Observation Operators for theAssimilation of Occultation Data into
Atmospheric Models: A Review
Stig Syndergaard
Ying-Hwa Kuo
Martin Lohmann
COSMIC Project Office
University Corporation for Atmospheric Research
OPAC-2 Workshop, Graz, Austria, September 13–17, 2004
Overview
• Characteristics of occultation observations
• Assimilation of GNSS radio occultation data—which
data product?
• Brief summary of past and present efforts to develop
an observation operator for GPS occultation data
• Linear non-local (LNL) observation operators
• Summary and prospects
Characteristics of occultation data
• Limb sounding geometry complementary to ground
and space nadir viewing instruments
• High accuracy
• High vertical resolution
• All weather-minimally affected by aerosols, clouds or
precipitation (GNSS)
• Requires no first guess sounding
• No instrumental drift
Assimilating occultation data into NWP
Challenges and potential problems:
• Occultation data (e.g., phase, amplitude, optical depth, bending
angle, refractivity, absorption,. . . ) are non-traditional meteorolog-
ical measurements (e.g., wind, temperature, moisture, pressure)
• The long ray-path limb sounding measurement characteristics are
very different from the traditional meteorological measurements
(e.g., radiosonde) or the nadir-viewing passive microwave/IR mea-
surements
• The GPS radio occultation measurements are subject to various
sources of errors (e.g., residual ionospheric effects, tracking er-
rors, super-refraction, optimization of bending angle profiles up
high,. . . )
Assimilating occultation data into NWP
The purpose of data assimilation is to extract the maximum infor-
mation content of the data, and to use this information to improve
analysis of model state variables (u, v, T , q, p,. . . ).
Minimization of a cost function (objective function):
J(x) = (x−xb)TB−1(x−xb)+(y −H(x))T(O + F)−1(y −H(x))
• Here we concentrate on the red part, in particular the obser-
vation operator H
GPS RO measurements & processing
L1 and L2 bending angle
Iono−free bending angle
Refractivity
L1 and L2 phase and amplitude
Spherical symmetrySatellite orbits &
Radio holographic methods, multipath
Ionospheric correction
Single path
High altitude climatology & Abel inversionL1 and L2 phase
Auxiliary meteorological data
A ,A
α
T,e,p
S ,S1 2
S ,S1 2
21
N
α ,α1 2
Choice of Assimilation Variable
Should consider the following factors:
• Use the raw form of the data, to the extent possible (e.g., the more
processing the less accurate the data due to additional assumptions or
auxiliary data used in processing)
• Ease to model the observables (and the adjoint)
• The need for auxiliary information (before the assimilation of the data)
• Ease to characterize observational errors
• Ease to characterize observation operator (representativeness) errors
• Computational cost
Assimilation of phases and amplitudes
Pros
• Most “raw” form of the data
• No assumptions are used
• Easy to characterize measure-
ment errors
Not practical
Cons
• Observation operator needs
to model wave propagation
(diffraction and multipath)
inside weather models
• Require precise GPS and LEO
satellite orbit information
• Require ionospheric model to ac-
count for ionospheric delays (we
do not have very accurate iono-
spheric models)
• Computationally very expensive
Assimilation of L1 and L2 bending angles
Pros
• Second most “raw” form of the
data
• Does not require precise orbit in-
formation
• Relatively easy to characterize
measurement errors
Not practical
Cons
• Assumption of spherical symme-
try introduced in the processing
• Need to consider uncertainty in
the “independent” variable (im-
pact parameter) which is derived
from observations
• Require ionospheric model to ac-
count for ionospheric bending
• Computationally very expensive
with ray tracing
Assimilation of iono-free bending angles
Pros
• Still quite close to the “raw” form
of the data
• Does not require precise orbit in-
formation
• Does not require ionospheric
model, but still extrapolation
above the uppermost NWP level
• Reasonably easy to characterize
measurement errors (still chal-
lenging for lower troposphere)
A possible choice
Cons
• Assumption of spherical symme-
try introduced in the processing
• Need to consider uncertainty in
the “independent” variable (im-
pact parameter) which is derived
from observations
