We theoretically and numerically investigate the problem of assimilating
multiwavelength lidar observations of extinction and backscattering
coefficients of aerosols into a chemical transport model. More specifically,
we consider the inverse problem of determining the chemical composition of
aerosols from these observations. The main questions are how much information
the observations contain to determine the particles' chemical composition,
and how one can optimize a chemical data assimilation system to make maximum
use of the available information. We first quantify the information content
of the measurements by computing the singular values of the scaled
observation operator. From the singular values we can compute the number of
signal degrees of freedom,

Atmospheric aerosols have a substantial, yet highly uncertain impact on
climate, they can cause respiratory health problems, degrade visibility, and
even compromise air-traffic safety. The physical and chemical properties of
aerosols play a key role in understanding these effects. The aerosol
properties are determined by a complex interplay of different chemical,
microphysical, and meteorological processes. These processes are investigated
in environmental modelling by use of chemical transport models (CTMs).
However, modelling aerosol processes is plagued by substantial biases and
errors

Measurements from satellite instruments provide consistent long-term data
sets with global coverage. However, it is notoriously difficult to compare
measured radiances to modelled aerosol concentrations. An alternative to
using radiances is to make use of satellite retrieval products. For instance,
one of the products of the CALIPSO lidar instrument (Cloud-Aerosol Lidar and
Infrared Pathfinder Satellite Observations) is a rough classification of the
aerosol types (i.e. dust, smoke, clean/polluted continental, and
clean/polluted marine). This retrieval product is based on lidar
depolarization measurements

A systematic class of statistical methods for solving this inverse problem is
known as data assimilation. Recent studies have applied data assimilation to
aerosol models with varying degrees of sophistication, ranging from simple
dust models

In all such approaches the choice of control variables is based on ad hoc
assumptions. Numerical assimilation experiments by

In numerical weather prediction (NWP) modelling, several studies have discussed the
information content of satellite observations for meteorological variables.
For instance,

The two main goals of this paper are (i) to apply a systematic method for
analysing the information content of aerosol optical properties with regard
to the particles' chemical composition, and (ii) to test an algorithm for
making an automatic choice of control variables in chemical data assimilation
such that all control variables are signal related, while the noise-related
variables remain unchanged by the assimilation procedure. The main hypothesis
is that by constraining the data assimilation algorithm to acting on the
signal-related variables only, the output will be less noisy than in an
unconstrained assimilation. The focus of our study will be on spectral
observations of extinction and backscattering coefficients, which can be
retrieved from lidar observations.

In addition to lidar measurements
from ground-based and aircraft-carried instruments

The paper is organized as follows. Section

This study consists of two parts. In the first part we quantify the information content of extinction and backscattering coefficients at multiple wavelengths. In the second part we perform a numerical test to investigate to what extent the concentrations of different chemical aerosol components can be constrained by observations of extinction and backscattering coefficients. The modelling tools required for this study are (i) a chemical transport model, (ii) an aerosol optics model, and (iii) a data assimilation system.

We employ the chemical transport model MATCH, which is an off-line Eulerian
CTM with flexible model domain. It has been previously used from regional to
hemispheric scales. Here we use a model version that contains a
photochemistry module with 64 chemical species, among them four secondary
inorganic aerosols (SIAs) – namely ammonium sulfate, ammonium nitrate, other
sulfates, and other nitrates. It also contains a module with 16 primary
aerosol variables – namely sea salt, elemental carbon (EC), organic carbon
(OC), and dust particles, each emitted in four different size bins. Thus, the
model contains 20 different aerosol variables. The particle-radius ranges of the four
bins are as follows:

size bin 1: 10–50

size bin 2: 50–500

size bin 3: 500–1250

size bin 4: 1250–5000

The model reads in emission data, meteorological data, and land use data and computes transport processes, chemical transformation, and dry and wet deposition of the various trace gases and aerosols. As output, it provides concentration fields of gases and aerosols, the deposition of these chemical species to land and water-covered areas, as well as the temporal evolution of these variables.

We mention that there exists another model version that includes aerosol microphysical processes, such as nucleation, condensational growth, and coagulation. In that model version the aerosol size distribution evolves dynamically. The model has 20 size bins and seven chemical species (EC, OC, dust, sea salt, particulate sulfate (PSOX), particulate nitrate (PNOX), and particulate ammonium (PNHX)), although not all species are encountered in all size bins. The total number of model variables currently in that version is 82.

