Using sigma and curvilinear model coordinates
Question:
How do I handle sigma coordinate output in Ferret?
How do I handle models based on curvilinear coordinates?
Example:
Explanation:
Ferret provides both graphical methods and analytical methods for dealingwith sigma (and curvilinear) coordinates. The analytical methods involveregridding the sigma coordinates to a grid based on a true depth axis.The results of this method can then be combined in calculations with otherdepthgridded fields (The same procedures apply to pressuregridded fields.)The graphical methods are "faithful" in the sense that they donot involve regridding the data to another grid.
In this FAQ we will show two examples:
I) Open a dataset containing model data that uses a sigmacoordinate "stretching variable". From the datset define variables to use for two techniques, a purely graphical one, and a second which regrids the data to a fixed Z axis.
Start by defining the "depth" functionas the product of a bathymetry variable and the sigmaz stretching variable.
We will then demonstrate two methods for working with the data:
 purely graphical method using 3argumentSHADE and FILL commands
 Z axis replacement technique using ZAXREPLACE
II) begin by:creating an artificial sigma coordinate dataset.
define the "depth" functionas the vertical integral of layer thickness and produce some referenceplots from it. (In your application you may already have a depth variable.)
We will then demonstrate three methods for working with the data:
 purely graphical method using 3argumentSHADE and FILL commands
 Z axis replacement technique using ZAXREPLACE
 older technique based on the @WEQ transformation (prev4.9).
Open a sigma coordinate data set:
Your model data might look like this:
yes? use "http://address/thredds/dodsC/directory/roms_forec/ROMS_Ocean_Model_best.ncd"
yes? show dat
currently SET data sets:
1> http://address/thredds/dodsC/directory/roms_forec/ROMS_Ocean_Model_best.ncd (default)
name title I J K L M N
HC Scoordinate parameter, critica ... ... ... ... ... ...
CS_R Scoordinate stretching curves ... ... 1:30 ... ... ...
H bathymetry at RHOpoints 1:294 1:194 ... ... ... ...
SALT salinity 1:294 1:194 1:30 1:6626 ... ...
TIME_RUN run times for coordinate = time ... ... ... 1:6626 ... ...
Have a look at the attributes of the salinity variable
yes? show attr salt
attributes for dataset: http://address/thredds/dodsC/directory/roms_forec/ROMS_Ocean_Model_best.ncd salt.units = 1e3 salt.long_name = salinity salt.time = ocean_time salt.coordinates = time_run time s_rho lat_rho lon_rho salt.field = salinity, scalar, series salt._FillValue = NaN salt.standard_name = sea_water_salinity
The coordinates attribute tells us that the Z axis is to be given by the s_rho variable.
Define the "depth" function, and plot with a purely graphical method
The variable H is the bathymetry on the XY grid of our variable. We can define depth using the bathymetry and the sigmacoordinate stretching variable. Depth is the depth in meters, at the XY grid of SALT, and at each unitless z level of the grid of SALT.
Now use the 3 argument SHADE and FILL commandsthat directly plot data fields on "curvilinear coordinates". Todisplay the field we need only create multidimensional fields specifyingthe horizontal and vertical positions associated with each data point tobe plotted.
yes? let depth = cs_r* h
yes? shade/y=100/L=1 salt[l=1], lon_rho, depth
Z axis replacement technique (using the ZAXREPLACE() function)
Define a Z axis and use ZAXREPLACE to regrid the data to this axis. Note: this demo is graphical but the procedure is valid for analysis as well.
This is the z axis that is used in the Levitus Climatology dataset. You can define any Z axis, with depth in meters matching the units of variable h.
yes? define axis/z/units=meters/depth zaxlevitr = {0,10,20,30,50,75,100,150,200,300,400,600,800,1000,1200,1500,2000,3000,4000,5000}
Variable cs_r has negative values between 1 and 0, and our z axis refers to depth as positive down, so we'll change our definition of the depth variable, making its values positive depths as well.
yes? list cs_r VARIABLE : Scoordinate stretching curves at RHOpoints DATA SET : Regional Ocean Modeling System (ROMS): FILENAME : ROMS_Ocean_Model_best.ncd FILEPATH : http://address/thredds/dodsC/directory/roms_forec/ SUBSET : 30 points (Z) 0.983 / 1: 0.9802 0.95 / 2: 0.9260 0.917 / 3: 0.8593 0.883 / 4: 0.7862 0.85 / 5: 0.7109 0.817 / 6: 0.6366...
yes? let depth = 1* cs_r* h
yes? let/title="`salt,return=title`"/units="`salt,return=units`" salt_on_depth = zaxreplace (salt, depth, z[gz=zaxlevitr])
yes? fill/Y=100/L=1 salt_on_depth ! The image below
yes? shade/z=10/L=1 salt_on_depth, lon_rho, lat_rho ! a single depth in meters, curvilinear in longitude/latitude
OR, Create an artificial sigma coordinate data set:
(Only needed for demonstration purposes)
Create variables to define a bottom bathymetry on a grid and a verticalsigma coordinate system of 10 layers. We will invent a (nonphysical) flowfield in a channel.
The variables we create will be
 bathymetry: the bottom bathymetry for the model
 a channel with a rise along the axis of the channel
 h: layer thickness
 varies in X, Y, Z, and T in this example
 flow: the flow field