• Residual ionospheric observation
error
• Computationally expensive with
ray tracing
Assimilation of atmospheric refractivity
Pros
• Simple observation operator (lo-
cal operator on model variables)
• Does not require precise orbit in-
formation
• Does not require extrapolation
above the uppermost NWP level
• Less sensitive to uncertainty in
independent variable (height)
• Computationally inexpensive
(operationally feasible)
A possible choice
Cons
• Interpreting retrieved profile as
model local refractivity
• Residual ionospheric observation
error
• Requires initialization by clima-
tology for upper boundary
• Representativeness errors must
include effects of horizontal re-
fractivity gradients
• Bias in lower troposphere due to
super-refraction
Assimilation of retrieved T , q, and p
Pros
• Requires little or no work in the
development of observation oper-
ator (as T , q, and p are model
state variables)
• The retrieved data can be assimi-
lated by simple analysis or assim-
ilation methods
• Computationally inexpensive
Not a good choice
Cons
• Far from the “raw” data
• Auxiliary information is needed
for retrieval (e.g., 1DVar), and
additional errors are introduced
• Representativeness errors must
include effects of horizontal re-
fractivity gradients
• Errors in retrieved T , q, and p
are correlated
• Bias in lower troposphere due to
super-refraction
First attempt on an observation operator
2D bending angle operator suggested by [Eyre 1994]
• Based on geometric optics and Bouguer’s law for a ray path in cylin-
drical co-ordinates
• Pointed out important issue: the tangent point for an asymmetric at-
mosphere will not be the same as that calculated from the measurement
geometry assuming spherical symmetry
• Pointed out that the “independent” variable (impact parameter) is
derived together with the bending angle
• Suggested further development on the operator
“neglecting the horizontal gradients in the calculation of refracted angle
can lead to errors which are comparable with the changes in refraction
expected from typical errors in short-range forecast temperature”
2D ray tracing bending angle operators
Ongoing work by [Zou et al. 1999, 2000, 2002, 2004; Liu et al. 2001; Shaoand Zou 2002; Liu and Zou 2003]
• Non-linear observation operator based on 2D ray tracing
• Applied to simulation studies and 3Dvar assimilation of real data
• Very computationally expensive (∼ 0.01 s per ray on a super computer,and about six times that for the adjoint of the operator)
– One occultation ∼100 rays
– Six satellites (COSMIC) will provide ∼750 occultations within a six hour forecast window
– Data assimilation requires repeated calculations to find optimal solution (∼ 100 iterations)
– 100× 750× 100× 6× 0.01 s ≈ 2 days
[Gorbunov and Kornblueh 2003]
• End-to-end description of a 2D ray tracing operator and its derivativeswith respect to model state variables
Other (faster) observation operators
• Simple horizontal averaging of model refractivity [Zou et al. 1995; Kuoet al. 1997; Healy et al. 2003]
– Gaussian weighting gives slight improvement over local refractivity [Healy et al. 2003]
• 1D bending angle operators [Palmer et al. 2000; Healy and Marquardt2004 (developed within the EUMETSAT GRAS-SAF)]
– Using the “forward” Abel integral transform to obtain the bending angles from modellocal refractivity
– Horizontal gradients must be accounted for in the observation error budget
– Non-linear: independent variable (impact parameter) depends on refractivity
• Non-linear 2D bending angle operator [Healy et al. 2003]
– Implementation of approximations to speed up the calculations, without introducing ad-ditional errors that degrade quality
– Takes into account most of the effects from horizontal gradients
– Getting the impact parameter right is essential [Healy 2001]—should be modelled to matchdefinition of observed impact parameter [e.g., Gorbunov and Kornblueh 2003]
Error of fast 2D bending angle operator
[Healy et al. 2003]
• Be aware: fractional errors in bending angle are significantly largerthan corresponding errors in refractivity (almost a factor of 10 nearthe surface)
• Does not mean that assimilation of refractivity would be 10 times better
Linearized non-local operators
A new class of linearized non-local (LNL) observation operators have
been developed recently that have the following features:
