Upload
andrea-baldwin
View
218
Download
0
Embed Size (px)
Citation preview
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