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