1. Introduction
As our understanding of the complex processes in the atmosphere has evolved, comprehensive atmospheric chemistry models have been developed. At the same time, our ability to measure the concentration of various chemical species in the atmosphere has significantly improved over the past decades. A forecast model and observational data are combined in the data assimilation process in order to provide an optimal estimate of the true atmospheric state. Since available observations are usually distributed unevenly and sparsely, the spatiotemporal distribution of the observations plays an essential role in the data assimilation process. While the location of many observations is a priori fixed (such as ground stations and satellite observations), it is often possible to include in an analysis a few additional observations whose locations may be selected in a flexible manner. For example, if additional observations are to be taken using dropsondes from an aircraft mission, a strategy to select the times and locations of these observations must be specified. Adaptive methods search for spatiotemporal locations of the observations that minimize the error in the analysis and subsequent forecasts. In practice the design of adaptive strategies is constrained by the limited number of resources available and physical considerations (e.g., only a limited area may be covered in a given interval of time).
Significant research has been dedicated to the problem of adaptive location of the observations in numerical weather prediction. The experiments performed in the data assimilation context show that the success of the adaptive strategies relies on the identification of the areas where the errors in the initial conditions are large and/or are rapidly growing (usually referred to as target areas). Berliner et al. (1999) provide a rigorous statistical formulation and mathematical framework for the adaptive design problem. The Fronts and Atlantic Storm Tracks Experiment (FASTEX; Joly et al. 1997) and the North Pacific Experiment (NORPEX-1998) provided real-life applications where several methods for targeting observations were tested. A targeting technique based on adjoint sensitivity was formulated by Langland and Rohaly (1996). Palmer et al. (1998) and Buizza and Montani (1999) describe adaptive techniques using the dominant singular vectors of the integral tangent propagator associated with a nonlinear dynamical system. Implementation of the gradient and singular vector methods relies on adjoint modeling. Bishop and Toth (1999) use an ensemble transform technique based on nonlinear ensemble forecasts to approximate the prediction error covariance matrices. Recently, an ensemble transform Kalman filter method was proposed by Bishop et al. (2001) for the adaptive observations problem. Bishop (2000, manuscript submitted to Quart. J. Roy. Meteor. Soc., hereafter BIS) introduced an extended observation gradient targeting technique and provided a comparative analysis of several adaptive observing methods in the context of estimation theory. Baker and Daley (2000) consider an adjoint-based technique to determine the sensitivity of the forecast to the observations and the background field and apply this method to the adaptive targeting problem.
In the past few years variational methods have been successfully used in data assimilation for comprehensive 3D atmospheric chemistry models (Elbern and Schmidt 1999; Errera and Fonteyn 2001). Satellite observations of the chemical species are beginning to provide global datasets that facilitate an improved understanding of the natural and human influence on the changes in the atmosphere. In addition, valuable measurements are performed during extensive field experiments using aircraft missions, ships, and balloons. The Aerosol Characterization Experiment (ACE-Asia, spring 2001) and National Aeronautics and Space Administration (NASA) Transport and Chemical Evolution over the Pacific Experiment (TRACE-P, spring 2001) have recently produced datasets of the trace gases and aerosol distribution over East Asia and the western Pacific. At the same time, these experiments revealed the necessity of developing computationally feasible methods for targeting observations in air quality modeling. The adjoint sensitivity technique we consider in this paper in the context of the 4D variational data assimilation was inspired by the work of Fisher and Lary (1995) for atmospheric chemistry models. Associated with a single observation, they introduced an “influence function” and used it to provide information about the sensitivity of the cost functional with respect to intermediate states of the model. Their study of a stratospheric photochemical box model with trajectories showed that observations of some chemical species may provide useful information about unobserved species. Similar experiments were performed later by Elbern et al. (1997) using a tropospheric chemistry box model. We extend this technique by taking into account the spatial dimension and a general set of parameters, and we use the influence functions to develop an algorithm for the adaptive location of the observational system. An adjoint-based gradient method for a transport-chemistry model is implemented and it is shown that the set parameters can be naturally extended to include emissions and certain types of boundary values with minimal additional cost.
