(10.47)
and one then finds . Ehhalt et al. (1998) show that this model is most realistic when chemical loss and dilution take place at comparable rates, whereas a situation where chemical loss is faster than dilution will tend toward the first limiting case of the Lagrangian mixing model β ≈ τj/τi.
Figure 10.23 (bottom panel) applies these ideas to the C2H2–CO relationship by examining the log–log correlation in the observations and the model. In this case, τCO/τC2H2 = 3–3.5 and , where the variability reflects the temperature dependence of the rate constants. We find β ≈ 1 in the fresh Asian outflow off the China coast, indicating that the correlation is driven by dilution; this confirms that the slope of the linear relationship provides a measure of the emission ratio. For the more remote regions, we find that the slope exceeds , indicating that the correlation mostly reflects chemical loss. This means then that simulation of the log–log slope provides a test of model [OH]. Sensitivity simulations presented by Xiao et al. (2007) to fit the observed slopes imply that tropical OH concentrations in the model are 50% too high.
This relatively simple example illustrates how observed correlations between species can provide constraints on emissions, mixing, chemistry, and other processes, but interpretation is often not obvious and mistakes can easily be made. Simulation of observed relationships with a 3-D model, complemented by model sensitivity studies, can thus be a powerful tool to advance knowledge. It is particularly satisfying if knowledge gained from the complex 3-D model can be distilled into a simpler model illuminating the fundamental processes driving the observed relationships. As the saying goes, “no model should be more complicated than it needs to be.” But starting from a complicated model can provide the best guide to judicious simplification.
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11
Inverse Modeling for Atmospheric Chemistry
11.1 Introduction
Inverse modeling is a formal approach for using observations of a physical system to better quantify the variables driving that system. This is generally done by statistically optimizing the estimates of the variables given all the observational and other information at hand. We call the variables that we wish to optimize the state variables and assemble them into a state vector x. We similarly assemble the observations into an observation vector y. Our understanding of the relationship between x and y is described by a model F of the physical system called the forward model:
y = F(x, p) + εO
(11.1)
Here, p is a parameter vector including all model variables that we do not seek to optimize as part of the inversion, and εΟ is an observational error vector including contributions from errors in the measurements, in the forward model, and in the model parameters. The forward model predicts the effect (y) as a function of the cause (x), usually through equations describing the physics of the system. By inversion of the model we can quantify the cause (x) from observations of the effect (y). In the presence of error (εO ≠ 0), the solution is a best estimate of x with some statistical error. This solution for x is called the optimal estimate, the posterior estimate, or the retrieval. The choice of state vector (that is, which model variables to include in x versus in p) is totally up to us. It depends on which variables we wish to optimize, what information is contained in the observations, and what computational costs are associated with the inversion.
Because of the uncertainty in deriving x from y, we have to consider other constraints on the value of x that may help to reduce the error on the optimal estimate. These constraints are called the prior information. A standard constraint is the prior estimate xA, representing our best estimate of x before the observations are made. It has some error εA. The optimal estimate must then weigh the relative information from the observations y and the prior estimate xA, and this is done by considering the error statistics of εO and εA. Inverse modeling allows a formal analysis of the relative importance of the observations versus the prior information in determining the optimal estimate. As such, it informs us whether an observing system is effective for constraining x.
Inverse modeling has three main applications in atmospheric chemistry, summarized in Table 11.1:
1. Remote sensing of atmospheric composition. Here we use radiance spectra measured by remote sensing to retrieve vertical concentration profiles. The measured radiances at different wavelengths represent the observation vector y, and the concentrations on a vertical grid represent the state vector x. The forward model F is a radiative transfer model (Chapter 5) that calculates y as a function of x and of additional parameters p that may include surface emissivity, temperatures, spectroscopic data, etc. The prior estimate xA is provided by previous observations of the same or similar scenes, by knowledge of climatological mean concentrations, or by a chemical transport model.
2. Top-down constraints on surface fluxes. Here we use measured atmospheric concentrations (observation vector y) to constrain surface fluxes (state vector x). The forward model F is a chemical transport model (CTM) that solves the chemical continuity equations to calculate y as a function of x. The parameter vector p includes meteorological variables, chemical variables such as rate coefficients, and any characteristics of the surface flux such as diurnal variability that are simulated in the CTM but not optimized as part of the state vector. The information on x from the observations is called a top-down constraint on the surface fluxes. The prior estimate xA is an inventory based on our knowledge of the processes determining the surface fluxes (such as fuel combustion statistics, land cover data bases, etc.) and is called a bottom-up constraint. See Section 9.2 for discussion of bottom-up and top-down constraints on surface fluxes.
