 3.

Pointwise deviations from data

For data of temperature, salinity, oxygen, phosphate, nitrate, silicate, total carbon, and total alkalinity, deviations of model elds Xmod (or dependent parameters ~p) to measurements Xdata are computed pointwise. denotes measurement errors. 4.

Systematic deviations from data

For the same properties, the bias is calculated by computing the systematic deviation within the neighborhood of a box.

X

2 j[next neighbors] (2.63)

5.

Deviations from sediment trap data

^y

Particle uxes are now explicitly calculated in the model. Flux parameters

2.8 Costfunction 33

;;s give uxes for the whole model domain and in boxes where trap data exist, the deviations to data are calculated. A detailed discussion of sediment trap dataJdataand model uxesJmodis given in Section 4. The general form of this term is analogous to other terms penalizing deviations to data. denotes measurement error.

X

Jmod^,Jdata

J

2

(2.64) 6.

Smoothness constraints (linear)

The second derivative of parametersp and ~pis used to enforce spatial smooth-ness of vertical velocities ~w, surface heat uxes, and gas exchange rates. Here, pdenotes the respective parameter and pe;pw;pn;ps are the parameter values of the next neighbors in eastern, western, northern, and southern direction, respectively. X

[(pe^,2p+pw)²+ (ps^,2p+pn)²] (2.65) 7.

Smoothness constraints (squared)

^y

The second derivative is used to enforce spatial smoothness of biogeochemical parameters. The parameters are squared prior penalizing the second deriva-tive because all biogeochemical parameters contribute with their square value to the model elds. This simply reects the 'true' appearance of export pro-duction, remineralization, and accumulation.

X[(p_2e^,2p²+p_2w)²+ (p_2s^,2p²+p_2n)²] (2.66) 8.

Deviations from a priori values (global)

For all independent parameters p deviations from a priori knowledge can be penalized (i.e., if one knows that a parameter value must be close to a certain numberP).

X

p^,P P

4

(2.67) 9.

Deviations from a priori volume transports

Deviations from a priori volume transports are penalized by integrating the horizontal velocities over prescribed surfaces and calculating the dierences to a priori transports^T.

(^X

i ~uAi^,Ti)²+ (^X

j ~vAj^,Ti)² (2.68)

2.8.2 Weighting of costfunction

F(p;p~)

F(p;p~) is a measure for the 'quality' of a model solution with respect to all indi-vidual terms. A large value of F(p;p~) indicates that the model solution does not fulll the features desired by the penalty terms. The individual terms of F(p;p~) are multiplied by weight factors to force the model in special directions. Table 2.3

34 The adjoint model summarizes weight factors used in this study. The weights are very dierent, rang-ing from 0.1 to 10⁶. This does not mean that, for instance, a priori transports are much more important than salinity data. The minimization respects the absolute value of weight factor multiplied with the penalty term. Low weight factors for deviations to data are mostly compensated by a high density of measurements. The absolute values are within the same range. Most of the weight factors were kept constant throughout all model runs. Changes due to the variation of weight factors are discussed in Section 4.2.

Term Weight Notes

Dev. from initial geostr. shear 0.01

Smoothness of ~w 1.0

Dev. from a priori vol. transports 10⁶ See Table 2.4

Dev. from initial pK^h 10⁴ Initial mixing coecients were Dev. from initial pK^v 10⁴ taken from Schlitzer (1995) Dev. from heat-ux data Qdata 10

Smoothness of heat-ux Qmod 2

Smoothness of parameter ^Corg 1, var Dierent weight factors were Smoothness of parameter ^Corg 1, var used to smooth out export pro-Smoothness of parameter ^CaCO3 1, var duction and remineralization Smoothness of parameter ^Corg 1, var in experiments SLAT and Smoothness of parameter ^Opal 1, var SLANT (see Section 2.9) Smoothness of parameter ^Opal 1, var

Dev. from data (Xmod^,Xdata) 0.1 Modeled properties

Bias of Xmod^,Xdata 0.1 T;S;O2;PO4;NO3;Si;TALK Dev. from ^PCO2 data 0.5 Weight factor high because Bias of ^PCO2;mod^,PCO2;data 0.5 data density of ^P_CO² is low Dev. from TALK data 0.5 Weight factor high because Bias of TALKmod^,TALKdata 0.5 data density of TALK is low Dev. from trap data Corg var Dierent weight factors were Dev. from trap data CaCO3 var used in the experiments, Dev. from trap data Opal var see Section 2.9

Table 2.3: Weight factors in costfunction F(p;p~). Var: variable, see text

Prescribed top-to-bottom ows were the same for all experiments and are summa-rized in Table 2.4.

Im Dokument vertical particle uxes (Seite 38-42)