The paper is organized as follows. In section 2 we briefly review the four-dimensional variational data assimilation (4DVAR) for transport-chemistry models. The adjoint sensitivity method is presented in section 3. The problem of the adaptive location of the observations is formulated in section 4 in the 4DVAR context. In section 5 we describe an adjoint sensitivity approach and propose an algorithm for targeting observations. Computational and implementation issues are addressed. Preliminary numerical experiments performed with an Eulerian two-dimensional atmospheric chemistry model are presented in section 6. An outline of the results and further research directions are given in section 7.
2. 4DVAR data assimilation for air quality models
3. Adjoint sensitivity analysis
Mathematical foundations of the adjoint sensitivity for nonlinear dynamical systems and various classes of response functionals are presented by Cacuci (1981a,b). Marchuk (1995) and Marchuk et al. (1996) describe in detail the adjoint modeling and the construction of the adjoint operators for linear and nonlinear atmospheric dynamics. Pudykiewicz (1998) shows the derivation of the continuous adjoint operator for a tracer transport model and an application to source parameters estimation. Further applications of the adjoint sensitivity analysis to variational data assimilation are presented by Le Dimet et al. (1997).
4. The adaptive observations problem
5. An adjoint method for adaptive observations
Baker and Daley (2000) noticed that traditional adaptive strategies are based on the a priori evaluation of the sensitivity of some aspect of the forecast at the verification time to intermediate states and are completely ignorant of any existing observations. Since 4DVAR data assimilation takes into consideration all observations available in the assimilation window, targeted observations strategies must account for uncertainty magnitude, uncertainty growth, and the details of the data assimilation scheme (Bergot 2001).
The adjoint targeting strategy we propose is based on the evaluation of two sensitivity fields: the first sensitivity field, Γυ, is associated to the verification cost functional
a. The influence function at the verification time
The sensitivity field we associate to the verification cost functional (32) is intimately related to the evaluation of the sensitivity of forecast errors to initial conditions as described by Rabier et al. (1996). To eliminate the explicit dependence on
b. The influence function of a single observation
The second sensitivity field we define is associated with the cost functional (13) to be minimized in the data assimilation process.
c. Adjoint computation of the influence function
The complexity of evaluating the influence function is dominated by the computation of the sensitivity values ∇
Using relation (29), while computing the influence function with respect to the model state, we obtain the influence function Γ
In practice, a discrete adjoint model is often implemented directly from the numerical method used in the integration of the forward model. In this approach reverse automatic differentiation tools, for example, Tangent Linear and Adjoint Model Compiler (TAMC) (Giering 1997), may be used to facilitate the adjoint code generation. If x* has the grid coordinates x*(ix, iy, iz), let e be the (j, ix, iy, iz) vector of the canonical base of Rs × R
d. Domain of influence and multiple observations
The influence function was defined in the case when a single observation was taken into account and its computation required only one integration of the adjoint model. However, in practice it is often the case that multiple observations are available at a moment tn. In this section we extend the definition of the influence function (42) to include a set of observations. By a simple analogy, the extension applies also to the definition (37).
