Calculating the depth from pressure in seawater

Objective

To identify how deep an instrument is deployed in the ocean, the ambient pressure is measured. With respect to the gravity variation with latitude and geopotential anomaly this pressure can be converted into meters depth. Saunders and Fofonoff (Saunders & Fofonoff 1976) developed an accurate equation for pressure to depth conversion using the hydrostatic equation and the Knudsen-Ekman equation of state. Here a function is given to compute depth in meters from pressure in decibars, latitude in degrees and geopotential anomaly in meters.

Equation

The exact formula is not convenient for routine applications due to logaritms and integrations. For practical use a fitted fourth order polynomial is used to calculate the specific volume V of a standardised ocean with a practical salinity of 35 and 0°C at depth p in decibars. The adopted equation given in the UNESCO handbook ‘Algorithms for computation of fundamental properties of seawater’ (UNESCO 1983) is:

The gravity variation with latitude Φ in degree is calculated by:

The depth z in meters can be calculated from the gravity variation g, the specific volume V and the geopotential anomaly D in J/kg:

Estimating the depth from pressure based on the Saunders and Fofonoff method (Saunders & Fofonoff 1976) results deviate by 0.08 meters at 5000 decibars and 0.44 meters at 10000 decibars (UNESCO 1983).

Algorithm

The below algorithm in object Pascal is equal to the original Fortran code (UNESCO 1983). For the given values of pressure (decibars), latitude (degree) and geopotential anomaly (dyn meters) the function returns depth in meters.

Function Depth (p, Lat, Del: Double) : Double;
// DEPTH IN METERS FROM PRESSURE IN DECIBARS USING
// SAUNDERS AND FOFONOFF'S METHOD.
// DEEP-SEA RES., 1976, 23, 109-111.
// FORMULA REFITTED FOR 1980 EQUATION OF STATE
// ----------------------------------------------------------
// UNITS:
//   PRESSURE     P      DECIBARS
//   LATITUDE     LAT    DEGREES
//   DEPTH        DEPTH  METERS
//   DTN. HEIGHT  DEL    DYN. METERS
// ----------------------------------------------------------
// CHECKVALUE:
//   1.) DEPTH = 9712.653 M for P=10000 DECIBARS, LAT=30 DEG, DEL=0
//   ABOVE FOR STANDARD OCEAN: T=0 DEG CELSIUS; S=35 (PSS-78)
// ----------------------------------------------------------
// Original fortran code is found in:
//   UNESCO technical papers in marine science 44 (1983) -
//   'Algorithms for computation of fundamental properties of seawater'
// ----------------------------------------------------------
// Translated to object pascal by:
//   Dr. Jan Schulz, 20. May 2008, www.code10.info
Var X          : Double;
    GR         : Double;
    DepthTerm  : Double;
Begin
  X := Sin (Lat / 57.29578);
  X := X * X;

  // GR=GRAVITY VARIATION WITH LATITUDE: ANON (1970) BULLETIN GEODESIQUE
  GR        := 9.780318 * (1.0 + (5.2788E-3 + 2.36E-5 * X) * X) + 1.092E-6 * P;
  DepthTerm := (((-1.82E-15 * P + 2.279E-10) * P - 2.2512E-5) * P + 9.72659) * P;
  DEPTH     := DepthTerm / GR + Del / 9.8;
end;

Literature

Saunders P.M., Fofonoff N.P. (1976): Conversion of pressure to depth in the ocean. Deep Sea Research 23:109-111.

UNESCO (1983): Algorithms for computation of fundamental properties of seawater. UNESCO technical papers in marine science 44:1-55.