Linear Inverse Modeling with an SVD treatment

(at least the extent that I’ve learned thus far)

Eleanor Middlemas

What is Linear Inverse Modeling (LIM)?

• Penland & Sardeshmukh (1995) [PS95]:

• What it looks like

• Compare to our linear model from class:

L

How does LIM work?• If accurately represents the dynamical

system, then given some state vector x at time t, this model can predict x at time t+τ :

• Where

• And

• So,

Covariance Matrix at lag τ0

L

L

L

L

How does one use LIM?

• 1) Calculate

• 2) Calculate • 3) Make a forecast!

L

L

L

Why would one use a LIM?• Uses covariance time-lag statistics

• Testing the linearity of a relationship between the growth of one variable and another variable, and how much it’s driven by white noise

• Penland and Sardeshmukh 1995: Can predict ENSO using this model; “constructive interference of several damped normal modes”

• Newman et al. 2009: Analyzes effect of air-sea coupling on tropical climate variability; concludes that the evolution of these parameters are “linear and stochastically driven”

• Shin et al. 2010: Investigates the relationship between SSTs among different tropical ocean basins, then hypothesizes about physical mechanisms

L

How does one use LIM?: Example• Example: Newman et al. 2009

• Determining importance of certain parameters on tropical SST evolution on different timescales (ENSO and MJO)

L

L

Covariance Matrix

How will I use LIM?• I am interested in finding the “least damped modes” of the

Community Atmosphere Model, version 4 coupled to a slab ocean model (CAM4-SOM)• Pre-industrial control run• What dictates the trends of the surface temperatures within this

model?

• I will attempt to implement a Linear Inverse Model, and then analyze it with Singular Value Decomposition

• Forewarning: My use of LIM should be taken lightly! Comments/suggestions welcome

How will I use LIM?• 1) Calculate

• 2) Calculate • 3) Make a forecast!

• 4) Calculate SVD on G

L

L

How will I use LIM?

• Input (to determine L)• “State vector”, x, 4 timeseries of 50 years, monthly data:

• Surface Temperature “st”• Sea Level Pressure “slp”• Surface solar heat flux “solar”• TOA net fluxes “total_TOA”

• Results in a matrix x = [600 4]

• Calculated L at 4 different lags: τ0=1,2,3,4 months

L

Results: Finding LSame as vector x (“state vector”)

L

τ0 = 1 τ0 = 2

τ0 = 3 τ0 = 4

L

Code credit to Kathy Pegion

Results: Finding LST solarSLP TOA

STSLP

SolarTOA

Shin et al. 2010

SLP

ST

Results: Making a Forecast

• icfile1: 20 different time steps for each of the 4 parameters

[4 20 120]=([4 4][4 20]) 120 times

• icfile2: a reshaped spatial map at a single time step for each of the 4 parameters

[4 288*192 120] = [4 4][4 288*192]

= 120 months

L

Results: Making a Forecast

As the lag used to calculate L grows, the longer it takes for the forecasts to approach zero

SS

T A

nom

aly

(deg

rees

K)

Time forecasted ahead of t0 (months)

Results: Making a Forecast

Notice the order of units on the colorbarForecasts’ pattern isn’t oscillating or changing – maybe a bug in the code?

Degrees K

Lag (τ0) used to calculate L = 1 month

Results: The Least-Damped Mode• SVD of G

Deg

rees

K

Forecasted Time (1-20 months ahead)

L calculated with τ0=1 L calculated with τ0=2 L calculated with τ0=3 L calculated with τ0=4

icfile1 (20 individual time realizations)

L

Results: The Least-Damped Mode• SVD of G

L calculated with τ0=1

L

Summary• I implemented a Linear Inverse Model (LIM) in order to

identify the least-damped modes of CAM4-SOM• But I am still learning…

• LIMs can answer a variety of important geophysical questions• Another perspective in forecasting• Can assess parameters’ relationships within observations and

models in a quantifiable way• A very powerful tool!

Future Work• Spend more time on producing/understanding forecasting

results • Add more or different parameters

• Try inputting PC’s instead of anomaly timeseries

• Try more methods mentioned in Penland and Sardeshmukh in 1995:• Investigate “optimal growth” (PS95)• Test the validity of the model (PS95)• The Tau Test

• Thanks to Dr. Mapes and Teddy Allen

References• Newman, M., P.D. Sardeshmukh and C. Penland (2009),

How Important is Air-Sea Coupling in ENSO and MJO Evolution? J. Clim, 22, 2958-2976.

• Newman, M., M.A. Alexander and J.D. Scott (2011), An empirical model of tropical ocean dynamics, Clim. Dyn., 37, 1823–1841.

• Penland, C., and P.D. Sardeshmukh (1995), The optimal growth of tropical sea surface temperature anomalies, J. Clim., 8, 1999-2024.

• Shin, S.I., P.D. Sardeshmukh, and K. Pegion (2010), Realism of local and remote feedbacks on tropical sea surface temperatures in climate models, J. Geophys. Res., 115, D21110, doi:10.1029/2010JD013927

Results: The Tau-Test• Is L independent of the time lag?

Results: The Tau-Test• Is L independent of the time lag? Nope…

Euc

lidea

n N

orm

of

L

Time Lag

Time Lag

Mag

nitu

de o

f L