Measurements of the large-dimensional chemical state of the atmosphere
provide only sparse snapshots of the state of the system due to their
typically insufficient temporal and spatial density. In order to optimize the
measurement configurations despite those limitations, the present work
describes the identification of sensitive states of the chemical system as
optimal target areas for adaptive observations. For this purpose, the
technique of singular vector analysis (SVA), which has proven effective for
targeted observations in numerical weather prediction, is implemented in the
EURAD-IM (EURopean Air pollution and Dispersion – Inverse Model) chemical
transport model, yielding the EURAD-IM-SVA v1.0. Besides initial values,
emissions are investigated as critical simulation controlling targeting
variables. For both variants, singular vectors are applied to determine the
optimal placement for observations and moreover to quantify which chemical
compounds have to be observed with preference. Based on measurements of the
airship based ZEPTER-2 campaign, the EURAD-IM-SVA v1.0 has been evaluated by
conducting a comprehensive set of model runs involving different initial
states and simulation lengths. For the sake of brevity, we concentrate our
attention on the following chemical compounds, O

In meteorology and atmospheric chemistry, both data assimilation and inverse
modelling seek to combine observations from a given observation network
set-up with a model to reduce forecast errors. In contrast, the objective of
targeted observations is to optimize the observation network for data
assimilation and ensuing simulations applying a given model

In numerical weather prediction, the optimal adaption of observations is a
commonly investigated problem

The successful application of singular vector analysis within numerical
weather prediction motivated the transfer of this analysis method to chemical
modelling, where studies addressing targeted observations are rare.

Initial values are not the only uncertainty when considering atmospheric
chemical modelling. Errors in boundary conditions, emission rates, and
meteorological fields add to the uncertainty of the chemical forecast

In this work, the approach of

The present paper is organized as follows: the theory of singular vector
analysis is presented in Sect. 2, where the application to initial
concentration uncertainties and emission factors is described as well as the
application of special operators. Singular vector analysis (SVA) is
implemented in the EURAD-IM

The application of singular vector analysis to atmospheric chemical modelling
allows for studying the influence of different kinds of uncertainties on the
chemical forecast evolution. Within this work, we target the largest
uncertainties in initial values and emissions, which both strongly determine
the chemical system's evolution. A brief outline of the theoretical
background of this application is presented in the following

A deterministic chemical forecast is processed by a typically nonlinear model
operator,

To allow for the calculation of
relative error growths and to place foci on limited sets of chemical
compounds and limited areas, we extend the analysis above by applying two
special operators, namely weight matrix,

The projection operator allows for analysis of a limited set,

With the help of both projection operator and weight matrix, we can consider
the relative impact of a limited set of perturbations at initial time,

Emissions,

Like initial values, emissions are subject to uncertainties or errors, since
their estimate is dependent on imperfect models and observation. Yet,
emissions vary in time, leading to uncertainties or errors,

Introducing the vector of emission factors,

As an analogue to Sect. 2.1, we further want to identify the most unstable
emission factor,

A focal set of initial and final perturbations can be examined with the help
of the projection operator,

For the design of a model enabling three-dimensional singular vector analysis
of chemical species and their temporal evolution, we implement the theory as
described in Sect.

The EURAD-IM simulates the chemical development in time and space based on
the following system of differential equations:

We augment the EURAD-IM to allow for the option of singular vector analysis
(SVA), yielding the EURAD-IM-SVA v1.0. In order to calculate targeted
singular vectors as described in Sect.

Newly coded tangent linear routines have been checked for consistency with
corresponding forward and adjoint modules. For consistency with the forward
model, the gradient check ratio

Consistency of tangent linear and adjoint model can be tested by inspecting
the validity of the following equation:

The central task of the EURAD-IM-SVA v1.0 is the detection of singular
vectors and their associated singular values. Two methods have been
implemented for solving the eigenvalue problems: the power method

We apply the set-up of the ZEPTER-2 measurement campaign

ZEPTER-2 deployed the ZEPPELIN NT airship as a platform to measure the distribution of different trace gases, aerosols, and short-lived radicals in the planetary boundary layer. During the campaign, 25 flights were carried out within a 100 km radius of the home base at Friedrichshafen airport (FDH), southern Germany. Vertical profiles of trace gases were measured above different surface types, including Lake Constance and surrounding forests.

ZEPTER-2 was supported by daily 3D-var analyses and chemical forecasts
modelled with the EURAD-IM. The ZEPTER-2 set-up of the EURAD-IM allows for a
practical application of the theory of targeted observations. Here, we apply
singular vector analysis to identify the most sensitive locations and
chemical compounds with respect to their impact on the final concentration of
ozone. This study is designed to give insight into example applications of
singular vectors in future campaigns by answering the following questions.

Which of the chemical compounds O

Where is the optimal location for observations of these components?