e. An algorithm for adaptive observations
The 4D variational data assimilation takes into account all observations available in the analysis time interval. Therefore, as pointed out in section 4, the interaction between observations must be considered in the observations selection algorithm. We assume that the data assimilation analysis interval is [t0, T] and at instant ti ∈ [t0, T], i = 1, I a set Oi of ni observations must be selected from the set of all possible observations at time ti. The target area
1) Algorithm for adaptive selection of the observations
6. Numerical experiments
a. The test model
The numerical experiments were performed with a two-dimensional test model based on the Carbon Bond Mechanism IV (Gery et al. 1989) with 32 variable chemical species involved in 70 thermal and 11 photolytic reactions. The spatial domain is [0 250] km × [0 250] km and a uniform grid Δx = Δy = 5 km is considered, such that there are 49 × 49 interior grid points and the dimension of the discrete state vector is N = 76 832. The wind field u(ux, uy) and the diffusion coefficient K(Kxx, Kyy) are taken constant ux = uy = 10 km h−1, Kxx = Kyy = 10−3 km2 s−1. The initial state distribution and emissions values are obtained using the box model urban and rural scenarios described by Sandu et al. (1997). An urban region is considered in the domain [50 150] km × [50 150] km. At the center of the urban area, (100, 100) km, we consider the initial state and emissions as in the urban scenario. Outside the urban area rural initial conditions and emissions are specified. Interpolation is done between the center of the urban area and the urban boundaries to obtain the initial state and emissions inside the urban region. Emission rates are constant in time and no deposition terms are considered. Boundary conditions are prescribed according to (3) and (4) with the inflow boundary values c← obtained from a box model integration with rural initial conditions. The advection operator is discretized using a limited k = 1/3 upwind flux interpolation as presented by Koren (1993) and the diffusion operator using central differences formula. The time integration of the semidiscrete model is performed using dimensional Strang operator splitting (Strang 1968) with a splitting time step 2Δt = 30 min. The advection–diffusion terms are integrated using a second-order explicit Runge–Kutta method and the chemistry-source terms using a variable step size second-order L-stable Rosenbrock method ROS-2 (Verwer et al. 1999). The reference state of ozone at the initial time t0 = 0430 local standard time (LST) and at the verification time tυ = 1100 LST are shown in Figs. 1a and 1b, respectively. An initial guess state for the model was obtained by shifting SW two grid points from the initial reference state. After shifting, random errors up to 20% are introduced in the ozone state. Isopleths of the relative errors (absolute values) in the initial guess ozone state and the corresponding forecast state at tυ are shown in Figs. 1c and 1d, respectively. We notice that large errors in the forecast state at tυ are located around the area [100 200] km × [100 200] km with maximal errors at the locations
A discrete adjoint model was generated using the adjoint model compiler TAMC (Giering 1997) for the advection–diffusion numerical integration and symbolic processing for the chemistry integration as described by Daescu et al. (2000). In the numerical experiments we present, the restriction of the verification time to tυ = 1100 LST was determined by the high storage requirements of the adjoint model implementation and by the limited computational resources available. For the two-dimensional example we consider, the discrete model state is a three-dimensional vector, ck(s, nx, ny), and the adjoint code requires manipulation of four-dimensional vectors as the complete forward trajectory is required and the sensitivity fields are time dependent. To overcome this difficulty, we used a two-level checkpointing scheme (Daescu et al. 2000) to store the forward trajectory such that there are two forward integrations per backward integration. During the first forward run we store the trajectory after each operator splitting step (30 min). During the second forward run, we store the trajectory inside each operator splitting step (the chemistry integration takes on average 8–10 steps for a 30-min interval). In double precision, our computational resources (HP-UX A 9000/778) allow only manipulations of vectors with dimension up to ∼106 and only 14 states may be stored in fast memory. With a 30-min operator splitting step, this limits our analysis interval to 6 h, 30 min, from 0430 to 1100 LST.
b. Validity of the model linearization
c. Examples of influence functions for ozone
d. Adaptive observations and 4DVar
Two methods are tested for the adaptive observations: the first method (M1) is based only on the a priori evaluation of the sensitivity field Γυ and fits in the traditional adjoint sensitivity framework. The influence function Γυ is evaluated once, with no update, and the observations at ti are always located at the points where the magnitude of Γυ(ti) is maximal. These locations are marked in Fig. 6 with “+” as time goes backward moving on the rows from the upper left corner (t = T) to the lower left corner (t = t0). The second method (M2) implements the algorithm presented in section 5d with a periodic update of the values of Γυ. The selected locations are marked in Fig. 6 with a solid dot (locations marked in Fig. 6 by “+” and “·” were selected by both M1 and M2 methods). The computational cost (as CPU time) of the forward and adjoint model integration and to implement each of the methods M1 and M2 is presented in Table 1. The CPU time required to implement method M1 is dominated by the expense of evaluating Γυ, which is roughly given by the cost of a forward–backward integration. The CPU time to implement method M2 requires in addition the evaluation of the influence function of the fixed observations Γ
7. Conclusions and further research
Strategies for targeting observations have been considered mostly in numerical weather prediction and applications to atmospheric chemistry are at a very incipient stage. The problem of the adaptive location of the observations in atmospheric chemistry research becomes increasingly important as transport-chemistry models begin to be used in forecast mode to enhance flight planning during large-scale field experiments. Expensive field-deployed resources can be utilized more effectively and the science success can be maximized by selecting an optimal observational path.
