3.11 Adaptive Data Assimilation to Include Spatially Variable Observation

Error Statistics

Rod FrehlichUniversity of Colorado, Boulder

andRAL/NCAR

Funded by NSF (Lydia Gates)

Turbulence

• Small scale turbulence defines the observation sampling error or “error of representativeness”

• Critical component of the total observation error

• Turbulence and observation sampling error have large spatial variations

• Optimal data assimilation must include these variations

Spatial Spectra

• Robust description in troposphere

• Power law scaling

• Spectral level defines turbulence (ε2/3 and CT

2)

Structure Functions

• Alternate spatial statistic

• Also has power law scaling

• Structure functions (and spectra) from model output are filtered

• Corrections are possible by comparisons with in situ data

• Produce local estimates of turbulence defined by ε or CT

2 for adaptive data assimilation

RUC20 Analysis

• RUC20 model structure function

• In situ “truth” from GASP data

• Effective spatial filter (3.5Δ) determined by agreement with theory

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

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 observation sampling pattern and

the local turbulence determines the sampling error

2 2 2x x xΣ = σ + δ

Sampling Error for Velocity and Temperature in Troposphere

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

1/3u vδ = δ = 0.688 (εL)

2 1/3T Tδ = 0.450 (C L)

Sampling Error for Global Model

• Rawinsonde in center of grid cell

• Large variations in sampling error

• Dominant component of total observation error in many regions

• Most 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, Doppler lidar)

• Requires calculation of sampling error • Calculation of analysis error

Adaptive Data Assimilation

• Assume locally homogeneous turbulence around the analysis point r

• forecast (first-guess)

• N observations

aN

b ok k

k=1

x (r) = c x (r) + d y (r )

oky (r )

bx (r)

Measurement Geometry

• Single observation at the analysis coordinate

• Multiple observations around the analysis coordinate

• Aircraft track

Analysis Error for u Velocity

• Instrument error is 0.5 m/s

• ( …... ) sampling• ( ___ ) all rawin• ( _ _ _ ) one rawin• ( _ . _ .) lidar• Turbulence is

important > 5%

SUMMARY

• Turbulence produces large spatial variations in total observation error

• Optimal data assimilation using local estimates of turbulence reduces the analysis error

NCAR Collaborators

• Bob Sharman

• Francois Vandenberghe

• Yubao Liu

• Josh Hacker

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 Temperature

• Instrument error is 0.5 K

• ( . . . . ) sampling error

• ( ___ ) all data• ( - - - ) single obs.• ( . - . - ) aircraft • Turbulence is

important > 50%

UKMO 0.5o Global Model

• Effective spatial filter (5Δ) larger than RUC

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

GFS 0.5o Global Model

• Effective spatial filter (5Δ) is the same as UKMO

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