• Makes use of simplified ray trajectories (can be straight lines orcurved lines) that do not depend on model refractivity
• Linearizes the assimilation problem: recalculation of ray paths atevery iteration is not necessary
• Abel-retrieved refractivity is not interpreted as local refractivity
• LNL operators are more computationally expensive than local re-fractivity operators, but significantly cheaper (perhaps 2 orders ofmagnitude) than 2D ray tracing
• LNL operators account for horizontal refractivity gradients and aremuch more accurate (perhaps a factor of 5) than local refractivityoperator
A forward-inverse mapping operator
[Syndergaard et al. 2003, 2004]
m
L
... 2 x
θ1
y
y
mθ
1Level m... Layer0 1 2
i
Mimicking the observations
and the Abel inversion using
finite straight lines
Somewhat similar to a 2D
weighting function [Ahmad
and Tyler 1998]
Basic requirement:
∫ L/2
−L/2
N(x, y)dx =
∫ L/2
−L/2
N(r)dx
• Discretized and solved for N(r) → N = AVN
• N(x, y) evaluated at (pressure) levels of NWP model
Example of observation operator
1. Horizontal interpolation (along pressure surfaces) of the tempera-
ture and specific humidity to the points used in the mapping
2. Evaluation of the refractivity at these points
3. Mapping the refractivity into a profile at the tangent points using
the mapping operator
4. Integration of the hydrostatic equation to obtain a precise rela-
tion between pressure and geometric height at grid points near the
tangent points
5. Horizontal interpolation of the geometric height to the tangent
point locations
6. Vertical interpolation of the mapped refractivity to the observation
points (observed tangent points)
Forward-inverse mapping in general
NWP data
forward−inverse
comparison
inverse
forw
ard
Real databy
nat
ure
w/constraints
shortcut
mappedvariables
atmospheric "retrieved"parameter(s) parameter(s) variables
model
modeledobservationspseudo−observables
Important: Near cancellation of otherwise crude approximations ⇒fast, but still reasonably accurate
• Useful for all kinds of occultation measurements (absorption too)
• Could perhaps be adapted for assimilation of radiances, etc. . .
Pseudo-phase observation operator
Introduced by [Sokolovskiy et al. 2004]
NWP data
inverse
forw
ard
Real databy
nat
ure
w/constraints
comparisonobservation
operator
forw
ard
oper
atio
nvariables
atmospheric
observations
retrievedrefractivity
retrieved modeled
modelvariables
pseudo−phases pseudo−phases
• Same advantages as refractivity mapping
• Simpler implementation of observation operator
• Extra step on the retrieval side
• Different representativeness and observation error covariances
Simulation of representativeness errors
[Sokolovskiy et al. 2004]
Mapping operator vs. phase operator
∇xJ = B−1(x− xb) + HT(O + F)−1(Hx− y)
refractivity mapping operator pseudo-phase operatorNWP data
forward−inverse
comparison
inverse
forw
ard
Real data
by n
atur
e
w/constraints AVxy xshortcut
mappedvariables
atmosphericvariables
model
modeledobservations
retrievedrefractivity refractivity
pseudo−observables
NWP data
inverse
forw
ard
Real data
by n
atur
e
w/constraints
comparison
observation
operator
forw
ard
oper
atio
n
−1y=A y~ Vxy x
variablesatmospheric
observations
retrievedrefractivity
retrieved modeled
modelvariables
pseudo−phases pseudo−phases
VTAT(O + F)−1(AVx− y) = VT(O + F)−1(Vx−A−1y)
(O + F)−1 = (A−1OA−T + A−1FA−T)−1 = AT(O + F)−1A
• If error covariances are consistent between the two operators, they
should lead to exactly the same assimilation result
FARGO-α and FARGO-N
Fast Atmospheric Refractivity Gradient Operator (FARGO)
Introduced by Poli [2004]
• FARGO-α: αFARGO = αlocal + ∆αFARGO
– αlocal is the forward Abel transform of Nlocal
– ∆αFARGO is a small correction term:
∆αFARGO =
∫path
cos θ[dn
dr(r, θ)− dn
dr(r, θ = 0)
]ds
– path is determined by 1D ray tracing; restricted to ±600 kmcentered at the tangent point
• FARGO-N : NFARGO = Nlocal + Abelinverse(∆αFARGO)
Case study with FARGO-NFAST OBSERVATION OPERATOR FOR GPS RO 25
Figure 7. Cross sections of (a) pressure, (b) temperature, (c) specific humidity, (d) refractivity inthe FARGO occultation plane. The hyperbolas are ray paths from the 2D ray-tracer run in the FARGOoccultation plane (only one ray out of five shown). Thick solid line at the bottom of each figure represents
the Earth’s surface.