More complete information about the mass transport model can be found in

For the sake of simplicity we here use the mass transport model without
aerosol microphysical processes (see next section). The model is set up over
Europe covering 33

We have two different optics models coupled to MATCH: one to the mass
transport module, and another to the aerosol microphysics module. The former
assumes that all aerosol species are homogeneous spheres, and that each
chemical species is contained in separate particles. Under these assumptions
the optics model is linear, i.e. the optical properties are linear functions
of the concentrations of the chemical aerosol species. The latter model
accounts for the fact that in reality different chemical species can be
internally mixed, i.e. they can be contained in one and the same particle.
That model also accounts for the inhomogeneous internal structure of black
carbon mixed with other aerosol components, and for the irregular fractal
aggregate morphology of bare black carbon particles

Table

Refractive indices at the three harmonics of the Nd:YAG laser assumed in the MATCH mass-transport optics model.

Data assimilation is a class of statistical methods for combining model
results and observations. The algorithm weighs these two pieces
of information according to their respective error variances
and covariances. As output the assimilation returns a result in model space
of which the error variances are smaller than those of
the original model estimate. In our case the model variables are the
mass mixing ratios of aerosol components in a three-dimensional discretized
model domain. These model variables are summarized in a vector

Data assimilation is commonly employed for constraining model results by use
of observations. However, one can also employ data assimilation as an
inverse-modelling tool, i.e. for retrieving a model state from measurements.
A summary of the theoretical basis of variational data assimilation is given
in Appendices B–D.

Many authors distinguish between data

The MATCH model contains a 3DVAR data assimilation module. This model uses a
spectral method, i.e. the model state vector is Fourier transformed in the
two horizontal coordinates. All error correlations in the horizontal
direction are assumed to be homogeneous and isotropic. The background error
covariance matrix is modelled with a method that follows similar principles
to the NMC method

The questions we ask are these:

Suppose we have an

Which are the

First we want to explain what we mean by

In a more general case we have to consider a state vector

The mapping from model space to observation space given in Eq. (

Another useful measure is obtained by expressing our incomplete knowledge of the
atmospheric aerosol state
by use of the Shannon entropy. The use of measurement information reduces
the entropy, and this entropy reduction

Both

By performing the transformation

We study the performance of the 3DVAR system by performing a numerical test. To this end, we first perform a reference run by driving the MATCH model with analysed meteorological data. These reference results are taken as the “true” chemical state of the atmosphere. We apply the optics model to the model output to generate synthetic “observations”, i.e. a vertical profile at a selected observation point of extinction and backscattering coefficients at three typical lidar wavelengths. Next we run the MATCH model again, this time driven with 48 h forecast meteorological data. The results are taken as a proxy for a background model-estimate that is impaired by uncertainties. Finally, we perform a 3DVAR-analysis of the “observations” and the background estimate in an attempt to restore the reference results. In this numerical test we have perfect knowledge of the true state, and we assume that our optics model is nearly perfect, thus providing nearly perfect observations (we assumed that the observation error standard deviation is 10 % of the measurement value). The only factor that may prevent us from fully restoring the reference state is a lack of information in the observed parameters. Thus, comparison of the retrieval and reference results gives us an indication of how strongly different model variables can be controlled by the information contained in the observations.

We perform this test (i) with the unconstrained 3DVAR algorithm and (ii) with the constrained 3DVAR algorithm. We compare both runs in order to make a first assessment of the impact of the constraints. In particular, we are interested in the prospect of reducing the risk of assimilating noise in such a highly under-constrained inverse problem.

We consider the set of parameters

Number of signal degrees of freedom

Table

This illustrates the pivotal importance of the observation error for the
amount of information that can be obtained from measurements. It is important
to understand that the observation error

A more realistic optics model,
such as the one investigated in

The strong impact of the observation errors on the information content of measurements suggests two conclusions.

In order to make the forward-model error

It is equally essential to accurately estimate the contribution of the uncertainties in the aerosol optics model, i.e. to estimate the forward-model error

In Table

From this we learn that the singular values

Let us now compare the different subsets of parameters in Tables

Signal degrees of freedom

We integrated the findings of Sect.

Vertical profiles of selected aerosol components in different size bins. From top to bottom: organic carbon in the third size bin (OC-3), OC in the fourth size bin (OC-4), elemental carbon in the third size bin (EC-3), and dust in the first size bin (DUST-1). The reference results are shown in black, and the background (first guess) estimate is shown in green. The unconstrained 3DVAR analysis results are presented in the left panels in blue, the constrained 3DVAR analysis results are shown in the right panels in red.

Figure

As Fig.

