Atmospheric Data Assimilation

Schär, ETH Zürich

Global Atmospheric Data Assimilation

Atmospheric Data Assimilation

Christoph Schär Atmospheric and Climate Science ETH Zürich, Switzerland http://www.iac.ethz.ch/staff/schaer Lecture “Numerical Modeling of Weather and Climate”

May 2007

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Steps in a deterministic weather forecast

Governing set of equations

Discretized form of governing equations

“Model”

Initial conditions

“Data Assimilation”

Forward integration

“Weather Forecast”

Observational data

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Summary of Data Sources

(ECMWF 2001)

Globale Observing System – Overview

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Global Atmospheric Data Assimilation

Surface data ^(70,000/d)

Sounding data (1200/d)

Commercial aircraft data (90,000/d)

Satellite data (10⁷/d) Observations

Global data assimilation system

Internally consistent global atmospheric data set

Best data set available today:

ECMWF / ERA-40

European Center for Medium-Range Weather Forecast

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Global Data Assimilation

Global data assimilation systems ingest a wide range of data from various instruments and observation times. They run a general circulation model (GCM) in hindcast mode. The resulting analysis is a spatially and temporally coherent description of the actual state of the atmosphere. In data sparse regions, where few observations are available, these systems in essence provide a mixture between a short-range (e.g. 6 h) forecast and the available observations.

Observations: soundings, surface observations, satellite data, etc.

Analysis: spatially and temporally coherent dataset on a grid

Initial conditions: starting point of numerical model integration Global data assimilation system

06 UTC 12 UTC 18 UTC 00 UTC 06 UTC 12 UTC 18 UTC

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Forecast and Data Assimilation Cycle

The assimilation system must be able to consistently handle data-sparse regions (e.g. the southern Pacific). The resulting

“analysis” is thus a mixture between available observations and a previous forecast (referred to as “first guess” or

“background”).

Different assimilation techniques are currently used. For instance:

• Optimal interpolation

• Variational assimilation in the space- domain (3D-Var)

• Variational assimilation in the space- time domain (4D-Var)

• Nudging

observations (sounding, etc)

forecast

"first guess"

analysis

initialized analysis

forecast + 10 days forecast

+ 6 hours

regional forecast proofed observations

data assimilation

initialization

four times daily twice daily quality control

regional models and/or interpretation

transmission to weather services,

Optimal Interpolation (OI)

Observations Gridpoint (z, z^fg )

€

z=z^fg+ A_i

(

)

i=1 m

∑

The analysis is defined as a weighted mean:

Optimal Interpolation: Choose weights A_i such, that the mean error over many cases is minimized (in the least square sense). To this end, the following functional must be minimized

where z^t denotes a true value at a grid point, and the outer brackets the mean over some time period.

J= 1

2

(

)

Here z^fg refers to the first guess (or background), which is derived from a previous short-range forecast.€

(z_i^obs, z_i^fg)

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The coefficients C_li represents the correlation of the differences between OBS and FG at neighboring gridpoints. They can be computed from past forecast performance. The matrix C is then referred to as covariance matrix.

The coefficients B_i are more difficult, as they involve the true value at a grid point, which is not known. To derive B_i, some simplifying assumptions are needed. For instance, it may be assumed that B_i only depends upon the distance between the grid point under consideration and the observation i, and is otherwise independent from the specific location of the grid point (see next slide).

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Role of observations in OI

Correlation of geopotential height at neighboring radiosonde stations as a function of distance.

Distance [1000 km]

Correlation

3D variational data assimilation

Model state: model vector x (n ≈ 10⁷ degrees of freedom)

“First Guess”: short-range forecast x^fg

Observations: vector y (m ≈ 10⁴ observations / time window) Observations operator H: simulated observations: y^sim = H(x^fg)

€

J(x) = 1 2

x_p−x_p^fg

( )

F_p^fg

p=1 n

∑

(

)

F_q^obs+F_q^H

q=1 m

∑

Minimization of penalty funktion J(x)

where

F^obs mean error of observations

F^H mean error of observations operator F^fg mean error of first guess

Statistical estimates are derived from past forecast performance

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Minimization in variational assimilation

€

∇J(x)

[ ]

fg

F_p^fg + H_q(x)−y_q F_q^obs+F_q^H

q=1 m

∑

p Minimization of J requires computation of gradient

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3D versus 4D Assimilation

21 00 03 06 09 12 15 18 21 Time [UTC]

21 00 03 06 09 12 15 18 21 Time [UTC]

3D-Var

Observations are summarized in a time window (e.g. 6 h)

4D-Var

Discrepancies (OBS- FG) are mapped back to the beginning of the assimilation window.