List of all singular vector simulations included in the ZEPTER-2
case study. Initial time (

We choose all spatial projections to contain grid points with ZEPTER-2
measurements and all compound-wise projections to focus only on chemical
compounds measured during the ZEPTER-2 campaign. In this manner, it is
revealed how singular vector analyses can support the set-up of an optimal
campaign design when the chemical compounds to be measured and an approximate
measurement route are already set. At final time, we focus specifically on
vertical measurement profiles, since measurement profiles grant a larger
magnitude of the optimal initial perturbation than single ZEPTER-2
measurement points (the location of the vertical measurement profile at final
time is denoted as “final profile VP

The configuration of the EURAD-IM-SVA v1.0 applied in this study is based on
the ZEPTER-2 set-up of the EURAD-IM. Here, RACM-MIM

CO emission source strength (mg m

Emission estimates of the ZEPTER-2 set-up are provided by the EMEP (European
Monitoring and Evaluation Programme) cooperative programme with a horizontal
resolution of 50 km. The data consist of annual emissions of CO, SO

Vertical placement of the first singular vector with respect to
initial value uncertainties for case 2a. Illustrated is the length of the
vertical singular vector per model level for passive tracer and ozone (left
panel) as well as for CO, OH, HONO, O

Initial concentrations of all simulations are taken from 3D-var assimilation
runs, conducted for the ZEPTER-2 campaign. Here, assimilation was
accomplished every 4 h, starting at 02:00 UTC, and observational data of
NO

In this section, elementary examples are demonstrated, illustrating
performance and interpretation of singular vectors for observation targeting.
The section is divided between initial value based singular vectors and those
determined by emission rates. For both measures, we identify both optimal
locations and optimal chemical compounds for additional measurements. Please
note that the analysis of initial value uncertainties includes results of
several leading singular vectors, while the analysis of emission factor
uncertainties is only concerned with the leading singular vector. The latter
is due to different implementations of eigenvalue problem solvers (see
Sect.

Vertical grid structure of the EURAD-IM-SVA v1.0 for the reference
state 47.85

Five largest singular values (SV) with respect to initial value
uncertainties for all 17 case studies. Case numbers are denoted according to
Table

Singular vector calculations are based on the tangent linear model assuming
that small perturbations evolve linearly within the simulation time. In order
to grant meaningful results, this assumption has to be validated first. We
apply Eq. (

For initial uncertainties, we have calculated the five largest singular
values for each of the considered cases using PARPACK (see Table

An evident point of interest for chemistry is the relation between singular
vectors resulting from passive tracer advection–diffusion, as merely
controlled by meteorological parameters, and those which are also affected by
reactive chemistry. Their differences can be visualized via horizontal and
vertical placement (for a definition of horizontal and vertical placement,
see Appendix

Vertical placement of the first and second singular vectors with
respect to initial value uncertainties for case 2a. Illustrated is the length
of the first and second vertical singular vectors per model level for ozone
(left panel) as well as the length of the second vertical singular vector for
CO, OH, HONO, O

We find the same properties to be true for the vertical profile of the second
singular vector. The left panel of Fig.

Horizontal placement of the first and second singular vectors with
respect to initial value uncertainties for case 8a. Left panel: 0.01
isopleths of the first horizontal singular vector for passive tracer (red
framed shading) and ozone (green filled shading). Right panel: 0.01 isopleths
of the first (green filled shading) and second (blue framed shading)
horizontal singular vectors for ozone. In both figures, the final profile
VP(

Examination of the horizontal placement (for a definition of horizontal
placement, see Appendix

Initial concentrations and horizontal placement of the first and
second singular vectors with respect to initial value uncertainties for
case 6. Illustrated are results for NO (left panel) and O

Results reveal furthermore that the horizontal placement of all considered
chemical compounds usually coincides. Remarkable differences within the
chemical placement are only discovered for cases 6, 7a, 8b, and 10, and can
be explained by varying initial concentrations within the otherwise
advection-controlled placement area. The horizontal distribution of the first
and second singular vectors at the lowest level for case 6 is displayed in
Fig.

The analysed ZEPTER-2 cases share a relative short simulation interval (the
longest simulation interval lasts 3 h 15 min) and a local projection on the final profile VP(

Relative ranking of the first (upper panel) and second (lower panel)
singular vectors with respect to initial value uncertainties. Illustrated are
results for O

Optimal compounds for additional measurements can be determined via the
relative ranking defined in Appendix

Figure

We also find that the measurement priority of NO is higher for simulations
starting during noon hours, while it is lower for simulations starting in the
morning or in afternoon/evening time frames. This feature is related to the
initial mixing ratio of NO which is close to 0 during night-time

Prior to analysing the singular vectors with respect to emission factors, the
linearity assumption is tested by inserting the calculated perturbations of
largest error growth into Eq. (

The optimization of observational networks with respect to measurements of
emissions themselves is somewhat artificial, as only for very special cases
are flux tower observations of CO

The subsequent analysis in Sect.

Figure

Optimal horizontal placement of emissions and initial values for
HCHO at surface level for case 5a. The 0.01 isopleths of the optimal
horizontal placement are indicated with a black line (initial values) and a
red line (emissions). The horizontal position of the final profile
VP(

Comparing the target area of emissions for different compounds, we find that the target areas differ quite substantially in some cases. This feature occurs due to different emission source strengths for different compounds and will be explained in more detail at the end of the next section.

