Estimating Aquadopp noise/removing bias in velocity variance

Hi there. 

A pesky reviewer was skeptical about Nortek's quoted Aquadopp noise floors, particularly whether they can be applied in the surfzone. 

I have to deal with the reviewer's comments, so I'm calculating noise levels using surfzone data. 

I wanted to check an assumption I made in the calculation.  First the algorithm...

It seems to me that variance(uj), where uj is velocity at j'th rangebin, is biased high by an amount variance(e), where e is noise.  

If noise is uncorrelated between 2 rangebins (numbered j & k), then covariance(uj,uk) is unbiased. 

If 2 rangebins are sufficiently close together, covariance(uj,uk) should roughly equal the true value of the variance of uj (which is also roughly the true variance of uk). 

Comparing the biased variance with the unbiased covariance gives a noise estimate...

0.5*[variance(uj)+variance(uk)] - covariance(uj,uk) = variance(e)

...plugging measurements into the left, noise levels can be calculated. 

Results look good (very good), but...

adjacent rangebins overlap, so...

...does this mean errors between adjacent rangebins are in fact correlated?

...considering area under product of 2 offset triangular weighting functions, I reckon

covariance(ej,ek)=variance(e)/4

for j & k adjacent rangebins.  If this is right, then cross-bin correlation can be accounted for in noise estimate.  But I don't really know what I'm doing, and messed up some similar calculations previously.  Is the formula

covariance(ej,ek)=variance(e)/4   for adjacent rangebins

correct?

Thanks very much. 

Steve H.

Dear Steve

In the case of a standard Aquadopp profiler, the transmit pulse and the receive window is of equal length.  Since the sampling volume is the convolution of the two, 1/4 of the return signal overlaps and your factor of four makes a lot of sense.  However, I am not quite sure if this is valid also for the energy content so I am not 100% sure if the right number is 1/4 or 1/16. 

We try to match the transmit pulse with the receive window also for the HR version of the Aquadopp profiler.

Best regards, Atle Lohrmann

Thanks Atle. 

I'll get some data in a still tank, and check. 

BTW, aquadopp performance in surfzone is looking better than expected, I'm looking forward to seeing what it tells us. 

Steve H.