faster near surface

faster at midchannel than near the edges

speeds up over the rise

speed increases with sinusoidal variation in time
See the Ferret script sigma_coordinate_demo.jnl for details of the variable definitions.
Define the "depth" function
The depth of each (x,y,k) grid point is computed by integrating h.We subtract h/2 because we want the depth of the midpoint of thelayer.
let depth = h[k=@rsum]h/2
set variable/title="DEPTH function"/unit=meters depth
Solutions:

purely graphical method

Z axis replacement technique (using the ZAXREPLACE() function)
As of Ferret v4.9, you can use the 3 argument SHADE and FILL commandsto directly plot data fields on "curvilinear coordinates". Todisplay the field we need only create multidimensional fields specifyingthe horizontal and vertical positions associated with each data point tobe plotted.
The following commands generate the graphic at the top of this page.
! regrid 'Y' to the data grid
let ygg = y[g=gg]
set variable/title="Y"/unit=kilometers ygg
! use the three argument 'shade' command
shade flow[x=0,l=1], ygg, depth[x=0,i=1]
(See the v4.9 releasenotes for more information on the threeargument SHADE command.)
See sigma_coordinate_demo.jnlfor details of the variable definitions.
Note: this demo is graphical but the procedure is valid for analysis, as well
Define flow field on a surface of constant 50 m depth
DEFINE AXIS/Z=0:180:2/UNIT=meters/DEPTH zdepth
LET flow_on_depth = ZAXREPLACE(flow,depth, z[gz=zdepth])
LET flow_at_50 = flow_on_depth[Z=50]
Plan views of flow. Right hand frame: SHADE/L=1 flow_at_50
Note: the missing points between the flow and the black mask occur becausethe deepest layer of flow is considered to be 1/2 grid box above the bottom
PLOT/X=0/Y=0/L=1:20 flow[k=8], flow_at_50
Right hand frame: SHADE/Z=0:180/Y=0/L=1 flow_on_depth
Right hand frame: SHADE/Z=0:180/X=0/L=1 flow_on_depth
(Compare this last plot with the graphic at the top of this page.)
preV4.9 technique (based on @WEQ transformation)
This presentation will solve precisely the same problems as the previoussection, which is based on the ZAXREPLACE() function. See the sigma_coordinate_demo_weq.jnlfor details of the variable definitions.
Visualize flow where z is a single fixed depth
Note: this demo is graphical but the procedure is valid for analysis,as well
Define flow field on a surface of constant 50 m depth
let kernel = depth[z=@weq:50] * flow
let flow_at_50 = kernel[z=@sum]
Note: the missing points between the flow and the black mask occur becausethe deepest layer of flow is considered to be 1/2 grid box above the bottom.
Time series of flow: Visualize flow where the "view" of the data requires a rangeof depths
Note: the procedure is valid for analysis as well as graphics
This procedure (borrowed from depth_to_density_demo.jnl) requires usto "borrow" an axis orientation to serve as the new depth axis.We can only access a single point location along whatever axis we borrow though that point can be any point of the axis. In this example we willborrow the TIME axis  thus we can produce plots only at L=1, L=2, etc..If we desired to produce a time series plot we would have to borrow theX or the Y axis, instead.
It takes just 5 commands to transform layers to depth in this manner:
! in a new grid replace the (borrowed) time axis with the desired depth axis
define axis/t=0:180:2/unit=meters tdepth
define grid/like=gg/t=tdepth ggdepth
! define a new variable, r0, with a value of zero wherever depth equals its
! coordinate on the tdepth axis (r0 is a 4dimensional variable with
! depth in the T axis slot)
! Note that "L=1" here and in the definition of kflow, below, determine
! the fixed value on the TIME axis at which this calculation takes place
let r0 = depth[l=1]  t[g=ggdepth]
! define a new variable, kflow, which, when summed along the Z axis, will give
! the (single) value of flow at the location where depth equals its own
! coordinate on the tdepth axis
let kflow = r0[z=@weq:0] * flow[l=1]
! sum the variable (integrate) along the Z axis. Since the Z axis reduces
! to a point in this operation the result is 3D  X,Y and depth.
let flow_on_depth = kflow[z=@sum]
Again, Crosschannel section of the flow; and Alongchannel sections of flow: