RECONSTRUCTING FINE-SCALE AIR POLLUTION STRUCTURES FROM COARSELY RESOLVED SATELLITE OBSERVATIONS

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1 RECONSTRUCTING FINE-SCALE AIR POLLUTION STRUCTURES FROM COARSELY RESOLVED SATELLITE OBSERVATIONS Dominik Brunner, Daniel Schau, and Brigitte Buchmann Empa, Swiss Federal Laoratories for Materials Testing and Research, Laoratory for Air Pollution and Environmental Technology, Üerlandstrasse 129, 8600 Düendorf, Switzerland ABSTRACT The distriution of short-lived air pollutants such as NO 2 is closely tied to geographically fixed emission sources. Variations due to changing weather conditions or emission strengths may e considered as noisy deviations from a mean picture. In this study we investigate to what extent fine scale details of (NO 2 ) air pollution structures can e reconstructed from coarsely resolved satellite oservations. For this we set up an idealized test environment where oth the original distriution to e reconstructed and the level of noise of the oservations are known, and apply a numer of iterative image reconstruction algorithms originally developed for application in computer tomography. In the case of noise-free oservations, the original distriution can e completely recovered except for spatial frequencies where the spectrum of the aperture function (the pixel rectangle) exhiits nulls. For noisy oservations the situation is more complicated and the success of reconstruction critically depends on the level of noise and the spatial density of the sample. 1. INTRODUCTION With the introduction of SCIAMACHY and OMI, satellite oservations of tropospheric air pollution have ecome availale at spatial resolutions suitale for studying air quality on the regional scale. Pollution patterns of short-lived species such as NO2 are more or less immoile as they are tied to stationary emission sources like cities, motorways, power plants, etc. By comining the views from multiple satellite overpasses it is therefore possile to resolve details eyond the resolution of the individual satellite footprint. Based on an idealized test case, we here present an analysis of the capailities and limitations of enhanced resolution imaging from simulated (OMI) satellite NO 2 oservations using a suite of iterative methods for image reconstruction. 2. IMAGE RECONSTRUCTION TECHNIQUES Let a(x,y) represent the true NO 2 column at point (x,y). A satellite measurement s k can then e modelled y hk ( x, y) a( x, y) dxdy + sk = noise (1) where h k (x,y) is the aperture response of the k th oservation. In our case this is a simple rectangle with height and width equal to the satellite pixel dimensions (e.g. 13 x 24 km 2 for OMI at nadir). More generally, we can write equation (1) as s = Ha + noise, (2) where H is a linear operator projecting the NO 2 pollution field onto the oservations. H is therefore frequently referred to as the oservation operator. We are interested in the inverse prolem a Hˆ 1 ˆ = s (3) where â is an estimate of a given the measurements. The inverse of the operator H, Hˆ 1 is exact only if H is invertile and the measurements are noise free, in which case, â = a. However, the prolem is often illposed as there may e fewer knowns (oservations) than unknowns (numer of cells of the reconstruction grid). In addition, the measurements contain noise which turns equation (2) into a minimization prolem (e.g. find a solution â such that the residuals (s Hâ) are minimized in a root mean square sense). Figure 1 presents three examples for the estimation of â from the oservations s. The first method (Fig. 1a) may e termed direct mapping or pixel area mapping. This method only inverts the sampling step ut not the smoothing effect of the aperture function. The second method () is similar to (a), ut instead of spreading an oservation over all grid cells covered y the satellite pixel the oserved value is only assigned to the grid cell in the centre of the pixel. Method (c) represents one of the iterative methods presented elow, which not only invert the sampling step ut, in addition, deconvolve the aperture function. A numer of iterative image reconstruction techniques have een presented in [1]. These methods include algeraic reconstruction techniques (ART) [2] such as additive AART and multiplicative MART which have een developed for application in computer tomography. We applied an additional algorithm (termed SIMPLE in the following) motivated y the iterative trajectory statistics method presented in [3]. Proc. Envisat Symposium 2007, Montreux, Switzerland April 2007 (ESA SP-636, July 2007)

a aperture Figure 1: Techniques for mapping satellite pixels onto a high resolution grid. (a) Pixel area mapping. â i is the value attriuted to cell i (i=1,.