Assimilation increment Observations Model trajectory

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3D versus 4D Assimilation

3D-Var

4D-Var

Observations are summarized in a time window (e.g. 6 h)

Discrepancies (OBS- FG) are mapped back to the beginning of the assimilation window.

€

J(x) = 1 2

x_p−x_p^fg

( )

F_p^fg

p=1 n

∑

(

)

F_q^obs+F_q^H

q=1 m

∑

€

J(x^o) = 1 2_i=0

T

∑

i −x_p^fg,i

( )

F_p^fg

p=1 n

∑

i=0 T

∑

( )

( )

F_q^obs,i+F_q^H,i

q=1 m_i

∑

The sum over i denotes the sum over all time windows (with a length of e.g. 1h).

x^o denotes the model state at t=t^o. xⁱ denotes the model state at t=tⁱ.

Tangent linear approximation in 4D-Var

Minimization of J(x^o) requires terms of the form

€

Aⁱ

( )

∂x^o_p

Thus, we need to know how the model state xⁱ depends upon x^o.

For simplification, one makes the tangent-linear approximation, which refers to the linearization around a particular non-linear model trajectory:

Tangent-linear approximation

Non-linear model trajectory

t

t

How do we need to change x^o to move xⁱ into the direction of the observation?

Obs

The tangent-linear model can be inverted to find x^o that matches the observation. The inverted model is referred to as adjoint model.

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Iteration in 4D-Var

To address the non-linear problem, an iteration is needed. The outer loop entails a non- linear model integration. The inner loop contains the minimization of J assuming a linearized (forward and backward = adjoint) model.

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Nudging

€

∂χ

∂t = physical terms + ε_i σ_i(x−x_i,t−t_i)

(

)

i

∑

“governing equations”

Nudging terms

where

εi = weight of observation i σ_i = weight in space and time

x σ

Location of observation

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Reanalyses

Reanalyses refer to objective analyses over extended periods:

• Typically they cover many decades,

• use a homogeneous numerical model and a homogeneous data assimilation system,

• use most of the data available for the respective periods (i.e the data stream itself is not homogeneous).

The most popular reanalyses are:

• ERA-40 Reanalysis (1958-2002):

see http://www.ecmwf.int/research/era/

High-resolution, “short”-term: 60 levels, T

159 (about 100 km)

• NCEP / NCAR Reanalysis (1948-present):

see http://www.cdc.noaa.gov/cdc/reanalysis/

Low-resolution, long-term: 28 levels, T62 (about 220 km)

Data in ERA-40

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Data coverage in ERA-40

(Uppala et al. 2005, QJRMS)

Radiosonde-coverage

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Example of a reanalyzed historic event

(Jung et al. 2003, ECMWF TM)

Reanalyses allow to study past events with today’s tools. The example relates to the reanalysis (top left) and re-foreasts of the storm of February 17, 1962. This

devastating storm created a storm surge that flooded the town of Hamburg and killed 340 people.

Analysis shows that with todays forecasting capability a resonable forecast was feasible with a lead of about 3-4 days.

The plotted field is the 10m wind gust in the Beaufort scale. The Beaufort scale is an empirical damage-oriented scale defined by Velocity = 0.836 B^3/2 m/s.

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2m-Temperature

(Simmons et al. 2006, JGR)

Reanalyses are also used to study interannual and interdecadal climate variations.

Satellite data in ERA-40

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Role of Satelite data

ECMWF 2003, SAC03

In these example, the quality of an assimilation system is quantified based on the scores of an associated medium-range forecast.

The differences between northern and southern hemisphere are due to vastly different coverage with conventional data.

Successful assimilation of satellite data became feasible only long after the first satellites. First (partially) successful results in about 1997.

As satellite data involves radiances (rather than primary model variables), variational data assimilation systems have systematic advantages.