into play in one-dimensional refractivity observation operator adjoints (Poli et
al. 2002) or two-dimensional radio occultation weighting functions (Eyre 1994).
The humidity field at 850 hPa exhibits a region of concentrated high amounts
of water vapor across the occultation plane, which contributes to high wet
refractivities. The geopotential fields at 700 hPa in Figure 6(c) and at 500 hPa
in Figure 6(d) indicate that the occultation is located perpendicularly to a
geopotential ridge emerging from continental Europe.
In order to understand how these horizontal meteorological features project
into the CHAMP occultation geometry, we now consider Figure 7(a)–(d) which
shows the cross-section of pressure, specific humidity, temperature, and refrac-
tivity in the FARGO occultation plane. Note that in this representation (1) the
arc of a circle obtained by intersection of the occultation plane with the Earth
[Poli 2004]
Representativeness error of FARGO-NFAST OBSERVATION OPERATOR FOR GPS RO 27
Figure 8. (a): Refractivity difference between observation (O) and calculated from background (B),using different observation operators for calculating B. Refers to the CHAMP occultation studied inFigures 5–7. (b): reduction in the O −B difference when local refractivity along TP, 2D ray-tracing,or FARGO-N are used, instead of the vertical local refractivity (in percents of the difference observed
minus vertical local refractivity).
example shown here. Using metrics similar to those of Figure 4, 2D ray-tracing or
FARGO-N reduces by up to 80% the difference between observed and calculated
refractivity in the lowermost 5 km altitudes, when compared to the difference
between observed and local refractivity (Figure 8(b)).
The detailed discussion of the example demonstrates how the derivation of
a fast observation operator helps understand the way the GPS RO observes the
horizontal gradients in the Earth’s atmosphere, and how these gradients affect the
calculation of bending angle and refractivity. The example analyzed here serves
also as an illustration of the accuracy of the FARGO operator.
[Poli 2004]
Connection between LNL operators
S
N
α
αN=F( )
N
_
_ _
_N=AS S=A N
_−1 α −1=F (N)
_
refractivitymapping FARGO−N
FARGO−αoperatorpseudo−phase
Connection between LNL operators
S
N
α
αN=F( )
N
_
_ _
_N=AS
?
S=A N_
−1 α −1=F (N)_
refractivitymapping FARGO−N
FARGO−αoperatorpseudo−phase
N = Nlocal + AV(N−Nlocal) N = Nlocal + F(∆αFARGO)
Connection between LNL operators
S
N
α
αN=F( )
N
_
_ _
_N=AS
?
S=A N_
−1 α −1=F (N)_
refractivitymapping FARGO−N
FARGO−αoperatorpseudo−phase
N = Nlocal + AV(N−Nlocal) N = Nlocal + F(∆αFARGO)
F(∆αFARGO) ≈∫ rtop
r
dy√y2 − r2
d
dy
[ ∫ L/2
−L/2(N−Nlocal)dx
]L
x
y
Lines ofintegrations
Atmospheric layers
Summary and prospects
• It is important to take into account horizontal gradients in occultationobservation operators
• Ray tracing is potentially very accurate but also very time consuming
• Trade-off between accuracy and speed
• LNL operators have small representativeness error and reduce compu-tational cost significantly
• Some of the proposed LNL operators are useful for assimilation of allkinds of occultation data
• Quantitative (statistical) representativeness error covariance estimatesare needed
• How fast are LNL operators really? Extraction of 2D refractivity fieldfrom NWP model may be the limiting factor
• Have we found the optimum trade-off between accuracy and speed?
References
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References
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