2 a aperture Figure 1: Techniques for mapping satellite pixels onto a high resolution grid. (a) Pixel area mapping. â i is the value attriuted to cell i (i=1,..,n) of the fine grid, s k is the value of satellite pixel k (k=1,..,k), and δ ik is 1 if pixel k (fully) covers cell i and 0 otherwise. () Pixel center mapping. The satellite pixel values are only attriuted to the grid cell in the centre of the pixel, i.e. (δc) ik is 1 if centre of pixel k is in cell i and 0 otherwise. (c) Iterative mapping. The distriution a i j from a previous iteration is used in some way to calculate the distriution a i j+1 in step j+1. Instead of assigning a constant value s k to all grid cells covered y a satellite pixel, this method uses the field â j from the previous iteration step as a-priori information for redistriuting s k inside a given satellite pixel. A description of the AART and MART algorithms can e found in [1]. Figure 2: Reconstruction of a simple function (superposition of three Gaussian, thick lack line) from a random set of oservations (lack diamonds) with an aperture window [x-2,x+2]. (a) Reconstruction using SIMPLE method. () MART algorithm. The reconstructed functions â j are shown as coloured lines for different iteration steps j. 3. RECONSTRUCTION IN 1-D In this section, the method is illustrated ased on a onedimensional prolem. The true distriution a to e reconstructed is a simple superposition of three Gaussian peaks (lack solid lines in Fig. 2). Randomly sampled oservations of an imaginary instrument with a wide aperture window (indicated y the horizontal ar in Fig. 2a) are overlaid as lack diamonds. The root mean square (RMS) difference etween the reconstructed â j and the original function a is shown in Fig. 3 as a function of the iteration step j for three different algorithms. The SIMPLE method quickly reaches a high score ut does not improve any further. Figure 3: Quality of reconstruction (in terms of RMS difference from original function) as a function of iteration step for AART (lack), SIMPLE (red) and MART (lue).

3 MART reaches the highest score after aout 300 iterations ut then overshoots (cf. Fig. 2). AART performs worse than the other two algorithms. 4. IDEALIZED 2-D TEST CASE To test the method in a controlled environment, we created an artificial set of satellite oservations y randomly placing 5000 OMI pixels over a highresolution NO 2 emission inventory for Switzerland [4] as shown in Fig. 4. Each OMI pixel was assigned a NO 2 concentration otained y averaging over all grid cells of the inventory covered y the pixel. The original emission inventory representing our target function a to e reconstructed is shown in Fig. 4a. The artificial set of OMI oservations placed over Switzerland is shown in Fig. 4. Only the pixel frames are plotted with colour representing the pixel mean concentration. Reconstructions ased on the MART algorithm are presented in Fig. 5 after the 0 th iteration (Fig. 5a) and after the 500 th iteration (Fig. 5). The 0 th iteration corresponds to the direct mapping illustrated in Fig. 1a. Fig. 5 demonstrates that in the noise-free case the original field (Fig. 4a) can e almost completely reconstructed given a sufficient numer of iterations. Individual polluted areas such as the cities of Zurich, Basel, Berne or Lausanne can e well identified in Fig. 5. Even the major traffic routes through the Alps as well as medium sized cities emedded in deep valleys like Chur in the Rhine valley and Sion in the Rhone valley can e identified. Note that for a complete reconstruction more oservations than only 5000 would e required ecause the original field contains cells (unknowns). The density of the oservation sample asically determines which (spatial) frequencies of the original image can e reconstructed. a a Figure 4: (a) NO x emission inventory. () Artificial set of 5000 noise-free OMI oservations. OMI pixels are placed randomly over the inventory and each pixel is assigned the mean value of the emissions of all grid cells covered y the pixel. Figure 5: (a) Image reconstructed from noise-free oservations after 0 th iteration (corresponding to direct mapping). () MART reconstruction after 500 iterations.