Figure

Figure

Observations (black solid line), and observation-equivalents of the background estimate (green), and of the unconstrained (blue) and constrained (red) 3DVAR analysis. The optical parameters and wavelengths are indicated above each panel.

Vertical profiles of the transformed model variables

We have seen that the analysis provides a reasonable, but, as expected, not a
perfect answer to the inverse problem. We have further seen that at the
observation site it relies more on the observations than on the background
estimate. Most importantly, we have seen that the constraints introduced in
the 3DVAR algorithm suppress noise in the analysis, especially in EC and OC.
However, the previous figures do not provide us with any direct insight of
how exactly the constraints accomplish this. To learn more about that we need
to inspect the analysis in the abstract phase space of the transformed model
variables

The first five rows (from top to bottom) of the matrix

Finally, we want to obtain a better understanding of how the aerosol
components

Comparison of the two columns clearly demonstrates that the elements of the
transformation matrix can vary considerably with vertical layer (or, more
generally, with location). This is because the error covariance matrix

However, in our case the SIA components consistently make a strong
contribution to the first signal-related element

We have quantified the information content of multiwavelength lidar
measurements with regard to the chemical composition of aerosol particles.
Different combinations of extinction and backscattering observations at
several wavelengths have been investigated by determining the singular values
of the scaled observation operator, by computing the number of signal degrees
of freedom

The observation error depends not only on the measurement error, but also on
the forward-model error. The latter depends on the uncertainties in the
aerosol optics model. This highlights the importance of developing accurate
aerosol optics models and of obtaining an accurate estimate of the
observation error, especially of the uncertainty in the aerosol optics model.
This is a prerequisite for extracting as much information as possible from
the measurements, while avoiding to extract noise rather than signal. More
often than not, computational limitations and lack of knowledge force us to
introduce simplifying assumptions about the particles' morphologies. However,
we know that aerosol optical properties can be highly sensitive to the shape
(

The singular values of the scaled observation operator provide us with an abstract
measure to compare the standard deviations of the background (prior) estimate
to those of the observations. The reason why this is a rather abstract
measure is because background and observation errors are, in general, in different
spaces and cannot be directly compared. However, we constructed a mapping that
transforms the state vector in physical (model) space to an abstract phase space
in which the components of the state vector can be partitioned into signal-related
and noise-related components. The singular values indicate to what extent the
signal-related phase-space variables can be constrained by the measurements.
We exploited this fact by constructing weak constraints in a 3DVAR data assimilation
code, which limited the assimilation algorithm to acting on the signal-related
phase-space variables only (hereafter referred to as the

When mapped into observation space, the analysis result closely reproduced the measurements. When viewed in the abstract phase space, we found that the constrained analysis did, indeed, yield noise-related components that were close to zero, as they should be. This was not so in the unconstrained analysis. Also, the magnitude of the signal-related phase-space components was generally larger in the constrained analysis than in the unconstrained analysis. This confirms that the constraints we introduced work as intended.

In our specific test case secondary inorganic aerosol components were most faithfully retrieved by the inverse modelling solution, followed by organic and black carbon. Dust and sea salt mass mixing ratios were more challenging to retrieve. We could explain this by inspecting the linear coefficients in the transformation from physical space to the abstract phase space. We found that those aerosol components that had the largest weight in the transformation were most faithfully retrieved by the analysis. However, these linear coefficients depend on the background error covariances (which can change with location), and on the observation error variances. Therefore, it is difficult to draw general conclusions about which aerosol components are most easily retrieved by a given set of measurements.

The results presented here suggest further questions for future studies. We have performed this investigation with a mass transport model, thus focusing on the information content of optical measurements with respect to the chemical composition of aerosols. When we include aerosol microphysical processes, then the model delivers the aerosols' size distribution, as well as their size-resolved chemical composition. This makes the problem quite different from that we investigated here. First, the dimension of the model space is considerably larger for an aerosol microphysics transport model. Constraining such a model with limited information from measurements becomes even more challenging than in the case of a mass transport model. On the other hand, an aerosol microphysics model delivers information on the particles' size distribution and mixing state. Therefore, this would require us to make fewer assumptions in the aerosol optics model, which may reduce the observation error. The present study could be extended to investigate the information contained in extinction and backscattering measurements for simultaneously constraining the chemical composition and the size of aerosol particles.

Another important issue concerns the choice of the aerosol optics model. In
the present study we employed a simple homogeneous-sphere model in which all
chemical components were assumed to be externally mixed. There is little one
can put forward in defence of this model other than pure convenience.
(Regarding the applicability of simplified model particles in atmospheric
optics see the review by

The data used in this study are included in the Supplement.

