PERFORMANCE PREDICTION

Sonar Equation Application

The sonar equations developed for handling the detection of sonar targets are the basis for predicting the performance of bathymetric systems. The sonar equation most generally used expresses the system performance as a measure of signal-to-noise ratio at the transducer for a specified depth. The applicable equation for bathymetric systems can be written in two forms and the choice of propagation loss model determines which form is used. The general form of the sonar equation used is

Equation -1: N_{E =} L_{S -} N_{W -} L_{N -} N_{RD -} N_D

where

N_E = Signal excess, expressed in decibels (dB) as the ratio of signal to noise at the system input above that required for reliable performance, and is a measure of detection reliability.

L_S = Source level, a sound pressure expressed in dB referred to 1 dyne/ cm² at a distance of 1 yard from the transducer

N_W = Total propagation loss (dB)

L_N = System noise level (dB)

N_RD = Required signal-to-noise ratio for display used (recorder, digitizer) (dB)

N_D = Two-way transmission loss in dome or other interface between the transducer and the water (dB).

Source Level (L_S)

The source level is the sound pressure level (on axis) expressed in dB relative to 1 µbar at a distance of 1 yard from the transducer. The source level is determined by:

n The amount of electrical transmitting power applied to the transducer

n The ability of the transducer to convert electrical energy to acoustical energy in the water (i.e., transducer efficiency)

n The ability of the transducer to concentrate the energy into a beam (i.e., transducer gain).

The above factors can be combined in a logarithmic equation to give the source level

Equation -2 L_S = 71.6 + 10 log P_E + N _DIT -E

where

P_E = Electrical transmitting power in watts (average pulse power).

E = Transducer efficiency expressed in dB

N_DIT = Transducer transmitting directivity index.

71.6 = Parameter conversion constant.

In addition, the directivity index for a circular piston transducer can be calculated with a good approximation by

Equation -3

where

A = Effective radiating surface area of transducer in square inches

Propagation Loss (N_W )Propagation Loss (N_W )

There are two basic propagation loss models, which can be used in the sonar equation to calculate the depth capability of a bathymetric system. The difference between the two models is based on the assumptions concerning the physics of the reflection from the bottom. The first model considers the bottom echo to be primarily a coherent signal reflection with a bottom loss factor to account for absorption and scattering losses. This model is sometimes referred to as the "lossy mirror" or "specula" model.

The second model assumes no specularly reflected component, and the bottom reflected signal is an incoherent addition of a large number of independent signal components. Each unit element of the illuminated area contributes a small amount of reflected signal power dependent on a back-scattering coefficient. In this case, the reflected signal is a function of the illuminated area. This model is referred to as the target strength or scattering model.

Specular Model

The propagation loss is given by

Equation -4

where

N_W = Total propagation loss in decibels

R = Depth, yards

N_BL = Bottom loss caused by absorption and scattering.

Typical values of bottom loss, N_BL , at an operating frequency of 12 kHz range from 10 to 30 dB, with value of 20 dB used for an average value. For slope angles less than one half the acoustic beamwidth, there will be a strong specular return and this first model applies.

Scattering Model

The propagation loss for this model is

Equation -5

where

S_B = Bottom back scattering strength per unit area, a function of bottom slope.

A = Effective illuminated area in square yards.

Measured values of back scattering coefficient S_B are listed in Table 1.