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? Ifaccurately 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 τ 0 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 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: τ 0 =1,2,3,4 months L

Results: Finding L Same as vector x (“state vector”) L τ 0 = 1 τ 0 = 2 τ 0 = 3τ 0 = 4 L Code credit to Kathy Pegion

Results: Finding L ST solarSLPTOA ST SLP Solar TOA Shin et al SLP ST

Results: Making a Forecast icfile1: 20 different time steps for each of the 4 parameters [ ]=([4 4][4 20]) 120 times icfile2: a reshaped spatial map at a single time step for each of the 4 parameters [4 288* ] = [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 SST Anomaly (degrees K) Time forecasted ahead of t 0 (months)

Results: Making a Forecast Notice the order of units on the colorbar Forecasts’ pattern isn’t oscillating or changing – maybe a bug in the code? Degrees K Lag (τ 0 ) used to calculate L = 1 month

Results: The Least-Damped Mode SVD of G Degrees K Forecasted Time (1-20 months ahead) L calculated with τ 0 =1L calculated with τ 0 =2 L calculated with τ 0 =3 L calculated with τ 0 =4 icfile1 (20 individual time realizations) L

Results: The Least-Damped Mode SVD of G L calculated with τ 0 =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, 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, 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: /2010JD013927

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

Results: The Tau-Test Is L independent of the time lag? Nope… Euclidean Norm of L Time Lag Magnitude of L