SOFTWARE METHODS FOR THE 3-D ANALYSIS

The operations involved in the procedure (Figs.2 and 3) can be summarized as follows:

STEP a

Selection of ROI, according to the availability of SARAGO (or STD) data (water samples and their respective density and quality) and to the temporal window of sampling (time/date information), in order to attain the best synopticity in the selected area

STEP b

Selection of the grid and depth steps ($dx,dy,dz$) for the volume integration and definition of the function intervals

STEP c

Definition of the base levels of the data and generation of the grid that will mask the 3-D function down to the final depth

STEP d

extraction of the data with the criterion of A and production of the 2-D grid for every selected depth interval as for B, mask with C and clip with ROI of A

STEP e

calculation of the volume integral of the masked and clipped 3-D function for any selected interval of salinity (or temperature, or other physical property) .

figure

Figure 2: Idealized sketch of the grid (with steps dlon,dlat) used for the 3-D function analysis (black lines), the encompassed area which of integration that is defined by the polygonal $lon_i,lat_i$, for $i=0,3$ (red box) and the SARAGO trajectories of the U2 cruise (blue lines).

figure

Figure 3: Idealized sketch for the 3-D function analysis.$dx,dy$ define the grid which is calculated for every $n^{th}~ dz$ slice down to $z_{max}$.

In a mathematical form we state that, given the 3-D function $f(x,y,z)$, represented by the $z_{max}$ rectangular grids defined by $lon_{min}, lat_{min}(0,0), lon_{max}, lat_{max}(nx,ny)$ of Fig.2 and 3, we can subdivide it categorically such as:

$$f(x,y,z) ~ = ~\cases{ f_0(x,y,z) ~~ \{ k_0 \leqslant f_0 \l... ...eqslant f_n \leqslant f_n ; ~\forall ~ x,y,z \in D ~ \} \cr }$$ (1)

where the $k_n$ parameters define the function intervals and where

$$D ~ = ~\{ x,y,x \in A : f(A) \ne NaN ~ \}$$ (2)

is the set of points that are delimited by the polygon defined by $lon_i,lat_i$ (Fig.2) and $x_i,y_i$ (Fig.3) for $i=0,n$, in this case $n=3$.

The volume integral for each $n^{th}$ interval of $f(x,y,z)$ is therefore calculated as:

$$F_n ~ = ~ \int \int\limits_D \int f_n(x,y,z) \, dx \, dy \, dz$$ (3)

The procedure was performed on a GNU/Linux computer using the Perl and C programming language and some of the GMT suite of programs [10],[11]. SQL queries provided access to the SARAGO position ( Lat, Lon, Depth), time, salinity and temperature data.

The core of the procedure is the gridding, filtering and masking of the data (ROI and final depth surface) (STEP D above).

We used GMT's surface, a global, adjustable tension continuous curvature surface gridding algorithm [12], and grdfilter, whereas the ROI masking was done with grdmath. The volume calculations in all the cases of Eq.1, was then performed on the masked surfaces by the sum of the non NaN values on every depth slice, using GMT's grd2xyz. The program is also able to output binary volume data which can be processed with other software for visualization, such as IDL or MATLAB. In addition to this, the IGM topographic library Tplib was used, alongwith a C program to interface a MySQL database and get data on output according to an SQL query on the command line. The PERL script listed in Appendix was developed to make the procedure as automated as possible. The only input required are the operator's choices of the geographical box, by providing a two point segment and left or right offset (to define a rectangular box), and a fine tuning on data (particularly on the base levels). Other parameters control the spacing of the grids, the filter parameters, etc. Even though not tested, we believe that the procedure could be easily tailored to the input of other data, such as STD samples, with the contraints and limitations due to the much lower degree of sinopticity.

The procedure is suitable to be performed in other computing environments like WINDOWS, provided that the GPL'ed GMT, Perl and C compiler and PostScript viewers like Ghostscript and ghostview are available.