GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-9-965-2016Assimilating compact phase space retrievals of atmospheric composition
with WRF-Chem/DART: a regional chemical transport/ensemble Kalman filter
data assimilation systemMizziArthur P.mizzi@ucar.eduArellano Jr.Avelino F.EdwardsDavid P.AndersonJeffrey L.PfisterGabriele G.https://orcid.org/0000-0002-9177-1315National Center for Atmospheric Research, Atmospheric
Chemistry Observation and Modeling Laboratory, Boulder, CO,
USAUniversity of Arizona, Department of Hydrology and Atmospheric Science,
Tucson, AZ, USANational Center for Atmospheric Research, Institute for
Applied Mathematics, Boulder, CO, USAArthur P. Mizzi (mizzi@ucar.edu)4March2016939659783June20158September201520January201610February2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/9/965/2016/gmd-9-965-2016.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/9/965/2016/gmd-9-965-2016.pdf
This paper introduces the Weather Research and Forecasting Model with
chemistry/Data Assimilation Research Testbed (WRF-Chem/DART) chemical
transport forecasting/data assimilation system together with the assimilation
of compact phase space retrievals of satellite-derived atmospheric
composition products. WRF-Chem is a state-of-the-art chemical transport
model. DART is a flexible software environment for researching ensemble data
assimilation with different assimilation and forecast model options. DART's
primary assimilation tool is the ensemble adjustment Kalman filter.
WRF-Chem/DART is applied to the assimilation of Terra/Measurement of
Pollution in the Troposphere (MOPITT) carbon monoxide (CO) trace gas
retrieval profiles. Those CO observations are first assimilated as
quasi-optimal retrievals (QORs). Our results show that assimilation of the CO
retrievals (i) reduced WRF-Chem's CO bias in retrieval and state space, and
(ii) improved the CO forecast skill by reducing the Root Mean Square Error
(RMSE) and increasing the Coefficient of Determination (R2). Those CO
forecast improvements were significant at the 95 % level.
Trace gas retrieval data sets contain (i) large amounts of data with limited
information content per observation, (ii) error covariance
cross-correlations, and (iii) contributions from the retrieval prior profile
that should be removed before assimilation. Those characteristics present
challenges to the assimilation of retrievals. This paper addresses those
challenges by introducing the assimilation of compact phase space
retrievals (CPSRs). CPSRs are obtained by preprocessing retrieval data sets
with an algorithm that (i) compresses the retrieval data, (ii) diagonalizes
the error covariance, and (iii) removes the retrieval prior profile
contribution. Most modern ensemble assimilation algorithms can efficiently
assimilate CPSRs. Our results show that assimilation of MOPITT CO CPSRs
reduced the number of observations (and assimilation computation costs) by
∼ 35 %, while providing CO forecast improvements comparable to or
better than with the assimilation of MOPITT CO QORs.
Introduction
There is increased international interest in chemical weather forecasting
(Kukkonen et al., 2012; MACC-II Final Report, 2014). Such forecasts rely on coupled
forecast model–data assimilation systems that ingest a combination of
remotely sensed and in situ atmospheric composition observations together
with conventional meteorological observations. Generally the remotely sensed
observations come in the form of trace gas retrievals. Examples include
carbon monoxide (CO) total and partial column or profile retrievals from the
Terra/Measurement of Pollution in the Troposphere (MOPITT) and Metop/Infrared
Atmospheric Sounding Interferometer (IASI) instruments. The associated data
sets are characterized by large numbers of observations with limited
information per observation. Such remotely sensed data have been assimilated
in various settings (e.g. Bei et al., 2008; Herron-Thorpe et al., 2012;
Klonecki et al., 2012; Gaubert et al., 2014), but there have been only a
few papers addressing data compression strategies. Two such papers were
Joiner and da Silva (1998) and Migliorini et al. (2008). This article is
inspired by their research and introduces an efficient assimilation strategy
that reduces the number of MOPITT CO retrieval observations by
∼ 35 %. Greater reductions are possible, for example, with IASI CO
and ozone (O3) retrievals, and depend on the number of (i) levels in
the retrieval profile and (ii) linearly independent pieces of information in
the retrieval profile.
Joiner and da Silva (1998) first proposed the idea of using information
content to reduce the number of retrieval observations. They suggested
projecting retrievals onto the eigenvectors of the observation error
covariance matrix and zeroing those coefficients with little or no
information. Their approach evolved from one-dimensional retrieval algorithms
(e.g. Twomey, 1974; Smith and Woolf, 1976; Thompson, 1992).
Retrievals are often obtained using the optimal estimation method of
Rodgers (2000) to obtain solutions to the retrieval equationyr=Ayt+(I-A)ya+ε,
where yr is the retrieval profile, A is the
averaging kernel, yt is the true atmospheric profile
(unknown), I is the identity matrix, ya is the
retrieval prior profile, and ε is the measurement
error in retrieval space with error covariance Em –
the measurement error covariance in retrieval space. Joiner and da
Silva (1998) proposed projecting Eq. (1) onto the trailing left singular
vectors from the singular value decomposition (SVD) of
(i) (I-A) (their method 1) or (ii) the smoothing error
Es=(A-I)Pp(A-I)T, where Pp is the
retrieval prior error covariance (their method 2). Those projections removed
the components of the retrieval that the instrument could not measure with
sufficient sensitivity. They called that approach null-space filtering.
Joiner and da Silva (1998) recognized that when the retrieval was strongly
constrained by the retrieval prior profile, the assumptions underlying
null-space filtering were invalid. For such retrievals, they proposed
filtering based on an eigen-decomposition of Em (their
PED (partial eigen-decompostion)
retrievals). Their analysis showed that PED retrievals were well conditioned
and independent of the retrieval prior profile.
Migliorini et al. (2008) noted that the Joiner and DaSilva (1998) filtering
depended on their truncation criteria and was therefore somewhat arbitrary.
They also showed it was possible to achieve similar filtering results with an
alternative approach that used a more well-defined truncation criterion.
Migliorini et al. (2008) rearranged Eq. (1) to obtain
yr-(I-A)ya=Ayt+ε.
Following their terminology, we call the left side of Eq. (2) a
quasi-optimal retrieval (QOR). Migliorini et al. (2008) noted that
Em was unlikely to be diagonal and likely to be poorly
conditioned. To address those issues, they applied a SVD transform to Eq. (2)
based on the leading left singular vectors of Em
(similar to that proposed by Anderson, 2003) – this step provided
diagonalization of Em and used a well-determined
truncation cutoff. They also applied a scaling based on the inverse square
root of the associated singular values – this step improved numerical
conditioning.
Migliorini et al. (2008) continued to reduce the dimension of
Ayt (i.e., the number of observations) by
neglecting those elements whose variability was smaller than the measurement
error standard deviation (unity in their rotated and scaled system). They
proposed identifying those elements with an eigen-decomposition of the
covariance of Ayt. Since that covariance is
generally unknown, they replaced it with the forecast error covariance and
showed that the resulting dimension was approximately equal to the number of
independent linear functions that could be measured to better than noise
level.
A more recent paper (Migliorini 2012) shows that retrievals can be
transformed to represent only the portion of the state that is well
constrained by the original radiance measurements when two requirements are
satisfied: (i) the radiance observation operator is approximately linear in a
region of state space centered on the retrieval and with a radius on the
order of the retrieval error, and (ii) the prior information used to
constrain the retrieval does not underrepresent the variability of the state.
Migliorini (2012) proves that when those conditions are met the assimilation
of radiances is equivalent to the assimilation of retrievals. The
Migliorini (2012) analysis shows that it is possible to use information from
the retrieval algorithm to compress information in the transformed retrievals
In this paper, we propose an approach that achieves results similar to
(i) Migliorini et al. (2008) without needing to approximate the covariance of
Ayt, and (ii) Migliorini (2012) without needing
information about the retrieval algorithm. Our goal is to compress the
retrievals and remove those components that are not dependent on the
measurements. In so doing we expect to make the assimilation of retrievals
more computationally efficient. The rest of this paper is organized as
follows: Sect. 2 introduces compact phase space retrievals, and Sect. 3
introduces the WRF-Chem/DART regional chemical weather forecast/data
assimilation system. Section 4 discusses our experimental design including
the study period, model domain, initial/boundary conditions, and relevant
WRF-Chem/DART parameter settings. Section 5 discusses the observations that
were assimilated in the various experiments described in Sect. 6. The results
from those experiments are presented in Sect. 7, and we end with a summary of
our thoughts and conclusions in Sect. 8.
Assimilation of compact phase space retrievals
We can rewrite Eq. (2) as
yr-(I-A)ya-ε=Ayt.
In Eq. (3) the averaging kernel A is singular and has low rank.
Therefore the information content for each component of
Ayt is relatively small, and the assimilation is
inefficient because one must assimilate the entire profile to get the same
information as can be compressed into a number of processed observations
equal to the rank of A. To compress Eq. (3), we propose
transforming it with the leading left singular vectors from a SVD of the
averaging kernel; i.e., A=USVT. We denote the truncated
system with the subscript zero so that A0=U0S0V0T; the truncated averaging kernel is
obtained by setting the trailing singular values (i.e., singular values that
were less than 1.0×10-4) and vectors to zero. The transformed
system has the form
U0T(yr-(I-A)ya-ε)=S0V0Tyt.
Migliorini et al. (2008) showed that subtracting (I-A)ya from Eq. (3) removes all contribution from the
retrieval prior profile. Equations (3) and (4) confirm their result because
the leading left singular vectors of A span its range, so the left
side of Eq. (3) should project completely onto U0T.
Following that transform Em becomes U0TEmU0, which may still be non-diagonal and
poorly conditioned. Therefore, we apply an SVD transform and inverse scaling
similar to that used by Migliorini et al. (2008). If the SVD of
U0TEmU0 has the form
U0TEmU0=ΦΣΨT, the transformed and conditioned form of Eq. (4) is
Σ-1/2ΦTU0T(yr-(I-A)ya-ε)=Σ-1/2ΦTS0V0Tyt.
Our approach compresses Eq. (3) so that the dimension of the compact phase
space retrieval (CPSR) profile on the left side of Eq. (5) is identical to
the number of independent linear functions of the atmospheric
profile to which the instrument is sensitive. This method is different from
that of Migliorini et al. (2008) because it compresses the quasi-optimal
retrieval observations based on a linear independence analysis and relies on
the assimilation system to decide how much weight to give the observations.
The approach of Migliorini et al. (2008) reduces the number of observations based
on an uncertainty analysis independent of the assimilation system. Our
approach identifies all linearly independent information contained in the QOR
profile (through projection of the QOR profile onto the left non-zero
singular vectors of the averaging kernel). The approach of Migliorini et al. (2008) may (i) discard some linearly independent information because the
left non-zero singular vectors of the observation error covariance are not
necessarily a basis for the space of QORs, and (ii) discard some linearly
independent information through their uncertainty analysis. Finally, our
approach relies on two transforms: (i) a compression transform (based on the
left non-zero singular vectors of the averaging kernel, and (ii) a
diagonalization transform (based on the left non-zero singular vectors of the
compressed observation error covariance). The approach of Migliorini et al. (2008)
uses two diagonalization transforms – the first based on the observation
error covariance and the second based on the transformed forecast error
covariance in observation space. Our diagonalization transform is analogous
to their first diagonalization transform except we apply it to the compressed
observation error covariance, and they apply it to the untransformed
observation error covariance. As in Migliorini et al. (2008), the final form
of our observation error covariance is the truncated identity matrix.
The assimilation of CPSRs should produce results similar to the assimilation
of QORs except for (i) the effect of assimilation sub-processes like
horizontal localization and inflation, and (ii) differences in the
observation error due to the CPSR compression transform. The QOR and CPSR
observation errors are different because the compression transform projects
the errors onto the leading left singular vectors of the averaging kernel and
retains only those components that lie in the range of the averaging kernel.
In summary the steps for obtaining CPSRs from trace-gas retrievals are as
follows (assuming the retrieval equation has the same form as Eq. 2):
Obtain the retrieval and retrieval prior profiles, the averaging kernel,
and the observation error covariance for a particular horizontal location.
Subtract the retrieval prior term (I-A)ya
from the retrieval profile yr. This yields the QOR as
defined by Eq. (3).
Perform a SVD of the averaging kernel. Form a transform matrix from the left singular
vectors associated with the non-zero singular values. Left multiply the QOR
by the transpose of the transform matrix. This yields the truncated QOR
profile as defined by Eq. (4).
Left multiply the observation error covariance by the transpose of the
transform matrix.
Right multiply that matrix product by the transform matrix. This yields the
truncated observation error covariance.
Perform a SVD of the truncated observation error covariance. Scale the left
singular vectors with the inverse square root of their respective singular
values. Left multiply the truncated QOR profile by the transpose of the
scaled left singular vector matrix. This yields the CPSR profile as defined
by Eq. (5).
As a check, left multiply the truncated observation error covariance by the
transpose of the scaled left singular vector matrix. Right multiply that
matrix product by the scaled left singular vector matrix. The result should
be an identity matrix with rank equal to the number of non-zero CPSRs from
the previous step.
Assimilate the non-zero CPSRs with unitary error variance.
WRF-Chem/DART – a regional chemical transport/data assimilation system
The Weather Research Forecasting Model with chemistry/Data Assimilation
Research Testbed (WRF-Chem/DART) system is the WRF-Chem chemical transport
model (www2.acd.ucar.edu/wrf-chem) coupled with the DART
(www.image.ucar.edu/DAReS/DART) ensemble adjustment Kalman filter
(Anderson, 2001, 2003) data assimilation system. WRF-Chem/DART is an
extension of WRF/DART (www.cawcr.gov.au and references therein).
WRF-Chem is the National Center for Atmospheric Research (NCAR) regional
Weather Research and Forecasting (WRF) model (www.wrf-model.org) with
chemistry. WRF-Chem is a regional model that predicts conventional weather
together with the emission, transport, mixing, and chemical transformation of
atmospheric trace gasses and aerosols. WRF-Chem is collaboratively developed
and maintained by the National Oceanic and Atmospheric Administration/Earth
System Research Laboratory (NOAA/ESRL), Pacific Northwest National Laboratory
(PNNL), and NCAR/Atmospheric Chemistry Observation and Modeling Laboratory
(ACOM). WRF-Chem is documented in Grell et al. (2005), discussed in Kukkonen
et al. (2012), and has been applied in various research settings (e.g.,
Pfister et al., 2011, 2013).
DART (Anderson et al., 2009) is a community resource for ensemble data
assimilation (DA) research developed and maintained by the NCAR/Data
Assimilation Research Section (DAReS). DART is a flexible software
environment for studying the interaction between different assimilation
methods, observation platforms, and forecast models. WRF-Chem and DART are
state-of-the-art tools for studying the impact of assimilating trace gas
retrievals on conventional and chemical weather analyses and forecasts.
Study period, domain, initial conditions, boundary conditions, emissions,
and initial ensemble generation
We conducted continuous cycling experiments with WRF-Chem/DART for the period
of 00:00 UTC, 1 June 2008 to 00:00 UTC, 1 July 2008 with 6 h cycling
(00:00, 06:00, 12:00, and 18:00 UTC). To facilitate a large number of
experiments, we used a reduced ensemble of 20 members and a horizontal
resolution of 100 km (101×41 grid points). We used 34 vertical
levels with a model top at 10 hPa and ∼ 15 levels below 500 hPa.
WRF-Chem ran with the Model for Ozone and Related Chemical Tracers (MOZART-4)
chemistry and Goddard Chemistry Aerosol Radiation and Transport (GOCART)
model aerosol options (Colarco et al., 2009; Emmons et al., 2010). Ideally
for chemical transport forecast experiments we would like an ensemble size of
at least 40 members, a horizontal resolution of no larger than 20 km, and a
vertical grid with at least 50 levels. We expect our small ensemble/coarse-resolution cycling results, as they pertain to the assimilation of QORs and
CPSRs, will apply to larger ensembles with higher resolutions. However, as
the vertical resolution increases, the sensitivity to vertical localization
may increase (because as the model's vertical resolution increases (i) the
vertical solution becomes less smooth and may exhibit greater vertical
variability and (ii) the fidelity of vertical localization becomes greater)
so that tuning of the vertical localization length may be necessary. For our
experiments we used a three-dimensional Gaspari–Cohn type localization with a
localization radius half-width of 3000 km in the horizontal and 8 km in the
vertical. We conducted sensitivity experiments to determine the appropriate
localization settings. Results from the horizontal tests are not discussed.
Results from selected vertical localization tests are discussed briefly in
Sect. 7.5.
We used NCEP Global Forecast System (GFS) 0.5∘ six-hour forecasts for
the WRF-Chem initial/boundary conditions. Our model domain extends from
∼ 176 to ∼ 50∘ W and from ∼ 7 to
∼ 54∘ N. We used the WRF preprocessing system (WPS) to
interpolate the GFS forecasts to our domain and generate the deterministic
boundary conditions. We used the WRF data assimilation system (WRFDA)
(http://www2.mmm.ucar.edu/wrf/users/wrfda/Docs/user_guide_V3.7/WRFDA_Users_Guide.pdf)
to generate the initial meteorology ensemble.
For the chemistry initial and lateral boundary conditions, we used global
simulations from the NCAR MOZART-4 model. The fire emissions came from the
Fire Inventory from NCAR (FINNv1; Wiedinmyer et al., 2011), and the Model of
Emissions of Gases and Aerosols from Nature (MEGAN; Guenther et al., 2012)
calculated the biogenic emissions as part of the WRF-Chem forecast. The
anthropogenic emissions were based on the US Environmental Protection
Agency's (EPA's) 2005 National Emissions Inventory (NEI-2005). We used or
adapted existing ACOM/WRF-Chem utilities
(https://www2.acom.ucar.edu/wrf-chem/wrf-chem-tools-community) to generate
the initial chemistry ensembles with a Gaussian distribution from a specified
mean and standard deviation. That distribution was truncated at the tails to
include 95 % of the distribution. Similar utilities were used to generate
the emission ensembles. We excluded the distribution tails to avoid the
potential for the extreme values to cause numerical problems in the chemistry
algorithms. Although we recognize that the assimilation cycling results may
be sensitive to the emission perturbation horizontal correlation lengths
(e.g. Pagowski and Grell, 2012), this was not particularly relevant to our
study so we set the horizontal and vertical correlation lengths to zero.
Meteorology observations and satellite trace gas retrievals
At each cycle time, depending on the experiment we assimilated meteorology
and/or chemistry observations with the DART ensemble adjustment Kalman filter
(EAKF) and then advanced the analysis ensemble to the next cycle time with
WRF-Chem. The 6 h forecast ensemble was then used as the first guess for the
next ensemble DA step.
We assimilated conventional meteorological observations and CO trace gas
retrievals from MOPITT. The meteorological observations were NCEP automated
data processing (ADP) upper air and surface observations (PREPBUFR
observations). They included air temperature, sea level pressure, surface
winds, dew point temperature, sea surface temperature, and upper level winds
from various observing platforms. We refer to those observations as the MET OBS.
We also assimilated MOPITT partial column/profile CO retrievals. MOPITT is an
instrument flying on NASA's Earth Observing System Terra spacecraft. MOPITT's
spatial resolution is 22 km at nadir, and it sees the earth in 640 km wide
swaths. MOPITT uses gas correlation spectroscopy to measure CO in a
thermal-infrared (TIR) band near 4.7 µm and a near-infrared (NIR)
band near 2.3 µm. TIR radiances are sensitive to CO in the middle and
upper troposphere while NIR measures the CO total column. Worden et
al. (2010), Deeter (2011), and Deeter
et al. (2012, 2013) showed that the
sensitivity to CO in the lower troposphere is significantly greater for
retrievals exploiting simultaneous TIR and NIR than for retrievals based on
TIR alone. MOPITT started data collection in March 2000. We used the
MOPITT v5 TIR/NIR products described in Deeter et al. (2013). We refer to the
MOPITT observations as the CHEM OBS.
Summary of the WRF-Chem/DART Forecast/Data Assimilation
Experiments.
The retrieval error covariance Er associated with each
MOPITT CO retrieval profile is provided as part of the data product. That
error covariance is derived by the retrieval process based on a specified
a priori error covariance Ea. Under the
optimal estimation theory of Rodgers (2000) Er is
related to Ea through the averaging kernel A
by Er=(I-A)Ea.
The measurement error in retrieval space Em is also
related to Ea and A by
Em=(I-A)EaAT. Generally for retrieval data sets, Ea,
Er, and Em are non-diagonal.
Experimental design
We conducted two basic experiments: (i) a control experiment where we
assimilated only MET OBS (MET DA); and (ii) a chemical data assimilation
experiment where we assimilated MET OBS and MOPITT CO partial column
retrievals in the form of QORs (MOP QOR). In addition we conducted an
experiment where we converted the CHEM OBS to CPSRs and assimilated the CPSRs
(MOP CPSR). We also conducted sensitivity experiments where we (i) zeroed the observation error covariance cross-correlations (MOP NROT) – as opposed to using a SVD transformation for diagonalization, and (ii) applied vertical localization (MOP LOC). The suite of experiments is summarized in Table 1.
For all experiments we used (i) DART horizontal and vertical localization –
Gaspari–Cohn localization with a localization radius half-width of 3000 km
in the horizontal and 8 km in the vertical, (ii) DART prior adaptive
inflation, (iii) no posterior inflation, (iv) full interaction between all
observations and all state variables – i.e., MET OBS update chemistry state
variables and CHEM OBS update meteorology state variables (joint assimilation
of MET and CHEM OBS), (v) DART clamping (i.e., the imposition of a minimum
threshold) on chemistry state variables to constrain the posterior ensemble
members to be positive, and (vi) the reported MOPITT retrieval error
covariance as the observation error covariance to account for unrepresented
error sources such as representativeness error.
(a) Shaded contours of CO in ppb for the MOP QOR (upper
panel) and MET DA (middle panel) experiments for the first model level above
the surface (∼ 1000 hPa) from the 6 h forecast valid on 18:00 UTC,
28 June 2008. The lower panel shows the difference contours for those
experiments (MOP QOR – MET DA). The shaded area represents the WRF-Chem
domain. (b) The upper panel shows the assimilated MOPITT CO
retrievals between the surface and 900 hPa for 18:00 UTC, 28 June 2008. The
lower panel shows the associated assimilation increment.
For the MOP QOR experiment, the MOPITT CO retrievals were converted to QORs
using an algorithm similar to that described by Migliorini et al. (2008)
except we did not perform their second forecast error covariance-based filtering.
ResultsThe control and chemical data assimilation experiments
The MET DA and MOP QOR experiments are intended to identify the impact of
assimilating chemistry observations. Figure 1a shows shaded contours of CO in
parts per billion (ppb) at ∼ 1000 hPa from the 6 h forecast valid at
00:00 UTC, 29 June 2008 and compares the MET DA and MOP QOR experiments. It
shows that over the course of MOP QORs the assimilation of CO retrievals
reduced the (i) positive CO bias found in polluted areas of MET DA (i.e.,
metropolitan areas with high-CO emissions – San Francisco, Los Angeles,
Chicago, and the northeast USA), and (ii) negative CO bias found in
nonpolluted areas in MET DA (Hawaii, east Pacific, southeast USA, and Baja).
The MET DA biases could result from model errors such as (i) emission errors
– CO emissions too high in polluted areas and too low in nonpolluted areas,
(ii) transport errors – insufficient CO transport away from polluted areas
and insufficient transport toward nonpolluted areas, and/or (iii) chemistry
errors – CO destruction too weak in polluted areas and too strong in
nonpolluted areas. The MET DA biases could also result from initial/boundary
condition errors that were corrected by the assimilation of MOPITT CO in MOP
QORs.
Figure 1b shows the assimilated CO retrievals for the 18:00 UTC,
28 June 2008 update cycle in the upper panel and the corresponding increments
in the lower panel. Comparison of those panels shows that the assimilation
step adjusted the CO concentrations primarily along the satellite observation
paths, which is a consequence of assimilating sparse observations. The DA
adjustments in Fig. 1b are generally consistent with the differences between
MOP QORs and MET DA in Fig. 1a (CO increases in nonpolluted areas – east of
San Francisco, the southeast USA, and Baja). However, that is a general
statement because the MOP QOR – MET DA differences are partially related to
the impact of assimilating CO observations during the preceding assimilation
cycle and partially related to the impact of assimilating all the CO
observations since the beginning of the cycling experiment
(∼ 100 cycles). Consequently, there are locations where the signs of
the MOP QOR – MET DA differences are different from the signs of the
increments (e.g. southwest of lakes Michigan and Huron and over the Ohio River valley and San Francisco Bay). The sense of those sign differences is
not an indication of relative forecast accuracy but that the (i) impact from
assimilating CO during the preceding cycle was similar to that from
assimilating CO throughout the cycling experiment (same signs), and
(ii) impact from assimilating CO during the preceding cycle was different to
that from assimilating CO throughout the cycling experiment (different
signs).
Figure 2 shows time series of the domain average CO from the MET DA and MOP
QOR experiments in retrieval and state space. The dots represent the
retrieval space results where the cool colors (blue and black) show the
forecasts, and the warm colors (red and magenta) show the analyses. The green
dots represent the MOPITT retrievals. The solid lines show state-space
results. Figure 2 has several interesting results. First, MET DA had a
negative bias of ∼ 10 ppb in retrieval space. Second, assimilation of
MOPITT CO reduced that bias by ∼ 5 ppb. Finally, in state space MOP
QORs increased the mean CO by ∼ 5 ppb. As discussed below, those
results are consistent with Fig. 1, which shows a large number of nonpolluted
areas in MET DA with a negative bias and a small number of polluted areas
with a positive bias.
Time series of the domain average CO from the MOP QOR and MET DA
experiments. The red and magenta dots show the domain average CO in retrieval
space for the MOP QOR and MET DA analyses denoted in the legend by “A”. The
blue and black dots show the domain average CO in retrieval space for the MOP
QOR and MET DA forecasts denoted in the legend by “F”. The green dots show
the domain average MOPITT CO retrievals. They are the same in both panels and
are included for reference. The solid lines show the domain average CO in
model space with the same color scheme as used for the analyses and forecasts
in retrieval space. The solid lines are the same in both panels are also
included for reference.
Figure 3 shows vertical profiles of the time (00:00 UTC, 25 June 2008 to
00:00 UTC, 29 June 2008) and horizontal domain average CO in retrieval
space. It shows that the MOPITT profile had greater vertical variability
(moderate CO near the surface, low CO in the middle troposphere: 500–400 hPa,
high CO in the upper troposphere: 300–200 hPa, and low CO near the
tropopause: 200–100 hPa) than the MET DA and MOP QOR profiles. It also
shows that the assimilation of MOPITT CO had positive impacts throughout the
troposphere with the greatest improvement in the upper troposphere. Figure 3
shows that there were differences in the MOP QOR/MET DA bias reduction
between: (i) the upper and lower troposphere (greater magnitude negative bias
reduction in the upper troposphere and lesser magnitude positive bias
reduction in the lower troposphere), and (ii) the forecast and the analysis
(greater bias reduction in the analysis than in the forecast). Those results
expand our understanding of the bias in Figs. 1 and 2. In Fig. 3 the forecast
and analysis show greater bias reduction in the upper troposphere. That
suggests that the domain averages in Fig. 2 were dominated by bias reductions
in the upper troposphere. Figure 3 also suggests that bias reductions in the
lower troposphere were dominated by the reduction of the positive bias in the
polluted areas of Fig. 1. Those results suggest that the following model
errors (as opposed to initial/boundary condition errors) caused the biases:
(i) the near-surface biases were likely caused by the CO emissions being too
high in polluted areas and too low in nonpolluted areas, (ii) the positive
biases in the lower middle troposphere (∼ 600 hPa) were likely caused by
erroneously large vertical CO fluxes from the near surface to the lower
middle troposphere and/or too little CO destruction, and (iii) the negative
biases in the upper troposphere were likely caused by erroneously small
vertical CO fluxes and/or too much CO destruction. We reach the conclusion
regarding model error versus initial/boundary condition (IC/BC) error because Fig. 3 shows that the bias
reduction in the lower troposphere is greater for the analyses than for the
forecasts. That suggests that following the assimilation of MOPITT CO in MOP
QOR,
the CO IC/BCs have improved relative to MET DA. Then during the course of
model integration the bias increases. Thus, we conclude that model error is a
more likely cause of the bias.
Vertical profiles of time/horizontal domain average CO from the MOP
QOR and MET DA experiments for 00:00 UTC, 25–29 June 2008. The results are
in MOPTT retrieval space. The red profiles represent the MOPITT retrievals.
Otherwise the color of the lines corresponds to the legend. forecast is
the assimilation prior, and analysis is the assimilation posterior.
Lastly, we tested the null hypothesis that the difference between the MET DA
and MOP QOR time series results was zero (H0: MOP QOR - MET DA = 0)
against an alternative hypothesis that the difference was not zero (HA: MOP
QOR - MET DA ≠ 0). We used the retrieval-space time series from
Fig. 2 and the large sample parametric test for the difference between two
means from a normal distribution. The test statistic was
Z=Y‾1-Y‾2σ12/n1-σ22/n2
where Y‾1, σ12, and n1 denote the sample mean,
sample variance, and number of samples for the MOP QOR experiment, respectively;
Y‾2, σ22, and n2 denote the analogous sample
statistics for the MET DA experiment; and n1=n2=104. The rejection
criteria was Cannot handle '' as spaceZCannot handle '' as space>zα/2, where α=0.05
and zα/2=1.96 for a two-tailed test at the 95 % confidence level.
We were able to reject the null hypothesis. Based on that result, we conclude
that assimilation of MOPITT CO retrievals significantly changed the
WRF-Chem/DART CO forecasts and analyses. When measured against MOPITT, those
changes were a significant improvement.
(a) Horizontal domain average of the full and truncated
terms in the retrieval equation for 18:00 UTC, 28 June 2008. MOP-Ret,
MOP-Trc, and MOP-Res are the MOPITT retrieval, truncated retrieval, and
residual profiles, respectively. QOR-Ret is the MOPITT QOR profile, QOR-Trc is
the truncated MOPITT QOR profile, and QOR-Res is the MOPITT QOR residual
profile. (b) Horizontal domain average of the MOPITT averaging
kernel profiles in the upper panel and leading left singular vectors of those
averaging kernels in the lower panel for 18:00 UTC, 28 June 2008.
Assimilation of compact phase space retrievals
Next we study the assimilation of CPSRs as described in Sect. 2 but first
review some CPSR attributes. Figure 4a shows vertical profiles of CPSR
characteristics averaged for the MOPITT retrieval domain at 18:00 UTC,
28 June 2008. The blue curves represent the MOPITT CO retrievals (MOP-Rets).
Those curves have reduced vertical structure due to the units
(log10(VMR) as opposed to VMR). After conversion from log10(VMR) to
VMR, MOP-Rets has greater vertical structure and resembles the MOPITT profiles
in Fig. 3. The black curves represent the MOPITT CO QORs (MOP-QOR) as defined
by Eq. (3). MOP-QORs differs from MOP-Rets in that they have maxima near the
surface and upper troposphere and a minimum in the middle troposphere. The
green curves represent the truncated profiles, which are obtained by
(i) projecting the full retrieval profile or the QOR profile onto the leading
left singular vectors of the associated averaging kernel to get the
projection coefficients (e.g., cr=U0Tyr, where cr is the projection coefficient
vector for the full retrieval and cqor=U0T(yr-(I-A)ya-ε)=U0Tyqor, where
cqor is the coefficient vector for the QOR profile – see
Eq. 4 in Sect. 2), and (ii) performing the inverse projection by
multiplying the leading singular vectors by their respective projection
coefficients and summing those dot products (e.g., y^r=U0cr is the truncated retrieval profile –
denoted MOP-Trc and y^qor=U0cqor is the truncated QOR profile – denoted
QOR-Trc). The forward transform in (i) is analogous to the first part of the
CPSR transform in Eq. (4). The inverse transform in (ii) brings the result of
forward transform in (i) back to state space. The inverse transform is not
part of the CPSR algorithm.
(a) Same as Fig. 1a except for the MOP CPSR experiment and
the middle panel from Fig. 1a, the MET DA experiment is not plotted.
(b) Same as Fig. 1b except for the MOP CPSR experiment.
In Fig. 4a the residuals are defined as the difference between the full and
truncated profiles (e.g., yr-y^r is
the full retrieval residual – denoted MOP-Res and yqor-y^qor is the QOR residual – denoted QOR-Res). If the
full profiles project completely onto the leading singular vectors, the
residuals are zero. The upper panel of Fig. 4a shows that the transform in
(i) has the greatest impact near the surface and the upper troposphere and
the least impact in the middle troposphere. When the truncation residuals are
nonzero, the original profiles contain components that are not in the range
of the averaging kernel. That always indicates a contribution from the
retrieval prior term (A-I)ya.
However,
a zero residual does not always indicate that the contribution from the
retrieval prior term has been removed. For QOR residuals, the retrieval prior
term contribution is completely removed. For the retrieval residuals, the
retrieval prior term contribution may not be completely removed. When
components of the retrieval prior term lie in the range of the averaging
kernel, they cannot be removed by the transform in (i) and are therefore not
included in the residual. For example in the upper panel of Fig. 4a the
similarity between MOP-Trc and QOR-Ret shows that the MOPITT retrieval was
strongly influenced by the retrieval prior and that most of the prior
contribution was removed by the transform in (i). That also shows that most
of the prior contribution was not in the range of the averaging kernel.
However, not all was outside the range, and the difference between MOP-Trc and
QOR-Ret shows that most was inside the range. This analysis shows that the
influence of the retrieval prior term cannot be completely removed by
projecting the retrieval onto the range of the averaging kernel. The results
show that it is necessary to use the Migliorini et al. (2008) quasi-optimal
subtraction in Eq. (2) to remove the retrieval prior contribution. Comparison
of QOR-Ret and QOR-Trc in the lower panel of Fig. 4a shows that QOR-Ret lies
completely within the range of the averaging kernel. That result was expected
from the discussion of Eqs. (3) and (4).
In summary Fig. 4a shows the state space impacts from applying the Migliorini
et al. (2008) quasi-optimal subtraction and the CPSR transform in (i). It
also shows that the quasi-optimal subtraction was necessary to remove the
influence of the retrieval prior. Thus, in CPSRs the quasi-optimal subtraction
removes the influence of the retrieval prior, and projection onto the leading
singular vectors of the averaging kernel provides the data compression.
In Fig. 4a the average number of leading singular vectors was ∼ 2.3.
CPSRs reduced the number of observations by ∼ 7.7 per MOPITT profile.
After thinning there were ∼ 30 000 MOPITT profiles per assimilation
cycle. That implies a CPSR reduction of ∼ 281 000 retrievals or
∼ 80 % per cycle. On application the actual reduction was less
because the number of non-retrieval observations was not reduced. As an
example when assimilating MET OBS and CHEM OBS we found a reduction of
∼ 35 % in the computation cost. That is a wall clock time reduction
based on NCAR's 1.5-petaflop high-performance IBM Yellowstone computer with
32 tasks, and 8 tasks per node. We expect similar reductions for other
computing configurations.
In Fig. 4b we examine the vertical structure of the CPSRs. The upper panel
shows the retrieval domain average of the MOPITT averaging kernel
profiles from 18:00 UTC, 28 June 2008. The lower panel shows the domain
average of the leading left singular vectors from SVDs of the averaging
kernel profiles in the upper panel. Comparison of the upper and lower panels
shows that while the singular vector and averaging kernel profiles are
similar it is not possible to associate a specific singular vector profile
with a specific averaging kernel profile or with a group of profiles.
However,
the averaging kernel of singular vectors show the sensitivity of the
associated CPSR to the true CO profile. The first singular vector shows
positive sensitivity to the entire CO profile with greater sensitivity in the
lower and middle troposphere and greatest sensitivity in the upper middle
troposphere. The second singular vector shows positive sensitivity in the
lower troposphere and negative sensitivity in the upper troposphere. Lastly,
the third singular vector resembles the first singular vector with greatest
positive sensitivity in the upper middle troposphere but with negative
sensitivity in the lower and upper troposphere. Those characteristics are
consistent with the MOPITT TIR and NIR joint sensitivities documented by
Worden et al. (2010), Deeter (2011), and Deeter et al. (2012, 2013). It should be
noted that the sign of the singular vectors in the Fig. 4b is arbitrary
because the left and right singular vectors can be jointly multiplied by
negative one and still qualify as singular vectors. However, when multiplied
by one sign the singular vector may have physical meaning, and when
multiplied by the other it may not. For our application, the sign that made
the vertical structure of the singular vectors most similar to that of the
averaging kernel had physical meaning. Therefore, in Fig. 4b we chose the
sign that made the singular vector profile most consistent with the averaging
kernel profiles.
To test the benefit of assimilating CPSRs we converted the MOPITT CO
retrievals to CPSRs and repeated the MOP QOR experiment (called MOP CPSR).
Those results are shown in Figs. 5–7. Conceptually the MOP CPSR results
should be similar to the MOP QOR results in Figs. 1–3. Practically, the
results are different due to (i) the effect of DA sub-processes like
horizontal localization and inflation, and (ii) differences in the
observation error caused by the CPSR compression transform. Comparison of the
contour maps in Figs. 1a and 5a shows that MOP CPSR provided similar
adjustments to MOP QOR but they were of greater magnitude and larger area
(the MET DA result was not plotted in Fig. 5a because it would be the same as
in Fig. 1a). The general trend from Fig. 1a that the assimilation of CO
retrievals reduced the positive CO bias in polluted areas and the negative
bias in nonpolluted areas appears in Fig. 5a. Comparison of Figs. 1b and 5b
shows that MOP CPSR generally assimilated the same CO retrievals as MOP QOR,
but the CPSR increments were of greater magnitude and more widely dispersed.
Comparison of the time series plots in Figs. 2 and 6 shows that there were
slightly greater bias reductions for MOP CPSR than MOP QOR. MOP CPSR reduced
the CO negative bias in retrieval space by ∼ 8 ppb and increased the
mean CO in state space by ∼ 10 ppb. Those improvements are also seen
from a comparison of the vertical profiles in Figs. 3 and 7, which shows that
MOP CPSR produced greater bias reductions for the forecast and analysis
throughout the troposphere. As in MOP QOR the MOP CPSR improvements were
greater in the upper troposphere than in the lower troposphere. The MOP CPSR
results from Fig. 7 provide further support for our suggestion that the
domain average bias reductions in Figs. 2 and 6 were due to bias reductions
in the upper troposphere because the greater bias reductions in the upper
troposphere of Fig. 7 (compared to Fig. 3) provided greater bias reductions
in Fig. 6 (compared to Fig. 2). In summary Figs. 5–7 confirm our analysis of
Figs. 1–3 and show that assimilation of CPSRs produced results that were
similar to or better than those from the assimilation of QORs at two-thirds
the computational cost. We also conducted significance testing for MOP CPSR
similar to that for MOP QOR and were able to reject the null hypothesis that
there was no difference between the MOP CPSR and MET DA time series in
Fig. 6.
Same as Fig. 2 except for the MOP CPSR experiment.
Verification against MOPITT retrievals
We calculated verification statistics (bias, root mean square error (RMSE),
and coefficient of determination (R2)) for the 6 h forecasts from all
experiments based on the time series results in Figs. 2 and 6. Those
statistics are plotted in Fig. 8. Generally, Fig. 8 shows that the
assimilation of MOPITT CO improved model performance for all metrics when
compared against the MOPITT retrievals. Figure 8 also shows that RMSE was
dominated by the bias and that the differences in the statistics for the
different treatments (assimilating QORs, CPSRs, cross-covariance zeroing, and
vertical localization) were generally negligible except for cross-covariance
zeroing. We now discuss the cross-covariance zeroing and vertical
localization experiments.
Same as Fig. 3 except for the MOP CPSR experiment.
Observation error covariance diagonalization through zeroing of the
cross-correlations
One method used to diagonalize the observation error covariance is zeroing of
the cross-correlations (see the Introduction to Migliorini et al., 2008). The
uncertainty of the error covariance and the practice of adjusting the
observation error variance to tune ensemble DA strategies are used to justify
the zeroing. As noted by Anderson (2001) and applied by Migliorini et
al. (2008), a more aesthetic and mathematically correct approach is to apply
a variance maximizing rotation based on a SVD of the error covariance. In
this section we compare those two error covariance diagonalization methods.
Recall that MOP QOR used an SVD-based rotation to diagonalize the error
covariance. We conducted a companion experiment MOP NROT where we used
cross-correlation zeroing. The assimilation/forecast plots are not shown
because it is not a central theme of this paper. However, we include the
verification statistics in Fig. 8. Significance testing and scores from
assimilation of MOPITT CO show that SVD-based diagonalization produced
significantly greater forecast skill compared to cross-correlation zeroing.
Based on that result we conclude that the second SVD-based rotation is a
necessary step in our definition of CPSRs.
Verification statistics for all experiments in MOPITT retrieval
space. The blue curve is the bias (model – observation), the red curve is
the root mean square error (RMSE), and the magenta curve is the coefficient
of determination. The experiments are described in the text and summarized in
Table 1.
