Introduction
Black carbon (BC) makes significant contributions to short-term climate
and human health as a
component of aerosolized fine particulate matter (PM2.5) in the
atmosphere. BC is emitted through incomplete combustion from natural and
anthropogenic burning of biomass and fossil fuels. Open biomass burning (BB),
which includes natural wildfires, deforestation, and agricultural waste and
prescribed burning, accounts for 40 % of total global BC emissions, while
anthropogenic energy related sources (e.g., on- and off-road diesel and
gasoline engines, industrial coal, residential cooking and heating) make up
the remaining 60 % . Future climate conditions
that increase drought and fire prevalence
e.g., and increasingly regulated
anthropogenic sources might lead to a reversal of these ratios in California
and globally . In
California, BB events have been shown to increase surface PM2.5
concentrations by a factor of 3 to 5 (×3 to ×5), compared to
non-fire periods . The heterogeneity in BC emission
and loss patterns and difficulty in replicating transport contribute to
prediction uncertainty.
Despite the recognized importance of biomass emissions, large discrepancies
remain in inventories in terms of biomass consumed and emitted chemical
species. considered two different inventories
during January and April 2006 over Southeast and East Asia, where the total
emitted organic carbon (OC) and BC throughout the month differed by
×12. found similar variability between
seven inventories in Africa during February 2010.
concluded that diffusion and loss mechanisms
limit the corresponding responses of domain-wide aerosol burden, AOD, and
2 m temperature to ×2–3. However, the inventory spread for
larger source magnitudes led to modeled column burden spreads of
×16–30 at hourly to daily grid scales.
The large range in inventories at fine scales results from the differing ways
in which they are built. In order to be globally applicable, fire locating
algorithms use remotely sensed hotspots from polar-orbiting satellites. Some
provide additional regional locational and diurnal information with
geostationary instruments. In all cases, daily emissions in a grid cell are
calculated as the product of activity (kgburned) and emission
factors for each species and vegetation class combination
(kgemitted(kgburned)-1). Bottom–up inventories combine rough
estimates of burned area with vegetation densities and percent biomass burned
associated with different land cover types (LCTs) to determine fire activity
e.g.,.
Top–down approaches use fire radiative power (FRP) measured by
polar-orbiting or geostationary satellites and the LCT-specific energy
content e.g.,, which
circumvents using uncertain estimates of burned areas
. A third approach combines the FRP with
top–down constraints of aerosol optical depth (AOD)
e.g.,. All three of
these approaches cross reference fire locations with biome lookup tables to
obtain the species-specific emission factors for each fire.
Improving short-term, local BC concentration predictions requires
characterization of fine-scale spatial and diurnal patterns of BB emissions.
The weakness of using only polar-orbiting data (e.g., Moderate Resolution
Imaging Spectroradiometer (MODIS) instruments aboard Terra and Aqua) in
bottom–up fire inventories is that there are nominally four overpasses per
day, often with missed detections due to cloud and smoke cover or fire sizes
beyond the instrument detection limits. Thus, these observations provide
little information about the diurnal pattern of fire counts and FRP.
and devise methods
for deriving climatological diurnal FRP patterns using geostationary
observations. Both provide new information to modelers, but the former is not
generalizable to grid-scale diurnal variability and the latter precludes the
possibility that diurnal FRP and emissions
patterns may be bimodal for specific LCTs and
fire regimes, or due to local meteorology.
In contrast to their BB counterparts, anthropogenic emissions of BC are
periodic across weekly and annual timescales. Their spatial distributions are
relatively well known in developed countries, and less so in developing
countries . Global estimates of annual
anthropogenic BC emissions vary by ×2 ;
national annual BC emissions in Asian countries and regions have
uncertainties from ×2 to ×5 . In
North America, including in California, uncertainties still persist in terms
of characterizing the magnitude of emissions in a particular year, seasonal
variability, and long-term trends in activity and control strategies
.
cite several inventories of annual US non-BB BC
sources, which are between 260 and 440 Ggyr-1, yielding a maximum
to minimum ratio of 1.7. However, like many other inventories, the US EPA
National Emission Inventory does not specify
uncertainty bounds either for the whole country or at state and county
levels.
These challenges in characterization of both BB and anthropogenic emissions
of BC and co-emitted species have led to the proliferation of top–down
constraint methods of varying complexity and utility. Several studies have
used adjoint-free methods for anthropogenic emissions in Los Angeles,
California, using aircraft measurements during the 2010 California Research
at the Nexus of Air Quality and Climate Change (CalNex) campaign.
constrained CO, NOx, and CO2, and
constrained CH4; both applied a Lagrangian
particle dispersion model (LPDM).
constrained CH4 using a mass balance approach and light alkane signatures
from multiple sectors. LPDM benefits from being able to resolve sources on as
fine of a grid resolution as is used in the underlying model. Both LPDM and
mass balance are limited to linear tracer problems where observations are
recorded under specific meteorological conditions.
used GEOS-Chem in an analytical inversion to
compare constraints from the CalNex aircraft measurements with those from
present and future satellite observations of CH4 throughout California.
Although an analytical inversion does not require an adjoint, the approach is
limited, computationally, to constraining only a few sources, which imposes
aggregation error . Adjoint-based four-dimensional
variational data assimilation (4D-Var) is able to account for nonlinear
behavior between the emission sources and observation receptors by
calculating exact gradients across physical processes. Such an approach does
not have the limitations imposed by mass balance, LPDM, or analytical
inversions, but does require development of an adjoint. The gradients are
usually calculated through an adjoint model, although recent work
performs 4D-Var on a limited area fire without
an adjoint. That new approach, while easier to implement, is limited to
solving for only a few spatially distributed sources due to computational
limitations.
In this study, we adapt the adjoint-based incremental 4D-Var used in the
WRFDA weather forecasting system
to solve tracer
surface flux estimation problems. The modifications to that system that are
required for this work are described in Sect. as well as in
(GH15). These include new linearized model
descriptions (GH15), memory and I/O trajectory management (GH15), a
log-normal emission control variable (Sect. ), calculation
of posterior variance (Sect. ), and improvements to the
Gauss–Newton optimization algorithm to handle nonlinearities
(Sect. ). As described in GH15, this approach of assimilating
chemical tracer observations in a regional numerical weather prediction and
chemistry model is unique in the context of previous 4D-Var flux constraints.
We apply the resulting tool, WRFDA-Chem, to constrain anthropogenic and BB
sources of BC throughout California during the Arctic Research of the
Composition of the Troposphere from Aircraft and Satellites in collaboration
with the California Air Resources Board (ARCTAS-CARB) field campaign. In June
2008, ARCTAS-CARB characterized aerosols and trace gases throughout
California with DC-8 aircraft flights on 20 (Friday), 22 (Sunday),
24 (Tuesday), and 26 (Wednesday) June .
used BC total mass measurements from a
single-particle soot photometer (SP2) and other simultaneous gas-phase
measurements to identify and characterize anthropogenic and BB plumes in
California. By using these observations and surface measurements from every
third day from the Interagency Monitoring of PROtected Visual Environment
(IMPROVE) network (), we provide top–down
estimates of BC surface fluxes using 4D-Var. The mixture of anthropogenic and
BB sources distributed across complex terrain and biomes is a difficult
system to characterize. Still, this scenario is typical of daily smoke
exposure forecasting during acute wildfire events, and is a relevant first
test case for the new 4D-Var system.
The approach taken in this work is described in Sect. ,
including the forward, adjoint, and tangent linear models, the prior
inventories and domain, and the adaptation of WRFDA. Section
describes the application of WRFDA-Chem to the BB and anthropogenic emission
inversion problem during ARCTAS-CARB. We conclude with a summary and
recommendations for future measurements and emission inversion research.
Method
Nonlinear, adjoint, and tangent linear models
Incremental 4D-Var requires forward nonlinear (NLM), adjoint (ADM), and
tangent linear (TLM) models. The NLM is nearly identical to WRF-Chem
, with the addition of emissions scaling factors. The
GOCART option facilitates 19 species, including 4 gas and aerosol species for
sulfate chemistry, hydrophobic and hydrophilic BC and organic carbon, 5 size
bins for dust, 4 bins for sea salt, and 2 diagnostic species for PM2.5
and PM10. While we use GOCART, the results presented are limited to BC.
The model configuration is the same as was used in
, and is summarized as follows: ACM2 PBL
mixing , the Pleim–Xiu land
surface model
and
surface layer mechanisms without soil moisture and
temperature nudging, Wesely dry deposition velocities
, GSFC shortwave and Goddard longwave
radiation, and microphysics turned off. Microphysical and radiative responses
to online aerosols are not taken into account for GOCART aerosols in
WRF-Chem.
We utilize the recently developed WRFPLUS-Chem
, which contains ADM and TLM code extending
the original WRFPLUS software . WRFPLUS-Chem
describes chemical tracers in the context of planetary boundary layer (PBL)
mixing, emissions, dry deposition, and GOCART aerosols. ADM and TLM gradients
have been verified against finite difference approximations. Second-order
checkpointing reduces the memory footprint to a feasible level for ADM and
TLM simulations over longer durations (>∼ 6 h) and/or that use many
chemical tracers (>∼ 10). applied
the ADM in calculating sensitivities relevant to the emission inversion
carried out here. Section includes a comparison of the
results of that study with the posterior emissions here.
The model domain is similar to that used by
. The spatial extent encompasses California
and other southwestern US states. We conduct two emission inversions, the
first on 22 June with a focus on biomass burning sources, and the second on
23–24 June with a focus on anthropogenic sources. We generated chemical
initial conditions by running WRF-Chem from 15 June 2008, 00:00:00 up until
the beginning of each inversion period. We used the default WRF-Chem boundary
condition for a BC concentration of 0.02 µgkg-1, which was
found to be consistent with observations with an upwind flight on 22 June.
Meteorological initial and boundary conditions are interpolated from
3 h, 32 km North American Regional Reanalysis (NARR) fields.
The horizontal resolution is 18 km throughout 80 × 80
columns, and there are 42 vertical levels between the surface and model top
at 100 hPa.
Our horizontal grid spacing was chosen to balance the wall-time and memory
requirements of 4D-Var with model accuracy, and the ACM2 PBL option was
chosen to reduce ADM and TLM development efforts.
recommend that the complex
terrain in California demands fine tuning of the WRF horizontal grid spacing,
PBL, LSM, and reanalysis initialization. Among other conclusions, those
authors found that at six surface sites near the land–ocean boundary, 4 and
12 km simulations with similar settings had mean wind speed biases of
(0.15 to 1.5) ms-1 and (-0.38 to 1.9) ms-1,
respectively. Supporting that conclusion, used a
36 km resolution chemical transport model (CTM), with offline
meteorology, and found significant negative mean fractional bias (MFB) in
modeled PM2.5 relative to surface observations of fires within narrow
northern California valleys in July 2008 (MFB = -34.95 %) and
during autumn 2007 Santa Ana winds (MFB = -110.22 %). During the
July 2008 episode, their CTM predictions had a smaller positive bias
(MFB = +21.88 %). Therefore, we would expect similar wind and
concentration biases at 18 km resolution, which may or may not be
improved by online meteorology. Incremental 4D-Var provides an opportunity to
utilize a different model configuration (e.g., resolution) for the NLM
comparisons of model to observations than that used for the ADM and TLM
simulations. The adaptation of that capability from meteorological
i.e., to chemical simulations and the
subsequent testing is reserved for future WRFDA-Chem developments.
Prior emission inventories
The prior includes sources of BC from anthropogenic activity and natural
wildfires. Anthropogenic emissions are taken from the US EPA's 2005 National
Emissions Inventory (NEI05) for mobile and point sources, including for
example diesel on-road and power production from coal. The individual sectors
are lumped together for each grid cell. We represent BB emissions using three
different wildfire inventories, FINNv1.0 and v1.5, both at
1 km × 1 km resolution
, and QFEDv2.4r8 at
0.1∘ × 0.1∘ resolution .
FINNv1.5 is readily available through NCAR
(http://bai.acom.ucar.edu/Data/fire/) to WRF-Chem users, while FINNv1.0
is no longer supported. However, we include FINNv1.0 in this study, because
it shows the equivalent value as a prior. FINN and QFED fall into the first
(bottom–up) and third (top–down constraint with AOD) categories of BB
inventories described in Sect. , respectively.
Any inverse modeling study that depends on an initial guess should start in a
region of high probability. In a Bayesian inversion, the first guess should
be unbiased. Here we address several known errors in our prior inventories
that we either fix or are unable to fix. All changes are consistent with
either observations or the intended physical descriptions of the inventories.
QFED scales global aerosol emissions from four biome types through multiple
linear regression between observed MODIS aerosol optical depth (AOD) and
modeled GEOS-5 AOD during the years 2004–2009. For temperate forests QFED
scales aerosols by ×4.5 throughout the world. That vegetation category
accounts for 80 % of the wildfire BC in California during 22–30 June
2008. The global scaling is problematic for the California fires, because the
GEOS-5 AOD is biased high in the western US during the summer fire seasons of
2006–2008 Fig. C14 of. In order to match the
regional climatological AOD scaling factors for the western US, we scale all
QFED BC sources by ×1/3. This
scaling is already taken into account in the prior emissions shown in
Sect. , and without it FINNv1.0 and QFED would differ by
×10 during the ARCTAS-CARB campaign.
The WRF preprocessor distributed with the FINN inventory is used to
distribute ASCII formatted lists of both FINN and QFED daily speciated fire
emissions to hourly netcdf files readable by WRF. The diurnal profile follows
the Western Regional Air Partnership profile – – and
is defined by a flux peak from 13:00 to 14:00 Local Time (LT), and flat
fluxes equal to 2.5 % of the peak value between 19:00 and
09:00 LT. Through modeling experience, we found two bugs with how the
FINN preprocessor interprets the WRAP profile and have fixed them for this
case study. The total FINNv1.0 emissions across the model domain before and
after fixing these bugs are plotted in Fig. along with MODIS
active fire counts . The first bug relates to how the time
zone of a particular fire is calculated from longitude. The preprocessor
converts a decimal longitude to integer time-zone bins; this allows a fire at
120.1∘ W to be an hour earlier in the diurnal profile than a fire at
119.9∘ W, even though they should be at nearly identical positions
in the WRAP profile. Such behavior might apply to anthropogenic emissions,
where cities near time-zone borders follow different daily cycles of
activity, but not to natural activity related to the 15∘ per hour
cycle of the Sun.
MODIS fire hotspot detections, excluding those with confidence
less than or equal to 20 % and double detections within 1.2 km of
each other (left axis) and domain-wide FINNv1.0 BB emissions during the
ARCTAS-CARB campaign, with and without fixes described in
Sect. (right axis).
The second bug, and the one most visible in Fig. , is in the
redistribution of UTC fire detections into LT emissions. MODIS Terra and Aqua
overpass times are distributed around noon and midnight LT globally, with
some adjustment as the image capture location moves farther from the Equator.
The fire hotspots are detected on UTC days, and their emissions are profiled
according to LT periods corresponding to the same UTC day as the detection.
In California, where the LT is UTC minus 8 h, the noon overpass corresponds
to 20:00 UTC, and 00:00 UTC corresponds to 16:00 LT
on the previous day (Sun cycle). Therefore, when a fire is detected during
nearly peak heat and emission fluxes at noon, a large fraction of the flux is
apportioned to the previous afternoon. For locations east of the
International Date Line, the LT reallocation is in the opposite direction. In
either case, some portion of the profile is shifted by 24 h. This error is
apparent as a temporal discontinuity in the case of transient fires that vary
significantly in magnitude from one day to the next, especially after a
recent ignition. Since the domain used here is nearly confined to a single
time zone, we simply move the emissions forward 1 day for times between 16:00
and 23:00 LT (00:00–07:00 UTC). A more robust fix will need
to be implemented in a future preprocessor.
Another error in the prior BB emissions is less easily resolved.
Figure shows where the MODIS active fires are located relative
to the inventory fire locations. Since QFED fires are provided on a lat–lon
grid, the fire centers do not coincide with its grid centers. When the
inventory is distributed to the 18 km model grid, some emissions are
shifted over by one column relative to the FINN locations. There are several
additional spurious emission locations in QFED, where no active fires were
detected on either 21 or 22 June. In a month long simulation, differences in
fire gridding between several inventories can be averaged out. In the
shorter-term inversions over California presented in Sect. ,
the locational differences do affect the results.
WRFDA-Chem inversion system
Incremental 4D-Var
The aim of data assimilation (DA) is to optimally combine uncertain
observations with uncertain model predictions to provide an improved estimate
of the state of a system than either gives alone. Here we apply incremental
4D-Var as first introduced by , utilizing the
existing software architecture in WRFDA, and extended to accommodate positive
definite emissions with large associated uncertainties
(Sect. ). In Appendix , we show (similar to
)
that incremental 4D-Var is equivalent to a Gauss–Newton (GN) optimization,
where a cost function,
minδxkJδxk=12δxk+xk-1-xb⊤B-1δxk+xk-1-xb+12Gk-1δxk-do,k-1⊤R-1Gk-1δxk-do,k-1,
is linearized around a current guess of a control variable vector (CV),
x∈Rn, minimized, then relinearized, and so on.
δxk is the CV perturbation sought in the kth linearization.
xb is the vector of prior CVs, B is the background
covariance matrix, and R is the model–observation error
covariance matrix. The nonlinear operator,
Gx=H1x⋮Hix⋮HNx,
is composed of the model–observation operators, with each Hi mapping
x to observation time i. The measurements at each acquisition time,
yio∈Rmi, are expressed independently for N acquisition
times by
yo=y1o⊤,…,yNo⊤⊤∈Rm,
where
∑i=1Nmi=m. The o superscript denotes that yo are
observations. do,k-1 is the innovation between observations and
model values in the previous linearization:
do,k-1=yo-Gxk-1.
The linearized cost function is derived under the assumption that
Gxk+δx≈Gx+Gk-1δx,
where G∈Rn×m is the Jacobian of G. The
superscript on Gk-1 denotes that it is linearized around the
state from the previous iteration, i.e.,
Gk-1=H1|xk-1⋮Hi|xk-1⋮HN|xk-1.
In an emission inversion for a single chemical species, n=nxnynt=O105-106, depending on the domain size and temporal
aggregation of posterior emissions. Since the number of members in
B is equal to n2, finding its inverse is computationally
unfeasible. To circumvent that challenge,
implemented the control variable
transform (CVT) through a square root preconditioner
, U, in WRFDA. The increment is
transformed as δxk=Uδvk, where
B=UU⊤,
U⊤B-1U=In, and
In∈Rn×n is the identity matrix. The
transformed minimization problem is
minδvkJδvk=12δvk-db,k-1⊤δvk-db,k-1+12Gk-1Uδvk-do,k-1⊤R-1Gk-1Uδvk-do,k-1,
where the background departure, summed over all previous outer iterations, is
db,k-1=-∑ko=1k-1δvko.
In addition to circumventing the calculation of B-1, the
preconditioner reduces the condition number of the problem, speeding up the
minimization process.
The solution to Eq. () is found by setting its gradient equal to
zero, i.e.,
∇δvJ=δvk-db,k-1+U⊤Gk-1⊤R-1Gk-1Uδvk-do,k-1=0.
This
yields the solution to the kth linearization:
δvk=In+U⊤Gk-1⊤R-1Gk-1U-1db,k-1+U⊤Gk-1⊤R-1do,k-1=-Hδv-1∇δvJ|δvk=0,
where Hδv=∇δv2J is the Hessian
of Eq. (). In addition to their large size, G and
G⊤ are not often known explicitly and can only be multiplied
by vectors through the integration of a TLM or ADM, respectively. As a
result, Hδv and its inverse are too large to store
and calculate explicitly. The inverse Hessian is generally approximated
through an iterative minimization (e.g., conjugate gradient), called the
inner loop, while the successive relinearizations are performed across outer
loop iterations. Finite precision and the problem dimension, n, prevent
Eq. () from being exactly equal to zero. Increasing the number
of inner loop iterations to approach such an objective does not necessarily
speed up convergence for the full nonlinear problem. Large innovations,
do,k, may remain after relinearization around the new state,
xk=xk-1+Uδvk. The balance between
computational expense and accuracy is chosen for each application.
Log-normal control variables
The positive definite nature of atmospheric chemical emissions combined with
uncertainties that are potentially greater than 100 % sets them apart
from most CVs sought in meteorological data assimilation. The cost function
in Eq. () is derived assuming unbiased Gaussian statistics in
both the background errors and model–observation errors. Emissions are more
likely to be log-normally distributed, since individual sources are found
from the products of variables which themselves are also positive definite.
In order to ensure positive definiteness, the ratios of modeled (posterior,
Ea) to tabulated inventory (prior, Eb) emissions in all grid cells are
gathered into a vector, β=exa, such that
Ea,j=Eb,jβj,
for CV member j. Each βj is a “linear scaling factor”, while
“exponential scaling factors” comprise the posterior CV vector,
xa. In this framework, xb is the background exponential
scaling factor. Setting xb=0 is equivalent to assuming that
the inventory emissions are the prior. Equation () is
expressed within the G operator and its Jacobian. x is resolved on
the grid scale and across hourly discretized emission rates; the temporal
resolution is customizable for particular applications.
Although other emission scaling forms have proven effective
, we stick with
exponential scaling factors here both as a first demonstration and to be
consistent with log-normal statistics for emission rates.
showed that a cost function utilizing this
exponential transform – which was previously applied to emission inversions
by, e.g., , , and
– converges toward the median of a multivariate
log-normal distribution for β. Our approach enables the use of
existing WRFDA optimization algorithms, with a simple modification described
in Sect. .
Model–observation concentration errors might also be treated as being
log-normally distributed, since concentrations are positive definite. Still,
the positive definite constraint on emissions ensures that the same applies
to modeled concentrations, and we find that treatment to be effective.
Introducing log-normality in the observations would corrupt the quadratic
form of Eq. (), which is necessary to derive the closed form
solution of the additive increment in Eq. (). Two alternatives
to our approach that include log-normal observations are proposed by
, who introduced a geometric incremental
formulation with a non-quadratic cost function, and ,
who devised a quadratic approximation to the additive incremental log-normal
cost function.
The scaling factor control variables necessitate a special treatment of prior
error variance. The common practitioner may have some intuition about
multiplicative emission uncertainties in β space (e.g., “factor of 2,
3, 4, etc.”), but not of the variance in exponential CV (x) space
that would populate the diagonal terms of B. A vector that follows
a multivariate log-normal distribution
(β∼LNμ,B) is simply
the exponential of a different vector that follows a multivariate Gaussian
distribution (i.e.,
x∼Nμ,B). According to,
e.g., , the sample mean and covariance of
β across many realizations are
Eβ0i=μβ0=expxb,i+12Bx,ii
and
Bβ0,ij=expxb,i+xb,j+12Bx,ii+Bx,jjexpBx,ij-1,
respectively, where i and j are general indices coinciding with
individual CV members, E is the expectation operator and exp is
the natural exponential function. The subscript β0 indicates a
variable is evaluated in log-normal space in the zeroth outer iteration, when
k=0, and the subscript x indicates an evaluation in Gaussian CV space. In
that Gaussian space, xb is the mean, median, and mode. As
Eq. () shows, the expected value, or mean, of β0
is not equal to its median, expxb,i, the latter being the central
tendency we find by minimizing Eq. ().
Equation () has not been used in previous emission
inversions to translate relative emission uncertainties into the exponential
space. When grid-scale relative emission uncertainties are less than
×3, there is not much error in assuming that
σβ0,i+12≈expxb,i+σxb,iexpxb,i-σxb,i=expσxb,i2,
which is equivalent to
σβ0,i+1≈expσxb,i
and its inverse
σxb,i≈logσβ0,i+1.
(σβ0+1) is the multiplicative uncertainty. For example,
σβ0=2 gives a factor of 3 (×3) relative emission
uncertainty. For our case, where xb=0,
Eq. () diverges from Eq. () by less than
3 % in terms of an error in (σβ0,i+1) for
σxb∈[0,log(2)], but reaches 100 % mismatch at
σxb=log(4.2). The procedure we describe below should be followed
when grid-scale uncertainties are probably above ×2, such as for
high-resolution inversions of BB sources. Simplifying Eq. ()
for a diagonal term, the ith prior variance of β0 is
σβ0,i2=exp2xb,i+σxb,i2expσxb,i2-1.
This is identical to the variance transformation between univariate
log-normal and Gaussian distributions. However, we want the inverse of this
relationship,
σxb,i=log1+σβ0,i2exp2xb,i+σxb,i2.
With an initial guess of σxb,i=0, Eq. ()
converges in recursion for reasonable ranges of σβ0,i. This
transformation only needs to be applied during preprocessing, and only once
for each unique value of prior relative emission uncertainty.
Gaussian covariance
The error covariance matrices in Eq. () are estimated using
existing knowledge of the underlying system. We assume that the off-diagonal
covariances in B are Gaussian in nature. These are defined through
the CVT in WRFDA-Chem, which only differs from that of WRFDA in order to
account for the temporal distribution of emissions. The transform
δxk=Uδvk is performed through two
separate operations as U=UtUh. Although the
horizontal transform (Uh) only deals with correlations in the x
and y directions, and the temporal transform (Ut) only does so
in the temporal dimension, they are both n×n, with sub-matrices
along the diagonal of dimension (nxny)×(nxny) and (nt)×(nt), respectively. The computational overhead of multiplying by either
transform is reduced by only handling the non-zero elements. Uh
is carried out using recursive filters and the scalar correlation length
scale, Lh .
The temporal transform, Ut, is constructed in a similar fashion
to the vertical transform in WRFDA for meteorological CVs
, except that herein we use all of its
eigenmodes. The user specifies the duration of emission scaling factor bins
(in minutes), the temporal correlation timescale (Lt, in hours), and the
grid-scale relative emission uncertainty, σx. WRFDA-Chem converts
these selections to a covariance sub-matrix
Bt=ΣCΣ∈Rnt×nt, where C is the temporal correlation matrix and
Σ=σxInt. Bt is square,
symmetric, and positive-definite. Similar to ,
C is defined using an exponential decay,
Cij=e-ΔtLt,
where Δt is the time elapsed between the beginning of two particular
emission steps. The covariance is decomposed into eigenmodes as
Bt=EtΛtEt⊤; these are
readily calculated, because the dimension of Bt is the square of
the number of emission time steps (e.g., 24 steps for hourly scaling factors
in a single-day inversion). Throughout the optimization, the temporal
transform is carried out through multiplication by
Ut=EtΛt1/2…0⋮⋱⋮0…EtΛt1/2
and its transpose.
The model–observation errors are also assumed to be Gaussian, and their
covariance matrix, R, is assumed diagonal. For each measurement,
p, the total variance is defined as the sum of observation
σp,o2 and model σp,m2
components, following the approach by .
σp,m is determined from an ensemble of 156 WRF-Chem model
configurations. Each member uses a unique combination of options for PBL
mixing, surface layer, LSM, and longwave and shortwave radiation,
and includes or excludes microphysics and subgrid cumulus convection.
σp,o accounts for instrument precision, representativeness error,
and averaging of measurements to the model resolution. We do not use the
weighting term previously defined by ,
because small residuals with low uncertainty do not appear to hinder the
inversion process. Refer to that work for more particular details of how
σp,m2 and σp,o2 are calculated.
Posterior error
Posterior uncertainty is a useful measure to diagnose the value of an
emission inversion. While areas where uncertainty has been reduced from the
prior include new information from the observations, areas without
uncertainty reduction are simply a new realization of the prior. In a region
of linear behavior of a nonlinear DA cost function, and when δx
is normally distributed, the posterior covariance, Pa, is
equal to the inverse Hessian of Eq. ()
e.g.,:
Pa=Hδx-1,
where
Hδx=B-1+Gk-1⊤R-1Gk-1.
Combining this with the expression for the Hessian of Eq. () we
used in Eq. () gives a conversion from the transformed
variable space
Hδv=U⊤HδxU.
Using a Lanczos recurrence to solve the inner loop optimization problem in
Eq. () has the benefit of producing the means to approximate
Hδv-1, which we demonstrate in
Appendix . The final result of that derivation is the posterior
error,
Pa=UHδv-1U⊤≈B+∑ki=1lλki-1-1Uν^kiUν^ki⊤,
in terms of the eigenvectors of Hδv,
ν^ki=Qlw^lki. Each inner
iteration, ki, leading up to the current iteration l of the Lanczos
optimization, produces (1) a new Lanczos vector in the orthonormal matrix
Ql=[q^1,..,q^l] and (2) a new row and
column in a tridiagonal matrix Tl, whose kith eigenpair is
λki;w^lki. Pa is
a low-rank update to B, because l≪n due to the wall-clock
requirements of running the TLM and ADM once per iteration.
Equation () is consistent with earlier publications
.
Land category types, MODIS fire hotspot detections on 21 and 22 June
2008, sized by FRP, and 18 km × 18 km gridded
FINNv1.0 and QFED emission locations.
Nonlinear optimization
For each outer loop iteration, δxk must be small enough to keep
the error associated with the TL assumption, Eq. (), below
some threshold. Otherwise the cost function may actually increase between
successive k's. However, the nonlinearity of the log-normal prior emission
errors contributes to failures in that respect. Violation of the TL
assumption and potential solutions are discussed in several DA works. The
prevailing strategy in chemical 4D-Var is to apply a quasi-Newton
optimization e.g.,,
eliminating the inner-outer loop structure of GN. Implementing this approach
in WRFDA with posterior error estimation would be a considerable additional
effort. Also, as we mentioned in Sect. , using the tangent
linear model in the inner loop presents computational advantages for dual
resolution 4D-Var.
There are several alternative approaches, which stem from the equivalence
between incremental 4D-Var and GN.
discuss application of GN in a trust region framework, which has the
limitation that a portion of the computationally expensive outer loop
increments will be rejected. Some authors have successfully applied the
Levenberg–Marquardt algorithm in EnKF
e.g., by adding a
regularization term to the cost function. That method requires one to perform
the inner loop approximation of
Hδv-1 multiple times, once for each
value of a scalar regularization parameter. A similar and cheaper approach is
damped GN (DGN), which changes the inner loop increment in
Eq. () to
δvk=-ηkHδv-1∇δvJ|δvk=0,
and uses a line search to find an optimal scalar ηk∈0,1
after the completion of each outer loop iteration
. DGN is based on the Armijo rule, which states
that the increment found by GN points toward a direction of lower J; if the
step size terminus is outside the linear behavior of the model, decrease the
step size. This strategy is implemented in WRFDA-Chem for log-normally
distributed emission errors.
ARCTAS-CARB case study
Inversion setup
From late May until 20 June 2008, the southwestern US experienced a very dry
period with little to no cloud cover appearing in MODIS true color imagery,
and no recorded rainfall for most of California. On 21 June, the Aqua and
Terra satellites recorded cloud cover for much of northern California, south
of San Francisco, and along the Sierra Nevada mountain range, and there were
widespread lightning strikes overnight. As is shown in Fig. , there
was a spike in fire detections during the night between 21 and 22 June. Thus,
from the morning through evening of 22 June, California experienced a
transient fire initiation event. The wildfires burned well into July,
exacerbating poor air quality throughout the state. The 20 June flight of
ARCTAS-CARB characterized northern California anthropogenic sources, but was
not influenced by fires. The 22 June flight embarked from Los Angeles,
transited the offshore Pacific inflow, flew directly through smoke from
forest fires in northern California, and then returned down the coastline.
That flight encountered anthropogenic sources of BC in the morning, and BB
sources for the remainder after returning to land. The 24 June flight passed
back and forth in the downwind region between Los Angeles and San Diego,
measuring the outflow from those cities and the transportation between them,
and 1-day old diluted BB outflow from the north. A fourth flight on 26 June
flew in the free troposphere from Los Angeles, north over the fires, and
exited the model domain to the east.
We use WRFDA-Chem 4D-Var to constrain BB and anthropogenic aerosols on 3 days
during ARCTAS-CARB using aircraft and IMPROVE surface observations. We
utilize aircraft measurements of absorbing carbonaceous aerosol at
10 s intervals from the single-particle soot photometer (SP2) on 22,
24, and 26 June . For this study, we assume
equivalency between the SP2 measurement and modeled BC, and re-average to the
90 s model time step using the revision 3 product, a process
described in . We also use 24 h average
surface observations of light absorbing carbon (LAC) on 23 and 26 June
, assuming an equivalence with modeled BC, and
ignoring the 7 % high bias relative to the SP2 found by
. All treatments of observations are identical to
those described in , including an analysis
of model–observation BC mismatch that feeds into the inverse modeling study.
Using measurements from 22, 23, and 24 June, the 4D-Var system constrains
anthropogenic and BB sources simultaneously. Data collected between 07:00:00
and 16:00:00 LT on 22 June are used in an inversion from 22 June,
00:00:00 UTC to 23 June, 00:00:00 UTC, during which time
WRF-Chem is run freely, without nudging. The emission scaling factors for
this 24 h time period for both source types are applied to subsequent days
from 23 to 26 June in a cross-validation experiment. The 24 and 26 June
aircraft and 23 and 26 June surface observations are used to analyze the
utility of observationally constrained scaling factors found on 1 day to fix
source errors on subsequent days. The 23 and 24 June surface and aircraft
data are used in a 48 h inversion from 23 June, 00:00:00 UTC to
25 June, 00:00:00 UTC, also without nudging. Cross-validation is
performed for these source estimates using 26 June surface and aircraft data.
Through preliminary testing, we found that horizontal correlation length
scales on the order of the grid spacing provide the lowest posterior cost
function. For both time periods, this length scale is set to twice the grid
scale, Lh=36 km. The emission scaling factors are aggregated in
each hour, which coincides with the emission file reading interval for both
source types. The correlation scale is set to Lt=4 h, following
. In addition to spreading error information
across adjacent grid cells, the correlation scales reduce the effective
number of CVs. Through sensitivity tests where we considered the smoothness
of the posterior and the stationary posterior cost function value, and after
consulting published values for regional emission uncertainties (see
Sect. ) in different global settings, we use a relative
grid-scale BB uncertainty of ×3.8. The BB uncertainty might also be
approximated from the ratio of prior domain-wide total emissions between
FINNv1.0 and QFED, which is given in Table as ×3.5. If
the median emission strength lies in the middle of QFED and FINNv1.0, then
the prior domain-wide relative uncertainty is ×3.5=×1.8.
The uncertainty would then need to be inflated further to account for spatial
and temporal disaggregation and the possibility that grid-scale sources from
the two inventories do not bound the true value (see
Sect. ). The prior anthropogenic grid-scale relative
uncertainty is set to ×2, which is within the reasonable bounds
discussed in Sect. .
Emission inversion scenarios.
Scenario
BB inventory
Lh
Obs. used (day)
22 Jun
FINN_STD
FINNv1.0
36 km
ARCTAS-CARB (22)
FINN_L18
FINNv1.0
18 km
ARCTAS-CARB (22)
QFED_STD
QFEDv2.4r8
36 km
ARCTAS-CARB (22)
QFED_L18
QFEDv2.4r8
18 km
ARCTAS-CARB (22)
FINN_V1.5
FINNv1.5
36 km
ARCTAS-CARB (22)
23/24 Jun
FINN_STD
FINNv1.0
36 km
IMPROVE (23)
ARCTAS-CARB (24)
QFED_STD
QFEDv2.4r8
36 km
IMPROVE (23)
ARCTAS-CARB (24)
ACFT
FINNv1.0
36 km
ARCTAS-CARB (24)
SURF
FINNv1.0
36 km
IMPROVE (23)
In addition to these standard settings, several sensitivity scenarios are
used to gauge the sensitivity of the posteriors during two time periods to
alternative inversion settings. The full set of scenarios are summarized in
Table , and are as follows. FINNv1.0 is used as the default
BB inventory in a scenario called FINN_STD for both inversion periods.
QFED_STD uses the QFEDv2.4r8 BB inventory. Both FINN_L18 and QFED_L18 use
Lh=18 km. FINN_V1.5 utilizes the FINNv1.5 BB inventory. For the
23/24 June inversion, we show results for both QFED_STD and FINN_STD, the
latter of which includes variations where either surface or aircraft
observations are excluded. The number of aircraft observations is
Nobs=241 on 22 June and Nobs=302 on 24 June. There
were Nobs=35 active surface sites on 23 June, 13 of them within
California. We use six outer iterations consisting of 10 inner iterations
each. Given the number of inner iterations used, and the wall-time of the
tangent linear plus the adjoint (×10 longer than the nonlinear model),
the cost of 4D-Var is approximately ×600 more than that of a single
forward simulation, which is much cheaper than using finite difference
methods to approximate derivatives instead of the linearized models when
n∼105.
Posterior model performance
For a linear model operator and Gaussian distributed errors, the cost
function can be used to evaluate the consistency of the statistics in
B and R. The χ2 criteria states that the
posterior cost function should be equal to 12NOBS
e.g.,. The convergence properties of the 22
and 23/24 June inversion scenarios are shown in the outer loop cost function
progression in Fig. . All of the 22 June scenarios led to
comparable cost function values at numerical convergence, as shown in
Fig. . The gradient norms are also reduced by nearly two orders
of magnitude in all cases. In all of the scenarios, J converges to
approximately NOBS, indicating that a portion of the model errors
are not fully spanned by prior emission errors. For the 23/24 June inversion,
QFED_STD reaches a lower cost function value, and both scenarios achieve
similar χ2 values as the 22 June cases. Scrutinizing other sources of
error (e.g., initial and boundary conditions for BC and meteorological
variables, transport, BB plume rise, and model discretization) either
independent from source strengths or simultaneously in the inversion
framework should elicit further cost function reductions. Considering the top
subplot in Fig. , the non-emission sources of error on 22 June
are evident when the prior and posterior predictions are on top of each
other, and remain on the edges of the low probability uncertainty region. For
example, the observations before 08:00 LT after takeoff from Edwards
are likely sensitive to the Pacific inflow, but not early morning emissions.
Since these locations have relatively small uncertainties, the posterior cost
function will never be reduced there.
Outer loop cost function and gradient norm evaluations for the 22
June (left column) and 23/24 June (right column) inversions.
Figure also shows that during the inversion period the
posterior is within the combined model–observation uncertainty (see
Sect. ) much more often than the prior. In the afternoon,
when the DC-8 passed over the wildfires, an increase in posterior emissions
captures several of the observed BC peaks. The posterior is able to match the
high-resolution variability of the observations at 13:30 LT, which
may support the validity of the temporal averaging scheme. The only time
during the inversion when the forecast degrades is for an observed peak at
22 June, 08:00 LT. The larger absolute observation uncertainty at
that time relative to that at 08:30 LT enforces a weaker constraint
in the inversion. Coupled with the assumed relative emission error
correlation length scale and the close proximity of these two measurements,
that stronger constraint at 08:30 LT dominates the morning
anthropogenic emission analysis increment.
Temporal variation of observed, prior, and posterior BC
concentrations during ARCTAS-CARB. The model values are obtained with the
FINN_STD inversion scenario. The shaded area encompasses 2 standard
deviations around the observations, which includes both model and observation
uncertainty.
Prior and posterior model versus 22 June ARCTAS-CARB observations
for the 22 June inversion. The left two plots are for FINN_STD and
QFED_STD. The plot on the right shows the progression of the slope and R2
from the prior, “0”, to the posterior, “a”, for similar linear
regressions in all scenarios.
The R2 coefficients and slopes for linear fits between the prior and
posterior and both aircraft and surface observations are summarized in
Tables and . Those results include
cross-validation data on non-inversion days, which is discussed in
Sect. . For both inversion periods, there are considerable
model performance improvements for observations that are used in the
inversion. FINN_STD improves R2 from 0.11 to 0.82 and slope from 0.26 to
0.8 on 22 June. QFED_STD improves R2 from 0.03 to 0.73 and slope from
0.34 to 0.71. Similar improvements occur for 23 June surface observations
during the 23/24 June inversion. The posterior match to 24 June aircraft
observations is improved, but not nearly as much as the other two data sets.
The 22 June inversion results are also shown in Fig. , where
the progression of the fit parameters is shown for the multiple scenarios.
While all scenarios show similar improvements, the FINN_STD and QFED_STD
results indicate the posteriors are still underpredicting many low and high
concentrations. A similar phenomenon occurs for the 24 June observations in
Fig. in the inversion that uses both surface and aircraft
observations. On both 22 and 24 June, the remaining low bias is either due to
large prior observation and model error (diagonal of R) or due to
the prior errors not being sensitive to emission increments.
Aircraft observation linear regression characteristics for the prior
(background, b) and posterior (analysis, a).
Obs. date →
22 Jun, Nobs=241
24 Jun, Nobs=301
26 Jun, Nobs=117
Inversion scenario
R2
slope
R2
slope
R2
slope
↓
b
a
b
a
b
a
b
a
b
a
b
a
22 Jun
FINN_STD
0.11
0.82
0.26
0.80
0.18
(0.15)
0.38
(0.25)
0.56
(0.52)
0.15
(0.49)
QFED_STD
0.03
0.73
0.34
0.71
0.15
(0.23)
0.43
(0.37)
0.59
(0.53)
0.39
(0.43)
23/24 Jun
FINN_STD
–
–
–
–
0.17
0.52
0.35
0.56
0.59
(0.16)
0.15
(0.11)
QFED_STD
–
–
–
–
0.11
0.52
0.36
0.55
0.63
(0.44)
0.41
(0.15)
ACFT
–
–
–
–
0.17
0.53
0.35
0.57
0.59
(0.29)
0.15
(0.08)
SURF
–
–
–
–
0.17
(0.17)
0.35
(0.40)
0.59
(0.13)
0.15
(0.17)
Distinct improvement (bold). Distinct degradation (italic).
Cross-validation (parentheses).
For the appreciable measured BC concentrations (> 0.25 µgm-3), which are likely caused by a source within the model domain and
simulation period, the lack of a source–receptor relationship is likely
caused by low resolution. Changing a point source to a grid-scale area source
changes its effective location. Temporal averaging of the observations will
not necessarily solve that problem since perfectly modeled transport could
still send a mislocated source in an entirely different direction than the
truthfully located source. This effect is evident for valley fires
, since placing the sources in the basin or
spreading them throughout the basin and the peaks will result in different
“downwind” concentrations. Downwind might be a very different direction if
the convective-scale winds contribute more information than the mesoscale
winds to the true source–receptor relationship. Since the emissions are
smoothed in the model and not in reality, the mislocation is more likely to
cause underprediction than overprediction.
Posterior emissions
Figure shows the prior and posterior BB emissions for
FINN_STD and QFED_STD during both simulation periods. In that figure there
are several outlined emission areas (EAs); each EA was chosen to identify
regions where a subset of the grid-scale analysis increment (δxEAX⊂δx) from both prior inventories
is of a similar sign. The coordinates of the EAs are listed in
Table . The two inversions do not reach identical total
posterior BC emissions, but they do converge in certain aspects.
Table gives the emission subtotals for the EAs. During both
inversions, each EA has emission increments of the same sign for both
scenarios. Therefore, while domain-wide sources seem to be bounded by the two
priors (as evidenced by their convergence), the same might not be true within
the individual EAs. EA3, which accounts for the smallest average posterior
total, is the only region where the magnitude of the log ratio between QFED
and FINN is smaller in the posterior on 22 June. The ratio is reduced in EA2,
but there the FINN posterior is ×2 larger than that for QFED. On
23/24 June, the two scenarios have less posterior spread in all of the EAs.
Although Table indicates large changes in source strengths
across the EAs, Fig. reveals that a majority of the absolute
emission increment (posterior minus prior) in both FINN_STD and QFED_STD
arose in only a few grid cells, often where the prior has the largest
magnitude. The linear scaling factor pattern is similar between the two
scenarios, with those for QFED_STD shifted toward decreases due to the high
prior bias.
Surface observation linear regression characteristics for the prior
(background, b) and posterior (analysis, a).
Obs. date →
23 Jun, Nobs=35
26 Jun, Nobs=36
Inversion scenario
R2
slope
R2
slope
↓
b
a
b
a
b
a
b
a
22 Jun
FINN_STD
0.06
(0.04)
0.26
(0.21)
0.03
(0.05)
0.10
(0.13)
QFED_STD
0.16
(0.14)
0.44
(0.41)
0.10
(0.11)
0.20
(0.21)
23/24 Jun
FINN_STD
0.04
0.75
0.25
1.04
0.03
(0.28)
0.10
(0.28)
QFED_STD
0.09
0.74
0.39
1.01
0.09
(0.15)
0.20
(0.16)
ACFT
0.04
(0.05)
0.25
(0.27)
0.03
(0.03)
0.10
(0.09)
SURF
0.04
0.74
0.25
1.02
0.03
(0.35)
0.10
(0.35)
Distinct improvement (bold). Distinct degradation (italic).
Cross-validation (parentheses).
Emission area coordinates. EA1–4 are used for BB totals and EA5–9
are used for anthropogenic totals.
LONmin
LONmax
LATmin
LATmax
EA1
122.5∘ W
120.5∘ W
35.7∘ N
38.5∘ N
EA2
123.8∘ W
122.1∘ W
38.9∘ N
40.4∘ N
EA3
124.3∘ W
122.9∘ W
40.4∘ N
41.7∘ N
EA4
122.1∘ W
120.0∘ W
38.5∘ N
40.4∘ N
EA5
117.8∘ W
116.9∘ W
32.1∘ N
33.4∘ N
EA6
121.0∘ W
117.8∘ W
33.4∘ N
34.6∘ N
EA7
123.0∘ W
121.0∘ W
36.6∘ N
38.8∘ N
EA8
120.6∘ W
118.6∘ W
35.2∘ N
37.0∘ N
EA9
118.0∘ W
116.5∘ W
34.0∘ N
36.0∘ N
EA10
116.9∘ W
115.0∘ W
32.1∘ N
33.4∘ N
The temporal distributions of prior and posterior BB emissions within the
four EAs are shown in Fig. across all inversion scenarios on
22 June. The FINNv1.5 prior is an extreme outlier on the local afternoon of
21 June for EA1, EA2, and EA4. The same is true all day on 22 June for EA2,
where the posteriors from other scenarios adjust toward the FINNv1.5 prior.
Meanwhile, at other times when FINNv1.5 appears to converge toward the
posteriors found using the other two priors, the prior relative uncertainty
of ×3.8 is too restrictive to allow full convergence, since the priors
differ by ×10. EA1 is characterized by decreases for all scenarios at
all times. EA2, EA3, and EA4 exhibit early morning peaks between 03:00 and
06:00 LT that were not captured in the prior. In separate sensitivity
tests, these peaks only appear when Lt>1 h, and become more
prevalent as Lt is increased. attributed
similar behavior in posterior estimates of the 2013 Rim Fire to persistent
large-scale burning. found similar,
less pronounced bimodal behavior for all of North America, which could be
more noticeable in a regional inversion. Another possibility on 22 June 2008
is that the early morning burning is caused by the transient fire initiation
event, which would explain the ramping of emissions for the QFED and FINNv1.5
posteriors in EA2. For both QFED and FINNv1.0, reducing the correlation
length to Lh=18 km reduces the analysis increment in all EAs. This
is especially apparent in EA4 for FINN_L18, where the increment is
negligible.
Prior and posterior model versus 24 June ARCTAS-CARB observations
for the 23/24 June FINN_STD inversion. The left plot uses both IMPROVE
(23 June) and ARCTAS-CARB observations in the inversion. The middle plot uses
only ARCTAS-CARB. The plot on the right shows the progression of the slope
and R2 from the prior, “0”, to the posterior, “a”, for similar linear
regressions.
Prior and posterior grid-scale BB emissions of BC per 24 h for
FINN_STD and QFED_STD on 22 June, 00:00–23:00 Z and 23 June
00:00–24 June 23:00 Z. All emissions are expressed for a 24 h
average. EA1–4 are outlined with black boxes.
The differing diurnal patterns in EA2 across scenarios could be attributed to
variation in plume heights, QFED regridding errors, and the regularization
term of the cost function. The observations most sensitive to EA2 sources
were captured within or very near fire plumes. Plume heights are calculated
hourly in an online 1-D vertical mixing scheme in WRF-Chem
,
which depends strongly on burned areas. With FINN, the areas are provided for
each fire independently, while for QFED the areas use a default value of
0.25 km2 per fire. In both cases, the maximum area burned per grid
cell per day is 2 km2. The regridding error discussed in
Sect. introduces fire locational errors, especially in EA2. A
small error in vertical or horizontal mapping of a discrete point source on
the model grid could hinder the optimization in distinguishing it from
others. The uniform relative uncertainty in the prior inhibits consolidation
of multiple posteriors when the prior spread is heterogeneous and sometimes
very large. Quantifying the heterogeneity of uncertainty could contribute to
posterior agreement between inversions using different priors, as well as to
reducing the cost function.
BB analysis increment (posterior minus prior) per 24 h and
posterior linear scaling factor (β) for the two primary BB scenarios on
22 June 00:00–23:00 Z and 23 June 00:00 Z–24 June 23:00 Z. EA1–4 are
outlined with black boxes.
The spread of local emissions provides some sense of that heterogeneity. Each
EA covers a region approximately the size of a grid box in a global
simulation with a chemical transport model. Due to the nature of variance
aggregation, uncertainty grows as the grid scale gets smaller. In individual
EAs, the spread between FINNv1.0 and QFED priors is ×2–×6 for
both hourly (Fig. ) and daily (Table ) strength
on 22 June. If the median emission strength lies in the middle, then a proxy
for prior EA relative uncertainty is ×2-×6=×1.4-×2.4. Since the two inventories use identical diurnal
patterns, the hourly estimate is missing information about uncertainties in
daily emission timing. Using the posterior spread in a similar way gives
approximate EA uncertainties of ×3-×10=×1.7-×3.2 on hourly scales and ×2-×7=×1.4-×2.6 on daily scales. This posterior estimate accounts for
contributions in the prior definitions, including regridding, plume rise, and
diurnal patterns. These ranges provide much more detail estimates than simply
taking the domain-wide ratio of total emissions for the campaign period.
However, the spread is itself missing information about uncertainty that
could be found through carrying out similar inversions across an ensemble of
model configurations and meteorological initial and boundary conditions
e.g.,, or by comparing many more
inventory priors e.g., and posteriors. All
this is to say that the BB inventories used in this study are not provided
with analytical estimates of uncertainty, and a lack of information for
deriving such values at hourly grid scales is a topic for future research.
Figure shows the total prior and posterior anthropogenic
emissions and Fig. displays the analysis increment and
linear scaling factor for FINN_STD on 22 June and separately on 23–24 June.
The only difference in QFED_STD, not shown here, is that anthropogenic
scaling factors are shifted in the negative direction in the posterior,
likely due to the higher bias in that BB prior. The increments found in a new
set of EAs are presented in Table .
The 23 and 24 June observations provide much more detailed information about
anthropogenic sources. The analysis increment reveals potentially
misrepresented city-level emissions in the NEI05 prior. Posterior BC near
Barstow, Victorville/Hesperia, Fresno, Edwards Air Force Base, and El
Centro/Calexico are increased, while sources near the three coastal cities
are decreased. Since Barstow is a crossroads for the BNSF and the Union
Pacific railroads, and since Fresno, Victorville/Hesperia, and El
Centro/Calexico lie at switching locations for major rail lines, the
inversion results may suggest that the prior is missing diesel rail sources
of BC. However, for locations where the prior magnitude of BB and
anthropogenic emissions are of similar magnitude, their posteriors are
subject to projection from one sector to another. It is more likely that the
low bias fire emissions north of Fresno are responsible for the prior
underpredictions of 23 June surface concentration measurements exceeding
2 µgm-3 see Fig. 5 of.
This is corroborated by the posterior BB emissions being scaled up near
Fresno on 23 and 24 June, and by the much smaller model bias for IMPROVE on
22 June before the fires started.
There are also small negative increments near Los Angeles (EA6) and San
Francisco (EA7) during both the 22 June and 23/24 June inversions, which are
likely attributable to on-road mobile sources. These results are consistent
with model bias in surface and aircraft observations on 20 June near both of
those cities .
found a decreasing trend in ambient
measurements of BC and in a fuel-based bottom–up inventory for both Los
Angeles and San Francisco from 1990 to 2010 that might not be captured for
the 2008 model year by the snapshot in NEI05. Using a similar fuel-based
approach, derived 2010 CO emissions in the South
Coast Air Basin surrounding Los Angeles that are ×1/2 the magnitude of NEI05. On-road and
other mobile sources make up 36 and 62 % of that difference,
respectively, and their bottom–up inventory matches more closely with NEI
2011. While not a perfect comparison to BC in 2008, the sign of error in
NEI05 relative to the coastal posterior and that study is consistent. An
inventory with sector-specific breakdowns of BC emissions, additional
inversions with more thorough speciated local observations, and higher
resolution would all be required to investigate sector-specific anthropogenic
pollution.
Total BB emissions for EAs and domain-wide during the 22 and
23/24 June inversions (averaged for a 24 h period). Absolute units are in
Mg. Note that the differences (Δ) may not sum due to rounding.
FINN_STD
QFED_STD
ΣEQFEDΣEFINN
ΣEb
ΣEa
Δ
ΣEb
ΣEa
Δ
b
a
22 Jun
EA1
14
4
-10
82
26
-55
×5.8
×6.4
EA2
6
30
+24
9
15
+6
×1.5
×0.5
EA3
6
4
-2
29
7
-22
×4.5
×1.6
EA4
18
22
+4
52
83
+31
×2.8
×3.8
DOMAIN
59
83
+34
209
171
-38
×3.5
×2.1
23+24 Jun
EA1
20
5
-15
70
12
-58
×3.5
×2.5
EA2
28
11
-16
96
29
-67
×3.5
×2.6
EA3
17
12
-5
37
20
-17
×2.2
×1.7
EA4
32
108
+77
107
107
0
×3.4
×1.0
DOMAIN
138
249
+111
471
354
-117
×3.4
×1.4
Total anthropogenic emissions for EAs and domain-wide during the 22
and 23/24 June inversions (averaged for a 24 h period). The posterior for
23/24 June is from an inversion using both the IMPROVE and ARACTAS-CARB
observations. Results shown are for the FINN_STD scenario. Absolute units
are in Mg. Note that the differences (Δ) may not sum due to
rounding.
22 Jun
23/24 Jun
ΣE23+24JunΣE22Jun
ΣEb
ΣEa
Δ
ΣEb
ΣEa
Δ
b
a
EA5
5
5
0
7
3
-3
×1.4
×0.7
EA6
12
8
-4
17
9
-8
×1.4
×1.2
EA7
10
6
-5
16
8
-8
×1.6
×1.5
EA8
3
2
-1
5
25
+20
×1.6
×9.9
EA9
5
4
-1
6
11
+4
×1.3
×2.7
EA10
2
2
0
3
8
+5
×1.4
×3.7
DOMAIN
81
68
-13
114
123
+9
×1.4
×1.8
Hourly BB diurnal emission patterns for the four EAs and all
inversion scenarios for 22 June, 00:00–23:00 Z, with the time shown in
LT. The priors are shown as black lines, while the posteriors from
specific inversion scenarios are shown in color. Note that FINNv1.0 did not
have any fires in EA4 on 21 June.
Error diagnostics
Analysis of posterior emissions uncertainties is useful for
understanding the value of the posterior emissions themselves. The diagonal
terms of Pa are the posterior variances,
σxa, which are always smaller than prior variances. The
variance reduction could instead be presented in β space, by utilizing
Eq. (). However,
σβk,i2<σβ0,i2 is not guaranteed when
xa,i>xb,i=0, because the posterior relative emission uncertainty
depends on xa,i. For this work, the reductions in variance are presented
in CV space. The low-rank estimate of Pa is only valid for
linear perturbations away from xa. The final outer loop estimate of
Pa is the most accurate, since it is linearized around the
state preceeding xa. A quantitative measure of error reduction in
the koth outer loop in the ith CV is
ρi,ko=1-Pi,iaBi,iko∈[0,1).
Values of ρi,ko closer to 1 reflect locations where the observations
provide a stronger constraint than the prior. This estimate may not reflect
the entire error reduction, since it does not capture potential reductions in
previous outer loops. Without propagating updated estimates for B
to subsequent outer loops e.g.,, we
also define ρagg, a qualitative metric that accounts for
increases in curvature (decreases in error) in all outer loops:
ρi,agg=1-∏ko=1kPi,iaBi,iko∈[0,1).
ρi,agg reveals additional information about observation
footprints not shown by ρi,ko=6. The nonlinear nature of the problem
means ρi,agg is not quantitative.
Both (ρko=6) and ρagg are presented in
Figs. and for the BB and anthropogenic
members of xa, respectively. Fifty inner loop iterations were taken
in the final outer loop to improve ρ estimates. ρko=6 is
<45 % across all scenarios, except for QFED_STD BB sources near the
IMPROVE sites on 23/24 June. If the inner loop were halted at 10 iterations,
the error reduction estimates would be reduced by up to ∼ 10 %
(i.e., 35 instead of 45 %) in the darkest grid cells. The BB error
reduction shown in Fig. has similar spatial distributions
for FINN_STD and QFED_STD scenarios, but differs significantly between the
two time periods due to the different spatial coverage of the observations.
The reductions in the north on 22 June are more disperse for QFED_STD, which
could be caused by the same regridding errors and
plume rise differences that
influence the posterior emissions. There is also more error reduction in the
south for the QFED_STD emissions. In general, the grid-scale uncertainty
improvement is confined to sources close to the observations.
The most obvious application of ρ is to evaluate the footprint of a set
of measurements. For example, the large relative BB emission increments in
EA1–EA3 on 23/24 June indicate that distant observations can have a large
impact on the posterior emissions magnitudes. However, ρi,ko=6 in
Fig. indicates there is nearly zero uncertainty reduction
for those emissions. Also, upon considering the last two columns of
Table , one might conclude that there is a missing weekend
(22 June) to weekday (23/24 June) variation in BC emissions within EA8–10.
However, Fig. shows that the 22 June observations only
weakly reduce uncertainty in emissions.
In a more tangible application, ρ can be used to assess existing and
future observing strategies in a similar way to how
used adjoint sensitivity information to plan future meteorological observing
sites to improve forecasts of extreme dust events in the Korean Peninsula.
Fig. presents anthropogenic ρ for different
combinations of surface and aircraft observations on 23/24 June. The surface
observations primarily resolve sources near Fresno, and to a lesser extent
near Los Angeles. Since the purpose of the IMPROVE network is to measure
background concentrations, it is mostly successful on 23 June in not being
influenced by anthropogenic sources of BC from the major cities. If the goal
were to measure anthropogenic sources, inflows, or domain-wide concentrations
on daily timescales, then ρ would suggest using a different surface
network distribution. Such a conclusion does not conflict with the success of
using IMPROVE observations to provide top–down constraints on both BB and
anthropogenic emissions on monthly timescales
e.g.,. That strategy is consistent with what is
generally known: further decreasing uncertainty requires observing the same
phenomena more thoroughly. For hourly to daily timescales, more observations
are needed close to and downwind of chemical sources, and at high spatial and
temporal resolution (e.g., from repeated aircraft overpasses, extra aircraft,
hourly-average surface sites, or satellites).
Prior and posterior grid-scale anthropogenic emissions of BC per 24 h
for FINN_STD on 22 June, 00:00–23:00 Z (top row) and 23 June, 00:00 Z to 24 June,
23:00 Z. EA5–10 are outlined with black boxes.
Anthropogenic analysis increment (posterior minus prior) per 24 h
and posterior linear scaling factor (β) for the
(a) FINN_STD (22),
(b) FINN_STD (23/24), (c) ACFT, and (d) SURF inversion scenarios.
EA5–9 are outlined with black boxes in the scaling factor plots.
BB error reduction in the final outer loop (ρko=6) and
aggregated across all outer loops (ρagg) for the two primary
BB scenarios on 22 June 00:00–23:00 Z and 23 June, 00:00 Z–24 June 23:00 Z. The
ARCTAS-CARB DC8 flightpath and IMPROVE sites at model grid centers are
overlaid.
Anthropogenic error reduction in the final outer loop
(ρko=6) and aggregated across all outer loops (ρagg)
for the (a) FINN_STD (22), (b) FINN_STD (23/24), (c) ACFT, and (d) SURF
inversion scenarios. The ARCTAS-CARB DC8 flightpath and IMPROVE sites at
model grid centers are overlaid.
Eigenvalue spectra for FINN_STD and QFED_STD in the final outer
loop on 22 June. The lines show the estimate of the spectrum
[λ1,…,λki=l] in every fourth inner loop iteration,
l. The black numbers in parentheses are the estimates of DOF that include
eigenvalues in the sets (converged to within 5 % of the previous
estimate, all available). The red numbers in brackets are the truncated
estimates of DOF using the most completely converged set of eigenvalues
available in the 50th iteration.
Another piece of information useful for comparing observing configurations
and inversion scenarios is the trace of the resolution matrix, or degrees of
freedom for signal, i.e.,
DOF=TrIn-PaB-1,
which is equal to the number of modes of variability in the emissions that
are resolved by the observations
.
Substituting the approximation for Pa from
Eq. (),
DOF≈n-TrB+U∑ki=1lλki-1-1ν^kiν^ki⊤U⊤B-1≈-TrU∑ki=1lλki-1-1ν^kiν^ki⊤U⊤B-1.
Since U is square,
U⊤B-1U=In, and
Trν^kiν^ki⊤=ν^ki⊤ν^ki=1, the expression
simplifies as
DOF=-Tr∑ki=1lλki-1-1ν^kiν^ki⊤U⊤B-1U≈∑ki=1l1-λki-1Trν^kiν^ki⊤≈∑ki=1l1-λki-1.
Therefore, the only information needed to compute DOF is the eigenvalues of
Tl. Each inner loop, ki, has the potential to constrain one
additional mode of variability in the emission scaling factors. For all of
our inversion scenarios, the leading eigenvalue is on the order of
102-103, which is equal to the condition number of the full-rank Hessian.
As the Lanczos optimization proceeds, each subsequent λki is
smaller, asymptotically approaching unity, and each eigenmode provides less
information than the one preceding it about scaling factor variability.
Figure gives three estimates of DOF at each level of
truncation in the final outer loop, that is, if higher degrees of eigenvalues
were ignored. In that figure, we plot eigenvalue spectra of the FINN_STD and
QFED_STD scenarios on 22 June. Similar to ρ, we use a 50-iteration
linear optimization to improve the bounds on DOF. The ki=l estimate of the
eigenvalue spectrum at each iteration is represented by a single colored
line. Each member of the eigenvalue spectrum, represented by vertical grid
lines in Fig. , converges toward an upper bound as more
iterations are taken. Initial guesses for the least dominant eigenvalues are
less than 1 for ki≥8 for FINN_STD, but they exceed 1 after an
additional iteration, consistent with the properties of the Lanczos sequence.
The first DOF value in parentheses adheres to the philosophy that only
converged eigenvalues should be used to estimate DOF; it excludes
λki,…,λl such that λki is more than
5 % changed from the previous estimate. The second DOF value in
parentheses uses all of the current estimates of the eigenvalues available in
iteration l. This is still a conservative estimate of DOF, because the true
eigenvalues of the full-rank Tn are always larger than their
current numerical estimate. After enough iterations, the numerical growth in
DOF is very small, and further computation is not warranted. As the
eigenvalue spectra in Fig. and the cost function reduction
in Fig. show, this is long after the cost function is
converged enough for practical purposes. The posterior CVs, which are the
primary result from inverse modeling, do not change significantly in the
final outer loop. Finally, the best estimates of DOF in red brackets are
evaluated at different truncations using the most-converged values of the
eigenvalues found in the 22nd iteration.
