Principles

The use of tracer distributions to determine atmospheric transport plays a relatively small role in studies of atmospheric dynamics. This is in considerable contrast to the situation in the oceans, where the classic ocean inverse problem is the inversion of indirect data, generally temperature and salinity, to determine circulation. Of course, in the ocean, temperature and salinity are not passive tracers – through their effect on density, they cause the large-scale circulation. The fact that the foci of inverse problems in oceanography and meteorology are different mainly reflects the atmosphere being better understood than the oceans.

First, there is an extensive atmospheric measurement network, primarily for prediction of weather, with observations at several thousand locations several times a day, which is supplemented by satellite measurements and regular vertical profiles from radiosondes. The closest analogue of the ocean tracer-inversion problem is that of assimilation of meteorological data (see Section 7.3), but the analysis of the atmosphere differs from that of the ocean by virtue of the much greater frequencies both of observations and of analyses. A second difference between the atmosphere and the oceans is that it is easier to model global circulation in the atmosphere than it is for the oceans. The smaller scale of oceanic eddies means that only very recently has it been computationally feasible to model the global ocean with sufficiently fine grids to resolve these eddies.

Model error

The effect of the ill-conditioning described in the previous chapter is to amplify errors both in the observations and in the models. This makes consideration of such errors a particularly important part of the analysis. Observational error has been discussed in Chapter 5. This chapter addresses the issue of ‘model error’.

The main contributions to model error are the following.

1. ▪ Errors in the atmospheric-transport model. This is the main topic of Section 9.2. The use of tracers to calibrate transport (Chapter 18) can also help characterise transport-model error.

2. ▪ Errors associated with discretisation of the source. This is discussed in Section 8.3.

3. ▪ Mis-match in spatial and temporal scales between models and observations, as discussed in Section 9.3.

4. ▪ Errors in the statistical model. These are discussed in Section 9.4

To discuss model error, we adopt a general formalism from Tarantola, using pmodel(m|x) to represent the probability (density) of the ‘true’ output being m given an input x. For an exact deterministic relation, m = G(x), this density becomes a deltafunction, i.e. it is zero unless the deterministic relation holds. The more general case with non-zero probability density form m ≠ G(x) allows for the possibility of error in the deterministic ‘model’, G(x). There may also be scope for characterising some aspects of ‘model error’ in terms of the prior distribution of model parameters as described by pprior(x).

My first contact with ill-conditioned inverse problems was while flying to the 1977 Australian Applied Mathematics Conference at Terrigal, NSW. By chance, I was seated next to Bob Anderssen, who showed me his recent report on regularisation expressed in terms of Fredholm integrals.

Ill-conditioned problems have now dominated the largest part of my scientific career. However, back in 1977, my (private) reaction to Bob's work was ‘why would any reasonable person want to work on such perversely difficult problems?’. The answer, ‘that this is the form in which we get most of our information about the real world’, completely escaped me. In any case, my idealised modelling of phase transitions in lattice systems was seemingly only loosely related to the real world – even my percolation modelling of bubble trapping lay many years in the future.

Two jobs later, I had left mathematical physics as a career and was working for CSIRO modelling the carbon cycle. In particular, I was trying to calibrate our model. My attempts at finding a best-fit parameter set were thrashing around in some poorly defined subspace, when a vague memory stirred. It took me about two days to find Bob's report again and start to realise what I was up against and what I should do about it.

One important point is that what Bob gave me was a technical report. This genre is much-maligned as ‘grey-literature’. It can, however, have the advantage of telling ‘the truth’ and ‘the whole truth’, even if the lack of anonymous peer review means occasional failures of ‘nothing but the truth’.

Green's functions

Much of the discussion in this book has simply assumed the validity of a linear relation between sources and concentrations, at least in a discretised form. This section aims to clarify the origin and significance of this ‘Green-function’ representation.

The approach follows the ideas introduced in Chapter 2: the process of modelling is regarded as going from the real world to a mathematical model and then to a computer implementation of the model. A sequence of mathematical representations of the transport equations is presented in this chapter, working backwards from those closest to numerical implementations of transport. This is followed by analyses of the representation of transport as sets of ordinary differential equations (ODEs) and partial differential equations (PDEs). This explicit consideration of the mathematical model is useful in establishing general characteristics of the solutions that are less apparent in the computer implementation.