Singular values (SV) with respect to initial values (iv) and
emissions (em). VP(

In response to question Q

Results for all considered levels and species are depicted in
Fig.

Relative ranking of the first singular vector with respect to
emission uncertainties. Illustrated are results for NO (top left), NO

Figure

The singular values of our calculations determine the relative error growths
of uncertainties in initial values and emissions, respectively.
Table

We find that the influence of singular values with respect to initial values
decreases with growing simulation length, whereas the influence of singular
values with respect to emissions increases (Table

Spatially dependent measurement priorities of the first singular
vector with respect to emission uncertainties. Illustrated are results for
HCHO (left) and CO (right) at surface level for case 2a. Please note that the
ranking is only depicted within the area of the relevance ranking. For each
panel, the horizontal position of the final profile VP(

Furthermore, Table

The EURAD-IM has been augmented to allow for singular vector analysis (SVA), resulting in the new EURAD-IM-SVA v1.0 model. The purpose of the EURAD-IM-SVA v1.0 is the calculation of the most sensitive chemical configuration with respect to initial values and emissions. The calculated sensitive configurations can be utilized to stabilize the chemical forecast by targeting sensitive system states for additional measurements. In this manner, the new tool can especially be applied for effective campaign planning.

In the framework of the model augmentation, newly coded or embedded routines
are tested for accuracy. Within the limits of numerical precision, single
routines as well as the complete model demonstrate correctness. Subsequently,
the EURAD-IM-SVA v1.0 is evaluated by conducting a set of case studies based
on the accomplished ZEPTER-2 campaign. Here, we evaluate the importance of
measurements with regards to their ability to improve the forecast for
locally predetermined ozone profiles. We investigate the influence of
additional measurements of O

Results of the singular vector decomposition with respect to initial values
reveal that the optimal placement for additional observations is linked to
height, with observations being more important at lower elevation where most
of the chemical production of ozone takes place. Here, optimal target areas
are controlled by mixing ratios of ozone precursors and their photochemical
lifetimes, as well as transport and diffusion processes. In terms of a
relevance ranking of chemical species, the measurement priority of species
differs location-wise, dependent on initial concentrations and the importance
of the precursor in the chemical formation of ozone. Overall, O

The singular vector decomposition with respect to emissions shows that the
optimal placement of measurements of emission factors is strongly dependent
on the location of emission sources. When considering the relevance ranking
of considered emitted species, we find that, for most cases, the influence of
emissions of NO is most important, followed by emissions of NO

Considering the error growth of uncertainties in initial values and emission strength, we find that the influence of singular values with respect to initial values decreases with growing simulation length, whereas the influence of singular values with respect to emissions increases. Due to short simulation intervals and focus on selected ozone profiles at the end of the simulation, the error growth is smaller than 1 in most of the cases, meaning that the final uncertainty is smaller in percentage than the initial uncertainty. Yet, there are also cases that show singular values greater than 1, proving the value of singular vector analysis even in the case of strongly restricted dynamics.

Altogether, our case study shows that the newly designed EURAD-IM-SVA v1.0 is a powerful tool which identifies critical chemical species and chemical locations with respect to initial values and emissions. Both optimal placement of measurements and relevance ranking of chemical compounds confirm the benefit of singular vectors for measurement selection guidance. This can be applied for effective campaign planning. Furthermore, the detected directions of largest error growth can be employed to initialize ensemble forecasts and to model covariances.

For three-dimensional chemical transport models, a singular vector

The optimal observation location for a given species

We analyse the optimal placement in terms of vertical and horizontal optimal
placement. The

Likewise, for the

A measurement priority of the associated chemical compounds can be
established for each grid point

Since the measurement priority of species

The code controlling the singular value decomposition is stored locally at the Rhenish Institute for Environmental Research as well as at the Jülich Supercomputer Centre (JSC) of the Research Centre Jülich. It is available by request via email (nadine.goris@uni.no, he@riu.uni-koeln.de).

The authors are especially grateful to Dirk Poppe for numerous fruitful discussions. Two anonymous referees and the editor provided very constructive and useful comments in order to improve the manuscript. This study was supported by the virtual institute for Inverse Modelling of the Atmospheric Chemical COmposition (IMACCO) in the framework of the Helmholtz-Impuls- und Vernetzungsfonds (grant VH-VI-117) and by the German Federal Ministry of Education and Research through the ZEPTER-2 project (grant 01LP0803A). All simulations utilized the Kinetic PreProcessor (KPP) developed by Valeriu Damian, Adrian Sandu, Mirela Damian, Florian A. Potra, and Gregory R. Carmichael. Many thanks to Adrian Sandu for his technical assistance. Technical and computational support was provided by the Jülich Supercomputer Centre (JSC) of the Research Centre Jülich. A large and central part of the case studies has been computed on JSC supercomputer JUROPA. Access given to these computational resources is highly appreciated. This work would not have been possible without the meteorological analysis obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) and the emission estimates provided by the EMEP (European Monitoring and Evaluation Programme) cooperative programme. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: T. Butler