Proper water properties calculator: vertical distance travelled by sound

0. Brief description. Very basic theory.

The speed of sound propagation in water depends on three parameters: temperature
t, hydrostatic pressure P and salinity s.
And almost every water body on the Earth never has a homogenous distribution of
all of them. For example, when we use echosounder to estimate the depth of a water
body, sound travels down and backwards through layers of water with different
speed.
As we know from elementary school, the distance travelled, in this case, can be
calculated as a sum of velocities ν, multiplied by equal time
intervals Δt:

s = ∑ν(t_{i}) · Δt

Where every next time interval t_{i+1} is calculated according to
t_{i+1} = t_{i} + Δt.
To make measurement more precise we have to decrease the time interval Δt
as small as possible, coming to the following idea:

s = lim ∑ν(t_{i}) · Δt, Δt→0

Which can be rewritten in a more common way:

s = ∫ν(t) dt

It means that to estimate actual distance travelled by sound in the vertical direction
we need to know the speed of sound in every point from the water surface to the
bottom. Which is quite unrealistic. The best that we can have is a temperature/
salinity profile - a few measurements of water salinity and temperature taken
in some points on different depths. So, the best we can do is to interpolate
TS-profile and calculate the sum mentioned above.
Corresponding speed of sound in a point with specified t and
s (and implicitly, P) calculates according to
the UNESCO Equation, that can be found in a work of Chen and Millero[1].

1. Calculation

To estimate the distance travelled by sound in the vertical direction (i.e. the
depth of a water body, and more precisely, the distance between echosounder's
transducer and the bottom), we need to measure propagation time τ
and should have the TS-profile for the current point in the water body.

Table 1. Input values

Parameter

Notation

Value

Range

Units

Measured propagation time

τ

0.001 .. 16

s

Number of time intervals

N_{t}

2 .. 99999

Geographic latitude

φ

-90 .. 90

°

Table 2. TS-profile parameters

Parameter

Notation

Value

Units

Number of points

N_{tsp}

Z coordinate step

Z_{s}

m

Table 4. Calculated distance travelled by sound in the vertical direction

Notation

Value

Units

Description

s_{ν=1500}

m

s = 1500 · τ

s_{νmean}

m

s = ν_{mean} · τ

s_{νsurf}

m

s = ν_{surf} · τ

s_{∑}

m

s = ∑ν(t_{i}) · Δt

First three values in the table 4 are given for comparison.
- s_{ν=1500} - is the distance, estimated assuming that speed of sound is constant and equals 1500 m/s.
- s_{νmean} - is the distance, estimated assuming constant speed of sound, calculated for mean t and s.
- s_{νsurf} - is the distance, estimated assuming constant speed of sound, calculated for t and s on the water surface.
- s_{∑} - is the distance, estimated by interpolating TS-profile and calculated the sum, discussed in the top of this document.