Pressure to depth conversion. Leroy '97 equation

Mathematical definition

$$\boxed{Z\left( {P,\phi } \right) = \frac{{9.72659 \times {{10}^2}P - 2.512 \times {{10}^{ - 1}}{P^2} + 2.279 \times {{10}^{ - 4}}{P^3} - 1.82 \times {{10}^{ - 7}}{P^4}}}{{g\left( \phi \right) + 1.092 \times {{10}^{ - 4}}P}}}$$

This is an equation for the "standard" ocean. Use corrective terms from the table above for particular applications.

Notation Definition Units Limits Conversion
$Z$ depth $\text{m}$
$P$ pressure $\text{kPa}$ $0\ < P < 1000$ $P\left[ {{\rm{MPa}}} \right] = P\left[ {{\rm{kPa}}} \right] \times {10^{ - 3}}$
$\phi$ latitude $\text{deg.}$ $-90\ < \phi < 90$

$$g\left( \phi \right) = 9.780318\left( {1 + 5.2788 \times {{10}^{ - 3}}{{\sin }^2}\left( \phi \right) - 2.36 \times {{10}^{ - 5}}{{\sin }^4}\left( \phi \right)} \right)$$

Notation Description Units
$g\left( \phi \right)$ gravity $\text{ms}^{-2}$

Corrective terms for various areas

$${Z_{corrected}} = Z\left( {P,\phi } \right) + \Delta Z\left( P \right)$$

Area of applicability Latitude Expression for $\Delta Z\left( P \right)$, $\text{m}$ Accuracy $\pm \text{m}$
Common oceans $60^\circ N - 40^\circ S$ $P/\left( {P + 1} \right) + 5.7 \times {10^{ - 2}}P$ $0.8$
North Eastern Atlantic $30^\circ N - 35^\circ S$ $P/\left( {P + 2} \right) + 3 \times {10^{ - 2}}P$ $0.3$
Circumpolar Antarctic $4 \times {10^{ - 2}}P - 2 \times {10^{ - 4}}{P^2}$ $0.1$
Mediterranean Sea $- 7 \times {10^{ - 2}}P + 2 \times {10^{ - 3}}{P^2}$ $0.2$
Red Sea $0$ $0.2$
Arctic ocean $0$ $0.1$
Sea of Japan $6 \times {10^{ - 2}}P$ $0.1$
Sulu Sea $8^\circ$ $0.9P/\left( {P + 1} \right) + 0.17P + 7 \times {10^{ - 4}}{P^2}$ $0.2$
Halmahera basin $0^\circ$ $0.8P/\left( {P + 0.5} \right) + 0.125P$ $0.1$
Celebes basin $4^\circ$ $1.2P/\left( {P + 1} \right) + 6.7 \times {10^{ - 2}}P + 2.2 \times {10^{ - 4}}{P^2}$ $0.4$
Weber deep $6^\circ$ $1.2P/\left( {P + 1} \right) + 6.7 \times {10^{ - 2}}P + 2.2 \times {10^{ - 4}}{P^2}$ $0.4$
Black Sea $43^\circ$ $1.1P$ $0.1$
Baltic Sea $60^\circ$ $1.8P$ $0.1$

Octave/Matlab implementation

function D = pressure_to_depth_sea_leroy_97(P,L)
% Inputs
%   P: pressure \ kPa
%   L: latitude \ degree \ -90 < L < 90
% Outputs
%   D: depth \ m

    P = P*1e-3;
    G = 9.780318*(1 + (5.2788e-3)*(sind(L).^2) - (2.36e-5)*(sind(L).^4));
    D = (9.72659e2)*P - (2.512e-1)*(P.^2) + (2.279e-4)*(P.^3) - (1.82e-7)*(P.^4);
    D = D./(G + (1.092e-4).*P);
end

Computational examples

References

  1. Leroy, Claude C; Parthiot, François, "Depth-pressure relationships in the oceans and seas", 1998