Suppose we have a system described by a set of variables

A pair of such problems is inverse

An equation

Data assimilation is usually employed for constraining models by use of measurements, but it can also be used to solve inverse problems. Here we focus on one specific data assimilation method known as three-dimensional variational data assimilation, or 3DVAR.

In a CTM we discretize the geographic domain of interest into a
three-dimensional grid. In each grid cell, the aerosol particles are
characterized by the mass mixing ratio of each chemical component in the
aerosol phase, such as sulfate, nitrate, ammonium, mineral dust, black
carbon, organic carbon, and sea salt. Suppose we summarize all these mass
mixing ratios from all grid cells into one large vector

In the remote sensing and inverse modelling community, the background estimate is more commonly referred to as the a priori estimate.

Suppose also that we haveThe optics model

We stress, once more, that the observation error must not be
confused with the measurement error

The true state of the atmosphere is, of course, unknown. Therefore, our
definition of the errors and their probability distribution is only of
conceptual use, but not of any practical value. However, we can reinterpret
the probability distributions by replacing

The observation
errors are often assumed to be uncorrelated (this is not always true). In
such a case the matrix

Equations (

We are interested in the most probable aerosol state of the atmosphere, i.e.
in that state

In practice it is common to introduce the variable

The solution to the equation

By solving
the equation

Our ultimate goal is to formulate the data assimilation problem in such a way
that the information contained in the measurements is fully exploited, but
not overused. To this end, we first need to know how many independent
quantities can be determined from a specific set of measurements. We
investigate this question by borrowing ideas from retrieval and information
theory – see

The main idea is to compare the variances of the model variables to those of
the observations. Only those model variables whose variance is larger than
those of the observations can be constrained by measurements. However, to
actually make such a comparison poses two problems. The first problem is that
one cannot readily compare error covariance

When we account for observation errors

The expectation value of a
discrete variable

To address the first problem, we diagonalize the covariance matrices by making
the following change of variables:

A matrix

We are still not in a position to make a meaningful comparison of model and
observation errors, since the first term,

We now make another change of variables:

In the above discussion we relied on plausibility arguments. We mention that
there are more systematic ways of approaching the problem. Here we merely state
some key results without going into details. The interested reader is
referred to chapter 2 in

One can compute the number of signal degrees of freedom

Another approach is based on information theory. Given a system described by
a probability distribution function

Our findings so far suggest a general strategy for how to optimize the amount
of information that can be extracted from measurements. First, we need to
compute the singular value decomposition in Eq. (

In the minimization of the cost function all elements of the control vector

For reasons we will explain later we formulate the constraints as weak conditions. However, for didactic reasons as well as for the sake of completeness, we will also mention how to formulate constraints as strong conditions.

Given

Compared to the unconstrained minimization problem, the introduction of

The

Determine a basis of the null space ker(

Solve the unconstrained (

In the approach described in the previous section the solution satisfies the
constraints exactly. Therefore, this approach is known as the minimization of
the cost function with

The formulation of the weak-constraint approach is conceptually quite simple.
One incorporates the constraints by adding an extra term to the cost function
Eq. (

We now want to incorporate the results of Appendix

The weak constraint approach is, arguably, more suitable in our case. We
have, in the preceding text, frequently used the terms

In order to apply the weak-constraint approach, we need to substitute the
constraint matrix

Clearly, how we set up the matrix

We will here discuss some practical aspects that are mainly interesting for model developers.

One of the main practical problems is the dimension

In variational data assimilation we encounter a similar problem in the
inversion of the matrix

In our case we are primarily interested in constraining the aerosol
components. Therefore, we formulate our weak constraints in a suitable
subspace of the physical space. Suppose, for simplicity, that we have reduced
all data to the vertical resolution of our model. Let

For those readers interested in spectral
formulations of 3DVAR we refer to Eqs. (28)–(30) in

Another aspect concerns the positive square root of the background error
covariance matrix, which appears in essential parts of the theory, namely in
Eqs. (

The Cholesky decomposition is, essentially, a special case of a LU decomposition, which applies to symmetric real (or Hermitian complex), positive definite matrices.

of theMichael Kahnert worked with the theoretical developments and numerical implementation, Emma Andersson performed the testing of the method.

The authors declare that they have no conflict of interest.

This work was funded by the Swedish National Space Board through project nos. 100/16 (MK) and 101/13 (EA). Edited by: M. Tesche Reviewed by: five anonymous referees