A technique to measure the volume of diving birds during voluntary dives

Abstract

Buoyancy has a substantial contribution to the total mechanical cost of diving in waterbirds. Most of the techniques currently employed to estimate buoyancy are based on measuring the volume of carcasses or forcibly submerged birds using the water displacement (Archimedes) principle. In such techniques, the voluntary control the bird might have over plumage and respiratory air volumes is neglected. Here, I propose an adjustment to the water displacement measurement that allows the measurement of buoyancy in real-time from unrestrained live birds diving voluntarily. The novelty and accuracy of the technique lie in using a pressure sensor to continuously measure the water level inside the dive tank while filtering out the interference from surface waves. The error of the volume measurement in the proposed technique was only ±2.4%. Feasibility of measurement is demonstrated on captive Great Cormorants (Phalacrocorax carbo sinensis). The measured volumes of Great Cormorants exceeded predictions made based on the volume of carcasses and on average resembled values measured from live restrained birds. However the technique highlighted the high variation in buoyancy of live birds (up to 30%) as a result of small changes in the air volumes taken with the bird underwater.

Appendix—the 1D low-pass Gaussian filter

The theoretical background for digital filter design is covered by any Signal-Processing text book (e.g. Oppenheim and Schafer 1975). A more specific description of the Gaussian filter can be found in Muralikrishnan and Raja (2009). Only the basic principles are summarized below.

A low-pass Gaussian filter has an impulse response (in the time domain) in the form:

$$ g(t) = {\frac{1}{{\sqrt {2\pi } \sigma }}}\,e^{{{\frac{{ - t^{2} }}{{2\sigma^{2} }}}}} $$

(5)

From the equation we can see that σ is the only parameter that adjusts the function. We seek the value of σ that for a given cutoff frequency (f _c) reduces the magnitude of the frequency response function to ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt 2 }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sqrt 2 }$}}$ (i.e. -3 dB). The Fourier transform of a Gaussian is a Gaussian as well. Hence the amplitude transmission of the filter in the time domain may be written as

$$ G(f) = e^{{ - 2(\pi \sigma f)^{2} }} $$

(6)

For the cutoff frequency (f _c):

$$ G(f_{c} ) = e^{{ - 2(\pi \sigma f)^{2} }} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt 2 }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sqrt 2 }$}} $$

(7)

which yields:

$$ \sigma = {\frac{{\sqrt {\ln 2} }}{{2\pi f_{c} }}} $$

(8)

Substituting Eq. 8 in Eq. 5 for the time points of the pressure signal data gives the coefficients of the filter.