The Green functions used in atmospheric-transport modelling represent specific cases of the general principle that an inhomogeneous linear differential equation (DE) with specified boundary conditions can be ‘solved’ by constructing an integral operator that is the inverse of the differential operator. The Green function is the kernel of this integral operator. An important property of this relation is that any particular Green-function solution is equivalent to the solution of the DE only for a specific set of boundary conditions.

General

In dealing with the real world, we very quickly find that it fails to exhibit the exact relations of applied mathematics. All of the quantities involved are subject to uncertainty and, in the case of ill-conditioned inverse problems, some quantities are subject to very great uncertainty. We need to represent our uncertainty, i.e. our lack of exact information, in statistical terms. Our analysis of trace-gas distributions will be in terms of estimates and these estimates will be regarded as being subject to statistical uncertainty. This chapter discusses the process of obtaining such estimates, first by setting up a general framework and then by considering several important special cases.

Our terminology for defining the general problem of interpreting indirect information is that there is a set of observations, denoted by a vector c, suggesting concentration data, from which we wish to infer some model parameters, denoted by a vector x. (These model parameters will often be more general than a set of source strengths, s.)

The simplest form of analysis is that we wish to use the data, c, to give us an estimate of the parameters, x, making use of a theoretical relation between the observations and the parameters. Since we do not expect the theoretical relations to predict the realworld values exactly, we distinguish these theoretical predictions by denoting them m, or more specifically m(x) when we wish to denote their dependence on the parameters.

Overview

Human activity is changing the composition of the atmosphere. This goes beyond the often obvious problems of local and regional pollution – even in remote locations there are changes in concentrations of minor atmospheric constituents such as carbon dioxide, methane and nitrous oxide. These and other long-lived gases affect the balance of radiation of the earth – they are the so-called greenhouse gases. Other long-lived gases are implicated in the decrease in concentration of ozone in the stratosphere.

The ability to understand the current atmospheric budgets of these trace gases is essential if we are to be able to project their future concentrations in the atmosphere. This book concentrates on one group of techniques that are being used to improve our knowledge – the interpretation of spatial distributions of trace-gas concentrations. An important theme of this book is the use of a statistical approach as being essential to obtaining realistic assessments of the uncertainties in the interpretation of trace-gas observations.

Modelling of the atmospheric transport of carbon dioxide (CO2), methane (CH4) and other greenhouse gases is used to interpret the observed spatial distributions of these gases. The spatial distribution of trace-gas concentrations represents a combination of the effect of spatially varying sources and sinks and the effect of atmospheric transport. Therefore a model of atmospheric transport is needed if the source/sink distributions are to be deduced from observed concentrations.

Inverse modelling of the atmospheric transport of trace constituents is a rapidly growing field of science. In a little over a decade it has developed from two-dimensional inversions of CO2 sources and sinks to encompass the range of activity described in the preceding chapters. Among the important developments has been the recognition of the ill-conditioned nature of most tracer-inversion problems, particularly in the case of estimating surface fluxes from surface concentrations. This recognition of the ill-conditioning, with associated amplification of errors, emphasises the need for a statistical approach that treats the deduction of trace-gas fluxes and other biogeochemical quantities as problems in statistical estimation.

The main applications of tracer inversion have been for the long-lived constituents, particularly CO2 and CH4. As a consequence, most development of inversion techniques has been of methods that are suitable for such cases and that are oriented to the use of monthly mean data from flask-sampling networks.

The two main types of inversion have been identified as ‘mass-balance’ and ‘synthesis’. The choice between these involves a number of trade-offs such as ease of inversion versus ease of analysis of uncertainty. However, the most important difference is the choice between either assumptions about the degree of spatial variation in the concentration fields (in the case of mass-balance inversions) or assumptions about spatial variation in the source fields (in synthesis inversions).

Measurement of trace constituents

Trace atmospheric constituents exist in the atmosphere in one of three phases: gases, liquids (where minor constituents dissolve inwater droplets) and solids (as components of aerosol particles). These different forms have a wide range of physical and chemical properties, leading to a range of atmospheric lifetimes ranging from milliseconds to millennia.