3. Chemical data assimilation. Here we construct a gridded 3-D field of concentrations x, usually time-dependent, on the basis of measurements y of these concentrations or related quantities at various locations and times. Such a construction may be useful to initialize chemical forecasts, to assess the consistency of measurements from different platforms, or to map the concentrations of non-measured species on the basis of measurements of related species. We refer to this class of inverse modeling as data assimilation. The corresponding state vectors are usually very large. In the time-dependent problem, the prior estimate is an atmospheric forecast model that evolves x(t) from a previously optimized state at time to to a forecast state at the next assimilation time step to + h. The forecast model is usually a weather prediction model including simulation of the chemical variables to be assimilated. The forward model F can be a simple mapping operator of observations at time to + h to the model grid, a chemical model relating the observed variables to the state variables, or the forecasting model itself.
Table 11.1 Applications of inverse modeling in atmospheric chemistry
Application State vector Observations Forward model Prior estimate
Remote sensing Vertical concentration profile Radiance spectra Radiative transfer model Climatological profile
Top-down constraints Surface fluxes Atmospheric concentrations Chemical transport model Bottom-up inventory
Data assimilation Gridded concentration field Atmospheric concentrations Mapping operator Forecast
Proper consideration of errors is crucial in inverse modeling. Let us examine what happens if we ignore errors. We linearize the forward model y = F(x, p) around the prior estimate xA taken as best guess:
y = F(xA, p) + K(x − xA) + Ο((x − xA)2)
(11.2)
where K = ∇xF = ∂y/∂x is the Jacobian matrix of the forward model with elements kij = ∂yi/∂xj evaluated at x = xA. The notation O((x – xA)2) groups higher-order terms (quadratic and above) taken to be negligibly small. Let n and m represent the dimensions of x and y, respectively. Assume that the observations are independent such that m = n observations constrain x uniquely. The Jacobian matrix is then an n × n matrix of full rank and hence invertible. We obtain for x:
x = xA + K−1(y − F(xA, p))
(11.3)
If F is nonlinear, the solution (11.3) must be iterated with recalculation of the Jacobian at successive guesses for x until satisfactory convergence is achieved.
Now what happens if we make additional observations, such that m > n? In the absence of error these observations must necessarily be redundant. But we know from experience that strong constraints on an atmospheric system typically require a very large number of measurements, m ≫ n. This is due to errors in the measurements and in the forward model, described by the observational e
rror vector ε0 in (11.1). Thus (11.3) is not applicable in practice; successful inversion requires adequate characterization of the observational error εΟ and consideration of prior information. A standard approach to do this is to use Bayes’ theorem, described in Section 11.2.
The chapter is organized as follows. Section 11.2 presents Bayes’ theorem and shows how it provides a basis for inverse modeling. Section 11.3 applies Bayes’ theorem to a simple scalar optimization problem in order to build intuition for the rest of the chapter. Section 11.4 introduces important vector-matrix tools for inverse modeling, including error covariance matrices, probability density functions (PDFs) for vectors, Jacobian matrices, and adjoints. Section 11.5 presents the fundamental analytical method for solving the inverse problem, Section 11.6 presents the adjoint-based method, Section 11.7 presents Markov Chain Monte Carlo (MCMC) methods, and Section 11.8 presents other optimization methods. Section 11.9 discusses means to enforce positivity in the solution to the inverse problem. Section 11.10 gives an overview of variational methods used in chemical data assimilation. Observation system simulation experiments (OSSEs) to evaluate the merits of a proposed observing system are described in Section 11.11. Inverse modeling has applications across many areas of the natural and social sciences, and a major source of confusion in the literature is the use of different terminologies and notations reflecting this diverse heritage. Here we will follow to a large extent the terminology and notation of Rodgers (2000), which we consider to be a model of elegance.
11.2 Bayes’ Theorem
Bayes’ theorem is the general foundation of inverse modeling. Consider a pair of vectors x and y. Let P(x), P(y), P(x, y) represent the corresponding PDFs, so that the probability of x being in the range [x, x + dx] is P(x) dx, the probability of y being in the range [y, y + dy] is P(y) dy, and the probability of (x, y) being in the range ([x, x + dx], [y, y + dy]) is P(x, y) dx dy. Let P(y|x) represent the conditional PDF of y when x has a known value. We can write P(x, y) dx dy equivalently as
Modeling of Atmospheric Chemistry Page 59