Natural Science Forum / Earth Science / Oceanography / January 2007

An important parameter here is the time T for one cycle of the sawtooth
(down and back up).
What is T?

AlexM

If the velocity of the probe is constant, plot the depth of the probe versus
time and the oxygen values versus time and then displace the two curves by
the time difference between the depth measurement and the oxygen
measurement.

There are several issues you need to deal with.  First, let's look at
generalized sensor response.

Idealized sensors whose speed of response is proportional to the
difference of their current reading and the current value of what they
are trying to measure, respond in a exponential fashion.  That is, as
e^-t/tau.  tau is called the time constant, and if you work out the
percentage of the true value of a step function change that the sensor
outputs as a function of time you get:

       time        % of true
       1  tau        63%
       2  tau        86%
       10 tau        99.995%

Life is easiest when the tau of all your instruments are the same.  A
well designed pressure sensor will have a tau an order of magnitude, or
more, faster than your oxygen sensor.  A badly designed one could have a
tau greater than your oxygen sensor.  You should, at least, know
something about its response time.

In the real world, of course, instruments are not ideal.  They usually
have a response that is only 1st order exponential.  Whether or not that
matters depends on the accuracy you require.  Many real instruments have
a response that is close to the sum of two exponential and that is an oft
used model when an exponential doesn't quite fit the instruments response
curve.  

(As a side note, you say that the oxygen sensor has a "settling time
(63%)".  I suspect you found that on the instrument specification sheet.  
Quite often instrument manufactures will give you a single value for the
response time to an arbitrary % of full reading.  They will call it
something which may lead you to belive it is an exponential time
constant.  It is usually not since the instrument deviates from an
exponential response.)

This means you may need to characterize the response curve.  This is
usually done by setting up a step function in the measured variable in
the lab, then plotting the instrument response and trying to match it to
an exponential or some other curve.  You can also try to get an actual
responsse curve from the instrument maker.

This leaves your saw tooth depth pattern.  I assume this is the sea and
swell superimposed on your instrument descent rate. Fourier transforms,
i.e. spectrum analysis, will pull out the frequencies with the greatest
effect on your depth pattern.  This will give you valuable information
about depth smearing in your data.  It is unlikely you will be able to
come up with a simple delay that will allow you to calculate an oxygen
level at every pressure point.  More likely, you will come up with an
average oxygen level over some depth bin size throughout the water
column.

Charly Coughran
ccoughran@DELETE_TO_REPLY_UCSD.EDU