The Importance of Atmospheric Variability for Data Requirements, Data Assimilation, Forecast Errors, OSSEs and Verification

Rod Frehlich and Robert SharmanUniversity of Colorado, Boulder

RAP/NCAR BoulderFunded by NSF (Lydia Gates) and

FAA/AWRP

In Situ Aircraft Data

• Highest resolution data

• Many flights provide robust statistical description (GASP, MOZAIC)

• Reference for “truth”

Spatial Spectra

• Robust description in troposphere

• Power law scaling• Valid almost

everywhere

Structure Functions• Alternate spatial statistic• Interpretation is simple (no aliasing)• Also has power law scaling• Structure functions (and spectra) from

model output are filtered• Corrections possible by comparisons with

in situ data• Produce local estimates of turbulence

defined by ε or CT2 over LxL domain

RUC20 Analysis

• RUC20 model structure function

• In situ “truth” from GASP data

• Effective spatial filter (3Δ) determined by agreement with theory

UKMO 0.5o Global Model

• Effective spatial filter (5Δ) larger than RUC

• The s2 scaling implies only linear spatial variations of the field

DEFINITION OF TRUTH• For forecast error, truth is defined by

the spatial filter of the model numerics• For the initial field (analysis) truth

should have the same definition for consistency

• Measurement error • What are optimal numerics?

Observation Sampling Error

• Truth is the average of the variable x over the LxL effective grid cell

• Total observation error • Instrument error = x• Sampling error = x• The instrument sampling pattern and

turbulence determines the sampling error

2 2 2x x xÓ = ó + ä

Sampling Error for Velocity and Temperature

• Rawinsonde in center of square effective grid cell of length L

1/3u vä = ä = 0.688 (åL)

2 1/3T Tä = 0.450 (C L)

UKMO Global Model

• Rawinsonde in center of grid cell

• Large variations in sampling error

• Dominant component of total observation error in high turbulence regions

• Very accurate observations in low turbulence regions

Optimal Data Assimilation• Optimal assimilation requires

estimation of total observation error covariance

• Requires calculation of instrument error which may depend on local turbulence (profiler, coherent Doppler lidar)

• Requires climatology of turbulence • Correct calculation of analysis error

Adaptive Data Assimilation

• Assume locally homogeneous turbulence around analysis point r

• forecast• observations

Nb o

k kk=1

x(r) = c x (r) + d y (r )∑

oky (r )

bx (r)

Optimal Analysis Error

• Analysis error depends on forecast error and effective observation error

• forecast error • effective observation

error (local turbulence)

2 22

2 2b eff

Ab eff

=+

beff

Analysis Error for 0.5o Global Model

• Instrument error is 0.5 m/s

• Large reduction in analysis error for b=3 m/s

Implications for OSSE’s• Synthetic data requires local estimates of

turbulence and climatology• Optimal data assimilation using local

estimates of turbulence• Improved background error covariance

based on improved analysis• Resolve fundamental issues of observation

error statistics (coverage vs accuracy)• Determine effects of sampling error

(rawinsonde vs lidar)

Implications for NWP Models• Include realistic variations in

observation error in initial conditions of ensemble forecasts

• Determine contribution of forecast error from sampling error

• Include climatology of sampling error in performance metrics

Future Work• Determine global climatology and universal

scaling of small scale turbulence• Calculate total observation error for critical

data (rawinsonde, ACARS, profiler, lidar)• Determine optimal model numerics• Determine optimal data assimilation, OSSE’s,

model parameterization, and ensemble forecasts

• Coordinate all the tasks since they are iterative

Estimates of Small Scale Turbulence

• Calculate structure functions locally over LxL square

• Determine best-fit to empirical model

• Estimate in situ turbulence level and ε

ε1/3

Climatology of Small Scale Turbulence

• Probability Density Function (PDF) of ε

• Good fit to the Log Normal model

• Parameters of Log Normal model depends on domain size L

• Consistent with large Reynolds number turbulence

Scaling Laws for Log Normal Parameters

• Power law scaling for the mean and standard deviation of log ε

• Consistent with high Reynolds number turbulence