Vertical localization and phase space retrievals
Ensemble data assimilation generally uses localization to remove spurious
correlations that may occur from under-sampling. Localization limits the
horizontal and vertical spatial scales on which the observations impact the
posterior. Vertical localization may be inappropriate when assimilating phase
space retrievals because ∼ 80 % of the vertical variation in the
retrieval is described by the first leading singular vector of the averaging
kernel (the basis function for the phase space transform) and that vector is
nearly independent of height (see Fig. 4b). Nevertheless, if vertical
localization is appropriate then the question becomes how to do it
because phase space retrievals are not associated with a unique vertical
location. One solution assumes that phase space retrievals are associated
with the level of maximum sensitivity in the transformed averaging kernel,
i.e., the averaging kernel after applying the compression and diagonalization
transforms discussed in Sect. 2. We applied such localization to MOP QOR (in
the MOP LOC experiment) and found that results from the two experiments were
similar. Therefore, we do not present the assimilation/forecast plots but
include the verification statistics in Fig. 8. Comparison of the verification
scores in Fig. 8 for MOP LOC with those from the other experiments (MOP QOR
and MOP CPSR) shows that vertical localization did not substantially alter
the results. We experimented with different vertical localization lengths and
found similar results. We are unsure whether this is a general result and are
continuing to investigate vertical localization.
Summary and conclusions
In this paper we incorporated WRF-Chem into DART and assimilated MOPITT CO
trace gas retrievals. We also introduced the assimilation of compact phase
space retrievals (CPSRs). CPSRs are preprocessed trace gas retrievals that
have (i) the influence of the retrieval prior removed, (ii) data
compression, (iii) SVD-based error covariance diagonalization, and (iv) unit
error variance scaling. We showed that assimilation of CPSRs is an efficient
alternative to assimilation of quasi-optimal retrievals (QORs) that provided
substantial reductions in computation time (∼ 35 %) without
degrading the analysis fit or forecast skill.
We presented results from month-long (00:00 UTC, 1 June 2008 to 18:00 UTC,
31 June 2008) cycling experiments where we assimilated conventional
meteorology and MOPITT CO retrievals. For MOP QOR the time series plots in
Fig. 2 showed that MET DA had a negative bias of ∼ 10 ppb. The
assimilation of MOPITT CO in MOP QOR reduced that bias by ∼ 5 ppb. The
vertical profile plots in Fig. 3 showed that assimilation of MOPITT CO
improved the CO analysis fit and forecast skill throughout the troposphere
when compared to MET DA. We also used traditional skill metrics (bias, RMSE,
and R2) to quantify the impact of assimilating CO retrievals. Those
results showed that bias dominated the RMSE and that assimilation of CO
retrievals improved WRF-Chem performance. Specifically, MOP QOR significantly
improved the WRF-Chem CO forecast skill for all three metrics.
Next we focused on making the assimilation of retrievals computationally
efficient and introduced compact phase space retrievals. CPSRs advance
the work of Joiner and DaSilva (1998) and Migliorini et al. (2008) by
describing an easily applied methodology to achieve data compression for
phase space retrievals. Conceptually, the assimilation of QORs and CPSRs
should yield similar results except for the effects of (i) assimilation
sub-processes like localization and inflation and (ii) different observation
errors due to the CPSR compression transform. Nevertheless, our CPSR approach
is different from that of Migliorini et al. (2008): (i) we perform two
transforms – a compression transform and a diagonaliztion transform, they
perform two diagonalization transforms; (ii) we identify and assimilate all
linearly independent information observed by the instrument, they may discard
linearly independent information – some because their transform vectors are
not necessarily a basis for the space of QORs and some because their
uncertainty analysis discards some information that lies in the range of
their transformed averaging kernel; (iii) our diagonalization transform is
analogous to their first diagonalization transform except we diagonalize the
compressed observation error covariance and they diagonalize the
untransformed observation error covariance; and (iv) we rely on the
assimilation system to decide how much weight to give the transformed
observations and require no information from the forecast ensemble, and they
use the forecast ensemble to decide which observations to discard.
MOP CPSR maps in Fig. 5 showed
that assimilation of CPSRs placed CO hot spots in the same locations as MOP
QOR but they were of greater magnitude and larger area. The time series and
vertical profile plots showed that those differences generally represented
analysis and forecast improvements. Skill metrics for MOP CPSR showed that
when compared to MOP QOR, the assimilation of CPSRs slightly improved the
forecast skill for all metrics, and when compared to MET DA it significantly
improved the forecast skill for all metrics. Based on those results we
conclude that the assimilation of CPSRs performed as well or better than the
assimilation of QORs at a substantially reduced computational cost
(∼ 35 % reduction in computation time).
Collectively our analysis of the MOP QOR and CPSR results in Figs. 1–3 and
5–7 suggested that (i) in the lower troposphere MET DA had a negative CO
bias in polluted areas and a positive bias in nonpolluted areas (Figs. 1 and
5) and (ii) bias reductions in the domain average retrieval space CO were due
to reductions in the negative CO bias in the upper troposphere (Figs. 2, 3,
6, and 7). We proposed three causes for the CO biases: (i) emission errors –
overestimation of CO emissions in polluted areas and underestimation in
nonpolluted areas, (ii) transport errors – too much CO transport from the
near surface to the lower troposphere and too little
transport from the lower to upper troposphere, and (iii) chemistry errors –
too little CO destruction in the near surface and lower troposphere and too much destruction in the upper
troposphere.
We expect that CPSRs have the potential for broad operational application.
CPSRs can be easily obtained from retrievals derived from any optimal
estimation algorithm. They can be used to assimilate retrievals with
correlated or uncorrelated errors for any sequential assimilation methodology
(both Kalman filter and variational-based algorithms). Due to their ease of
derivation, flexibility, and potential for large reductions in assimilation
computation time, CPSRs should facilitate the efficient assimilation of dense
geostationary observations.
Acknowledgements
NCAR is sponsored by the National Science Foundation (NSF). Any opinions,
findings and conclusions or recommendations expressed in this publication are
those of the authors and do not necessarily reflect the views of NSF. This
research was also sponsored by NASA grants NNX11A110G and NNX10AH45G. We
gratefully acknowledge the anonymous reviewers, Chris Snyder, and Louisa Emmons for their thorough
reviews of this manuscript and for providing many constructive comments. We
also acknowledge Helen Worden for assistance with the MOPITT data, and
Jerome Barre for assistance with coding the WRF-Chem initial and boundary
condition perturbations. Edited by: A. Lauer
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