Eigenvalue spectra for SURF, ACFT, and SURF+ACFT in the final outer
loop on 23 and 24 June. The lines show the estimate of the spectrum
[λ1,…,λki=l] in every fourth inner loop iteration,
l. The black numbers in parentheses are the estimates of DOF that include
eigenvalues in the sets (converged to within 5 % of the previous
estimate, all available). The red numbers in brackets are the truncated
estimates of DOF using the most completely converged set of eigenvalues
available in the 50th iteration.
Similar to ρ, the quantitative application of DOF is limited to the
final outer loop, when δxn is small enough that
(Hδv)-1|xn-1≈(Hδv)-1|xn.
Absent the need to estimate the posterior Hessian, the outer loop could be
ended an iteration earlier. In the inner loop, truncated estimates of
H-1 and its eigenvalue spectrum at earlier iterations will
provide conservative values for both DOF and ρ. The actual DOF value is
higher than any value shown in Figs. (22 June)
and (23/24 June). Therefore, the 22 June observations
constrain > 14 modes of hourly grid-scale variability through 4D-Var in
both the FINN_STD and QFED_STD scenarios. Just like for ρ, the
optimization constrains additional modes in the earlier outer loop
iterations, but that quantification is not straightforward since DOF are
defined for linear behavior. If all outer loops were similar, then the total
DOF value for the entire nonlinear optimization is on the order of 30 to 40.
As shown in Fig. , the DOF values on 23 and 24 June after 50
iterations are 10, 17, and 23 for the SURF, ACFT, and FINN_STD(23/24)
scenarios, respectively. The relative magnitudes show that using combined
surface and aircraft observations provides an additional value over using
either independently, although the two platforms might have some redundancy.
This conclusion is consistent with the maps of BB and anthropogenic ρ in
Fig. , where the footprints of SURF and ACFT have slight
overlap near Los Angeles, but are otherwise independent. Additionally, the
higher DOF value of ACFT is consistent with its more widespread and larger
magnitude ρ values. The slower eigenvalue convergence when both
observing types are utilized means that additional inner iterations could
yield higher estimates for DOF in that case. What is even more clear, and
intuitive, is that ρ and DOF estimates require more iterations as the
number of constrained CVs increases, which is directly dependent on the
number of observations. The sparse ρ map for SURF in
Fig. and the large spike near Fresno illustrate that while
near-source surface measurements can be a powerful constraint, measurements
of background concentrations provide relatively little constraints to
characterize CA anthropogenic emissions on 1-day timescales.
Cross-validation
As an additional evaluation of the robustness of the emission scaling
factors, we apply them in cross-validation tests. In two separate
evaluations, the 22 June scaling factors are applied to 23–26 June
emissions, and the 23/24 June scaling factors are applied to 25–26 June
emissions. The heterogeneous adjoint sensitivity signs and magnitudes for
each source sector we found on each day of the campaign
are an indication that corrective scaling
factors in each day will be unique. In that work, we found that the 24 June
observations were most sensitive to southern California anthropogenic sources
on 24 June and to northern and southern California coastal sources of both
sectors on 23 June. The 26 June observations were most sensitive to northern
California fires, and the adjoint sensitivities were of opposite sign than on
23 and 24 June.
As shown in Fig. , the cross-validated 22 June scaling
factors rarely generate improvements to model performance, when compared to
24 and 26 June aircraft observations. On 24 June, some of the high bias
predictions are corrected, or even over-compensated, but the low bias prior
locations are unaffected. Table shows the R2 and slope
of the linear trend lines. The scatter of the fit for QFED_STD on 24 June
and the slope for FINN_STD on 26 June are slightly improved, but all other
metrics degrade. The increase in slope for FINN_STD comes as a result of
better fit to very large concentrations above the PBL associated with fire
sources on multiple previous days. The posterior scaling factors generated
from the 23/24 June inversion degrade the forecast of aircraft measurements
on 26 June. Since the posterior primarily serves to reduce coastal
anthropogenic and BB emissions, it is not surprising that it does not improve
a low bias prior 2 days later.
Table includes cross-validated surface measurements on
23 June and 26 June. There is very little change to the modeled surface
concentrations as a result of posterior scaling factors derived from
inversions that only use aircraft observations. Assimilating surface
observations on 23 June (Monday) does improve model comparisons to surface
observations on 26 June (Thursday). Those small improvements imply that
errors are weakly correlated between weekdays. Although it is beyond the
information content provided by the observations used in this work, future
studies could compare the efficacy of using weak multiday correlation in
B and the hard constraint of 24 h periodic scaling factors
used herein. Aircraft and surface observations do not appear to be useful for
cross-validation of each other over the short timescales and limited set of
flights considered here. At least for this study period, when they are not
collocated, each provides some unique information to the inversion.
Conclusions and future work
We have presented the implementation and an application of
incremental chemical 4D-Var using an atmospheric chemistry model with online
meteorology in WRFDA-Chem. This work expands on our previous efforts to
develop the ADM and TLM in WRFPLUS-Chem .
This new inversion tool takes advantage of previous developments of
meteorological data assimilation in WRFDA
. That same framework
is applied to log-normally distributed emission scaling factors through an
exponential transform. We utilize the square root preconditioner for a CVT
using horizontal and temporal scaling factor correlations. The Lanczos linear
optimization algorithm in the inner loop allows for estimation of posterior
error and DOF for objectively evaluating observing systems. Outer loop
convergence is improved with a heuristic DGN multiplier, which allows the
incremental framework to handle the nonlinearity of the log-normal cost
function. While the optimizations herein focus exclusively on emissions,
which are known to be important drivers of model uncertainty in BC estimates
e.g.,, other factors
such as meteorology, plume rise and deposition mechanisms may also affect the
model's predictions of BC concentrations.
When applied to the ARCTAS-CARB campaign period, it is not clear which prior
emissions perform better. If assessment by initial cost function value alone
were meaningful, FINNv1.0 performs best. However, that could be due to
FINNv1.0 being biased low combined with the assumption of Gaussian
distributed model–observation errors. Positive residuals are weighted higher
than negative ones, even when relative errors are equal. There could be some
improvement to the posterior emissions by implementing the incremental
log-normal methods of or
. If the purpose of the inventory is to provide air
quality warnings to the major California cities, then FINNv1.0, FINNv1.5, and
QFEDv2.4r8 all have some built-in high bias that will err on the side of
caution. Their inability to reproduce high concentrations near sources points
to either a deficiency in the inventories, vertical mixing processes, or the
temporal observation averaging procedure followed herein, diagnosis of which
would require measurements of plume injection heights and widths. The
relative magnitudes of grid-scale fire and anthropogenic emissions make it
difficult to simultaneously constrain them without additional information.
More work should be done to improve both bottom–up and top–down estimates
of anthropogenic emissions outside of fire events. We also agree with
, who recommended multi-species inversions (e.g.,
BC and CO) to discern specific source sectors.
Through the setup and application of the 4D-Var system, we gained valuable
knowledge to guide future modeling and measurement efforts. We found two
errors in the diurnal distribution of BB emissions and identified a scaling
necessary to apply QFED to the western US. Additionally, the highly
heterogeneous posterior scaling factors during ARCTAS-CARB raise questions
that the limited BB observations during that time period do not answer.
(1) Are BB emission errors always heterogeneous, or only during a transient
initiation stage like that observed in June 2008? If heterogeneity is
consistent outside initiation events, then inversions should apply weaker
inter-day correlation than the hard constraint used herein or have
independent scaling factors for each day. (2) Are the temporally bimodal
posterior emissions realistic, or are they an artifact of the correlation
timescale used? (3) Are the BB plume heights reasonable, and should they
follow a diurnal pattern? The current 1-D plume rise mechanism in WRF-Chem
depends strongly on specified burned areas, which are diurnally invariant and
highly uncertain e.g.,. The last two
questions indicate that there is value in continuous night (between 20:00 and
06:00 LT) and day measurements of the same fire region. Since models
poorly predict shallow boundary layers, the use of nighttime observations in
4D-Var would require characterization and subsequent model tuning of those
vertical mixing processes. Furthermore, if it is accepted that
high-resolution models are required to accurately predict degraded air
quality events, then high spatial and/or temporal resolution concentration
measurements from research campaigns or geostationary satellites are
necessary to provide the sufficient constraints on inventory errors. The
error reduction estimation method provided herein will be useful for planning
these future missions.
Future applications of the WRFDA-Chem system developed here may consider
improvements such as the following. One possible way to reduce model
uncertainty would be to extend the multi-incremental 4D-Var available in
WRFDA to the new scaling factor CVs.
Multi-incremental chemical 4D-Var would use a high-resolution model forecast
to generate trajectory checkpoint files (see
), and could take advantage of
improvements to chemical transport at higher resolution realized by using
online meteorology demonstrated by and
. In addition, four-dimensional data assimilation (FDDA) nudging
has been shown to improve wind fields and was used successfully in an LPDM
emission inversion . Even after
exhausting methods to improve the posterior, the error contributions from
hard-coded descriptions of meteorology can be bounded using ensemble and
sensitivity tests
e.g.,.