# From "S-coordinate at RHO-points" to "depth

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Bai Diandian on 5 Dec 2023
Edited: akshatsood on 5 Jan 2024
Hello, everyone. I'm new to physical oceanography. Now, I wanna change "S-coordinate at RHO-points" into "depth. Please teach how to do this in detail. Thank you!

akshatsood on 5 Jan 2024
Edited: akshatsood on 5 Jan 2024
I understand that you seek guidance for converting "S-coordinate at RHO-points" into depth. This process involves leveraging the vertical stretching functions that map the S-coordinate, which is a dimensionless quantity varying between -1 at the bottom and 0 at the surface, to actual depth values. This process is most commonly used in ocean modeling systems, such as the Regional Ocean Modeling System (ROMS).
Here is a general outline of the steps that you could follow to achieve the conversion
STEP 1: Gather the required parameters which include critical depth for stretching (), stretching curve at RHO-points (), sea surface height at RHO-points (ζ), seafloor depth at RHO-points (H), surface stretching parameter (), bottom stretching parameter () and number of vertical levels (N)
STEP 2: As you have the S-coordinate, apply the Vertical Stretching function to transform it into actual depth values. A common function in use in ROMS is as follows
where `S` is the vector of S-coordinates at RHO-points.
STEP 3: Now, the depth of each grid can be computed by leveraging the below formula
where `Z_rho` are the depths at RHO-points.
Here is the code snippet for your reference
% calculate the S-coordinate (from -1 at the bottom to 0 at the surface)
S = linspace(-1, 0, N);
% apply the vertical stretching function
S_rho = (1 - theta_b) * sinh(theta_s * S) / sinh(theta_s) + ...
theta_b * (tanh(theta_s * (S + 0.5)) / (2 * tanh(0.5 * theta_s)) - 0.5);
% calculate the depths at RHO-points
Z_rho = zeta + (zeta + H) .* S_rho;
Please note that the exact stretching function and parameters depend on the specific ocean model and configuration you are using. The above workflow presents a general overview of the conversion process.
I hope this helps.