The main aspects of trace-gas measurement that need to be considered are

1. (i) the sampling technique;

2. (ii) the physico-chemical basis of the measurement technique; and

3. (iii) the laboratory requirements for achieving the requisite accuracy and precision.

Sampling

Some of the main divisions in the type of sampling are the following.

Flask sampling. One of the simplest strategies is that in which air is collected in containers and returned to a central laboratory for analysis. This has the advantage of avoiding differences in the measurement calibration between different laboratories and the additional advantage of requiring only simple technology in the field. The disadvantages of flask sampling are the low frequency of data and, for many compounds, the possibility of changes in composition either during the trapping process or during the period in transit from the sampling site to the measurement laboratory. The primary requirement is for flasks with inert surfaces, generally glass or stainless steel. However, other flask characteristics such as the valves and their lubrication have also proved to be important for some compounds.

Data bases

Many observational data sets for gas concentrations and a number of related data such as emission inventories are available on the internet. Since this is constantly evolving, URLs for institutions are given as well as URLs for specific data sets.

CDIAC The Carbon Dioxide Information and Analysis Center is at http://cdiac.esd.ornl.gov.

NOAA CMDL The Climate Monitoring and Diagnostics Laboratory is at http://www.cmdl.noaa.gov, from which links lead to data files. Data are also available by anonymous ftp from ftp.cmdl.noaa.gov with Globalview on path ccg/co2/GLOBALVIEW.

WDCGG TheWorld Data Center for Greenhouse Gases provides data to researchers via CD-ROM. The ftp data access is currently (2001) only for registered users. Details can be obtained via http://gaw.kishou.go.jp/wdcgg.html.

GEIA The Global Emissions Inventory Activity (GEIA) is an activity of the IGAC which is an IGBP core project and so GEIA can be accessed via the IGAC (or less directly via the IGBP). Currently the data base can be accessed more directly at http://weather.engine.umich.edu/geia.

EDGAR The EDGAR data base is at RIVM (http://www.rivm.nl (or specify file ‘index en.html’ for the English language). Currently the introduction to this data base is at http://www.rivm.nl:80/env/int/coredata/edgar/intro.html.

Concepts

Many factors can contribute to theway in which observational programmes to measure concentrations of trace gases are established. These include factors such as

1. ▪ perceptions of scientific need,

2. ▪ recognition of the potential of new measurement technology,

3. ▪ availability of researchers with specific skills; and

4. ▪ funding.

Apart from the choices of which gases are measured and by what technique, an important question is the locations at which air is sampled or measurements are made. Often initial sampling, particularly in a development phase, is based on opportunity. Samples are collected, or measurements made, as close as possible to the home base of the researchers involved. Beyond this, ad hoc criteria are often used: remote sites may have good signal-to-noise ratios, but may undersample important regions.

In view of the considerable cost and effort required to establish measurement programmes, it is desirable to assess the likely utility of such programmes in advance. As well as the question of where to sample, there are issues such as the extent to which new measurements might duplicate existing measurements. Particular cases are the relative role of isotopes and other multi-tracer studies.

This chapter concerns the use of modelling of atmospheric transport in this type of ‘experimental-design’ mode.

The structure and circulation of the atmosphere

The atmosphere exists as a thin, almost spherical, shell around the earth. Within the atmosphere differences in the thermal structure characterise a sequence of layers from the troposphere, adjacent to the earth's surface, through to the thermosphere and beyond. Figure 2.1 shows an average profile of the temperature of the atmosphere, indicating the layers and the boundaries between them.

In the study of greenhouse gases, the most important distinction is that between the troposphere and the stratosphere. The troposphere comprises about the lowest 85% of the atmosphere (in terms of mass) and is characterised by a negative lapse rate, i.e. temperature generally decreases with height. In contrast, in the stratosphere, temperature generally increases with height. While Figure 2.1 shows a mean distribution, the height of the tropopause varies significantly with latitude, being higher (typically 16 km) in the tropics than it is at high latitudes (typically 8 km).

Table 2.1 lists some of the components of the dry atmosphere. The main constituents, those in the upper group in the table, are distributed virtually uniformly in space and time, although it has recently become possible to measure the very small changes in oxygen content and use them to interpret aspects of the carbon cycle.