4 In the case of irregularly distriuted samples more oservations are required than in the case of samples on a regular grid for which the highest resolved frequency is given y the Nyquist theorem [5]. As shown in [1], the range of spatial frequencies which can e successfully reconstructed is not only limited y the sampling density ut also y nulls in the spectrum of the aperture function. Because the position of these nulls varies with the width of the aperture function, oservations with varying pixel sizes (as is usually the case for scanning instruments like OMI) is eneficial as it allows recovering frequencies that otherwise would e lost. to direct mapping). () SIMPLE reconstruction after 15 iterations. 5. THE NOISY REALITY The situation ecomes much more complicated in the presence of noise. Proaly more important than the actual measurement noise is real variations in NO 2 columns due to changing weather conditions and emission strengths which we here consider as noise, too. Figure 7: Quality of reconstruction (in terms of RMS difference from original distriution) as a function of iteration step for AART (lack), SIMPLE (red) and MART (lue). Fig. 6 presents reconstructions for the same test environment as in Fig. 4 ut in the presence of noise (a noise level of 40% of the signal was chosen). Depending on the level of noise and the numer of oservations availale, the iteration has to e stopped at an earlier or later stage ecause the reconstruction algorithms tend to amplify the (high-frequency) noise. Lowpass filtering of the oservation operator H used in the reconstruction is therefore mandatory. For the SIMPLE algorithm we applied a simple 9-point smoothing after each iteration step instead of lowpass filtering of H. As shown in Fig. 7 the SIMPLE algorithm performs the est for our idealized test case. However, further optimization of the different algorithms and application to real satellite oservations may alter this conclusion. It should e noted that the level of noise is much smaller in computer tomography applications for which these algorithms have een developed originally. Applying the same methods to noisy satellite oservations of air pollution is therefore not straightforward. 5. CONCLUSIONS Figure 6: (a) Image reconstructed from noisy (40% of signal) oservations after 0 th iteration (corresponding NO 2 is a short-lived species and therefore its distriution is closely connected to geographically fixed emission sources. Variations due to changing weather conditions and emissions may e considered as noisy deviations from a mean picture. If the mean picture is sufficiently well defined, fine scale details of the NO 2 pollution distriution can e reconstructed even from coarsely resolved oservations. The success of this reconstruction,

5 however, critically depends on the sampling density and on the level of noise. Methods for image reconstruction and resolution enhancement such as the Algeraic Reconstruction Techniques (ART) are availale. Because, different from other applications for which ART algorithms have een developed, the level of noise is very large in the case of air pollution oservations, a large oversampling of a scene will e necessary for successful reconstruction. The OMI instrument is particularly suited for this task as it not only offers a comparatively high resolution ut also a high temporal and spatial coverage. So far, the methods have only een applied in a controlled test environment where the target distriution and the level of noise of the oservations are oth known. Applicaility to real oservations yet has to e demonstrated. REFERENCES 1. Early, D. S., and Long, D. G. (2001). Image reconstruction and enhanced resolution imaging from irregular samples, IEEE Transactions on Geoscience and Remote Sensing, 39 (2), Gordon, R., Bender, R., and Herman, G. T. (1970). Algeraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography, J. Theor. Biol., 29 (3), Stohl, A. (1996). Trajectory statistics - a new method to estalish source-receptor relationships of air pollutants and its application to the transport of particulate sulfate in Europe, Atm. Environ., 30 (4), Keller, J., Andreani-Aksoyoglu, S., Tinguely, M., and Prévôt, A. S. H. (2005). Emission Scenarios : Their Influence on Ozone in Switzerland, PSI Bericht Nr , Paul Scherrer Institut, Villigen, Switzerland. 5. Gröchenig, K. (1992). Reconstruction algorithms in irregular sampling, Math. Comput., 59 (199),

Recent Results from Analyzing the Performance of Heuristic Search

Recent Results from Analyzing the Performance of Heuristic Search Teresa M. Breyer and Richard E. Korf Computer Science Department University of California, Los Angeles Los Angeles, CA 90095 {treyer,korf}@cs.ucla.edu