Using Synthetic Data to Test Downscaling Methods

John Lanzante (GFDL/NOAA)

CONCEPTS

Testing Downscaling:

Like Product Testing

My Product

STEP1: Recruit Test Subjects (Gather Data)CONCEPTS

STEP2: Feed Cereal For Several Decades (Apply Downscaling Method)

CONCEPTS

STEP3: How are subjects affected? How well did downscaling do?

Not so clear – Need more subjects? Need more data?

Real-world data may be limited? Can we generate synthetic data to fill the void?

CONCEPTS

STEP 4a: Snowmen most affected? Generate a new sample.

CONCEPTS

STEP 4b: Snow-women affected differently? Generate a new sample.

REALISTIC EXAMPLESCASE 1 – Linearity:

Simplest downscaling – linear regression

REALISTIC EXAMPLESCASE 1 – Strong Nonlinearity:

Simplest downscaling – linear regression

REALISTIC EXAMPLESSUMMARY CASE 1 – Nonlinearity:

Hard to test nonlinearity in real-world data ? (if we are just entering “non-linear regime”)Simulate various degrees of nonlinearityCompare linear & nonlinear downscaling methodsDetermine amount of degradationDetermine time in future when degradation becomes “too large”

REALISTIC EXAMPLESCASE 2 – Coastal Error:

Downscaling error maximizes along coastline

REALISTIC EXAMPLESCASE 2 – Coastal Error:

Obs gridpoint Entirely landModel gridpoint Partly land, partly ocean

REALISTIC EXAMPLESCASE 2 – Coastal Error:

Land more detail (extremes) than Ocean (damped)Missing peaks & troughs unrecoverable

REALISTIC EXAMPLES

SUMMARY CASE 2 – Costal Error:Simulate land & ocean pointsDownscale land from mixture (land + sea)Vary the proportions of the mixtureIs coastal effect due to mixture/mismatch?

SYNTHETIC DATA MODELOne Particular Synthetic Data Model: O= Observations M= Model y= year d= day Red = free parameter (user selects the value)

Oy d = Ōy + O’y d Yearly mean + AR1

O’y d = rlag1 * O’

y d-1 + ay d AR1

fvar = varŌ / varO [ varO = varŌ + varO]

My d = Oy d + by d corr = correlation(O,M)

a ~ N(0,vara) Proper choice of a & b b ~ N(0,varb) yields desired rlag1 & corr

SYNTHETIC DATA MODEL

STEP 1: Generate Base Time Series rlag1 day-to-day persistence fvar interannual vs. day-to-day variability corr strength of relation: model vs. obsSTEP 2: Historical Adjustment meanOBS characteristics of the distribution

meanMODEL

varOBS

varMODEL

STEP 3: Future Adjustment meanOBS characteristics of the distribution meanMODEL

varOBS

varMODEL

SYNTHETIC DATA MODEL

OUR APPLICATIONS OF THIS MODEL:Downscaling (just getting started)

No results yetApplied successfully to several related

issues (cross-validation, exceedance statistics, testing two distributions)

SUMMARYREAL-WORLD COMPLICATIONS:

Results may not be clear-cut: Sample size too small? Multiple factors may contribute?

Some conditions more interesting?

SOLUTION – GENERATE SYNTHETIC DATA: Advantages of Synthetic Data:

Unlimited sample size (enhance signal/noise)Change one factor at a time

Prescribe exact conditionsVary factor over a wide range (“turn the knob”)

Can extend outside the range of historical data Turn knob “all the way” for unambiguous results

A CAUTIONARY NOTE

No “One Size Fits All”:No single “best” synthetic data modelMust possess appropriate real-world characteristicsAbility to vary the relevant factors

Possible Models For Future Development:Skewed data (transform Gaussian data nonlinearly?)Precipitation (discrete Markov + bounded distribution?)

Model occurrence & amount separately?Multivariate model?

THE END

REALISTIC EXAMPLESCASE 1 – Weak Nonlinearity:

Simplest downscaling – linear regression

SUPPLEMENTAL

Causes of Nonlinearity?At highest T – model soil becomes excessively dry – T becomes excessiveOther possibilities: Water Vapor, Clouds, Sea-Ice, etc.

REALISTIC EXAMPLESCASE 2 – Coastal Error:

Land More extremesOcean Damped

REALISTIC EXAMPLESCASE 2 – Coastal Error:

X/Y Plot: Land (model) vs. Land (obs)

REALISTIC EXAMPLESCASE 2 – Coastal Error:

X/Y Plot: Ocean (model) vs. Land (obs)

SYNTHETIC DATA MODEL

STEP 4:Fit downscaling model to historical sample

STEP 5:Test downscaling in historical & future samples

OUR APPLICATIONS OF THIS MODEL:No results to show todayDownscaling (just getting started)Guidance in the use of cross-validationBiases in exceedance statisticsTesting difference between 2 distributions