Practical application of least square theory
* Function Fitting* Inverse problem
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
I want to compute SST anomalies
Which mean do I remove?
Pacific SST
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
OPTION 1: Remove the constant spatial mean
−15 −10 −5 0 5 10
−250 −200 −150 −10020
30
40
50
60Mean = 13.5 C
Pacific SST
−8 −6 −4 −2 0 2 4 6
−250 −200 −150 −10020
30
40
50
60
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
Function Fitting
OPTION 2: Remove the meridional gradient (better)
Pacific SST
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
Function Fitting
−250 −200 −150 −10020
30
40
50
60
How to remove the gradient with least square?
Spatial Mean Gradient removed
( ) ( )
( )ˆ1
T
T T
J−
= − − =
y Ex y Ex
x E E E y(1) Least Square Solution
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
What if we want to remove the gradient associated with the California Current?
( ) ( )
( )ˆ1
T
T T
J−
= − − =
y Ex W y Ex
x E WE E Wy
(2) Weighted Least Square Solution
−8 −6 −4 −2 0 2 4 6 8 10
−250 −200 −150 −10020
30
40
50
60
−8 −6 −4 −2 0 2 4 6
−250 −200 −150 −10020
30
40
50
60
Remove the meridional gradient (with different weighting)
weighted
not weighted
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
Equally weighted California Current and Cina Sea
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
weighted
Weighted Least square looks like this
−8 −6 −4 −2 0 2 4 6 8 10
−250 −200 −150 −10020
30
40
50
60
Anomalies (with different weighting)
−6 −4 −2 0 2 4 6
−250 −200 −150 −10020
30
40
50
60
weighted
weighted
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
What if we want to remove the gradient associated with the South Cina Sea?
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
What if we want to remove the gradient associated with the South Cina Sea?
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
Not so happy!
−250 −200 −150 −10020
30
40
50
60
0 5 10 15 20 25
−250 −200 −150 −10020
30
40
50
60
Limit the size of the model parameter, in particularThe parameter controlling the zonal gradient
weighted and tapered
weighted
( ) ( )
( )ˆ1
T T
T T
J−
= − − + = +
y Ex W y Ex x Sx
x E WE S E Wy
(3) Weighted and TaperedLeast Square Solution
Previous bad solution
New better solution