


 Model fitting by least squares  

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This book defines many geophysical estimation applications.
Many applications amount to fitting two goals.
The first goal is a datafitting goal,
the goal that the model should imply some observed data.
The second goal is that the model be not too big nor too wiggly.
We state these goals as two residuals, each of which is ideally zero.
A very simple data fitting goal would be that
the model equals the data ,
thus the difference should vanish, say
.
A more interesting goal is that the model should match the data
especially at high frequencies but not necessarily at low frequencies.

(9) 
A danger of this goal is that the model could have a zerofrequency component
of infinite magnitude as well as large amplitudes for low frequencies.
To suppress such bad behavior we need the second goal, a model residual
to be minimized. We need a small number .
The model goal is:

(10) 
To see the consequence of these two goals,
we add the squares of the residuals:

(11) 
and then, we minimize by setting its derivative to zero:

(12) 
or

(13) 
Let us rename to give it physical units of frequency
.
Our expression says
says matches except for low frequencies
where it tends to zero.
Now we recognize we have a lowcut filter with
``cutoff frequency'' .



 Model fitting by least squares  

Next: The planewave destructor
Up: UNIVARIATE LEAST SQUARES
Previous: Imaging
20141201