Strategies for targeting observations must take into account the properties of the data assimilation algorithm. With the current computing resources, variational methods based on adjoint modeling may be used to perform data assimilation for comprehensive atmospheric chemistry models. We described an adjoint sensitivity method and applied it to the problem of adaptive selection of the observations for a transport-chemistry model. Our results show that using the adjoint approach, sensivities with respect to various model parameters such as emission and deposition rates or boundary values may be obtained at a reduced computational cost. The influence functions associated with the observations and their domain of influence were shown to be essential tools in developing a strategy for adaptive observations in the 4D variational data assimilation context. At the same time, our results indicate that by using a periodical update of the sensitivity values to include the influence from all previously located observations, an observational path with significant benefits for the model forecast may be determined. The novel algorithm for adaptive observations we presented may be efficiently implemented at a computational cost equivalent with the cost of a few forward model integrations and our preliminary numerical experiments show promising results. Further research is needed to implement this algorithm for a comprehensive 3D Sulfate Transport Eulerian Model (STEM) model (Carmichael et al. 1986); test the algorithm performance on real observational datasets; and apply the new adaptive technique to future field experiments. Future work will also include a comparative study with targeting methods using the dominant singular vectors and an analysis of the interaction between the information provided by the “background” parameter estimation and adaptive observations.
Acknowledgments
This work was supported in part by funds from the National Science Foundation, under the Information Technology Research program.
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(a) The reference initial state of ozone, t = 0430 LST; (b) the reference state of ozone at tυ = 1100 LST; (c),(d) isopleths of the relative errors (absolute values) in the initial guess ozone state and in the corresponding forecast at tυ.>
Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0434:AASMFT>2.0.CO;2
The Θ statistics test for increasing longer forecast time. A random field is used to generate the initial perturbation and examples are shown for a magnitude of the intial perturbation δc0 up to 50% of the control initial state
Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0434:AASMFT>2.0.CO;2
Examples of influence functions for an O3 observation located at (tυ, ⊕): (a),(c) with respect to O3 and NO2 state at T, respectively; (b),(d) with respect to O3 and NO2 state at t0, respectively. Isopleths of the magnitude are shown and notice that a different scaling factor is used for each plot
Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0434:AASMFT>2.0.CO;2
Cumulative value of the influence function for an O3 observation located at (tυ, ⊕) with respect to NO2 emissions in the time interval [t0, tυ]. Isopleths of the magnitude are shown
Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0434:AASMFT>2.0.CO;2
The location and the influence function Γ
Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0434:AASMFT>2.0.CO;2
The adaptive location of the observational path. Observations selected by method M1 (marked +) and the observations selected by the adaptive method M2 (marked ;wd). The time is updated each half-hour and moves backward on the rows from the upper-left corner (t = 1030 LST) to the lower-left corner (t = 0430 LST).>
Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0434:AASMFT>2.0.CO;2
The optimization process. (a) Evolution of the cost functional. (b) Evolution of the ozone forecast error at tυ = 1100 LST. Normalized values are shown on a log10 scale with dashed line for method M1 and with solid line for the proposed adaptive method M2. While both methods provide the same relative reduction in
Citation: Journal of the Atmospheric Sciences 60, 2; 10.1175/1520-0469(2003)060<0434:AASMFT>2.0.CO;2
The CPU time (s) of the forward and adjoint integration and the CPU time (s) required to implement the methods M1 and M2. An additional forward integration time is included in the CPU time of the backward integration
