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

Dominik Brunner, Daniel Schaub, and Brigitte Buchmann

Empa, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Air Pollution and

Environmental Technology, Überlandstrasse 129, 8600 Dübendorf, Switzerland ABSTRACT The distribution of short-lived air pollutants such as NO2 is closely tied to geographically fixed emission sources. Variations due to changing weather conditions or emission strengths may be considered as noisy deviations from a mean picture. In this study we investigate to what extent fine scale details of (NO2) air pollution structures can be reconstructed from coarsely resolved satellite observations. For this we set up an idealized test environment where both the original distribution to be reconstructed and the level of noise of the observations are known, and apply a number of iterative image reconstruction algorithms originally developed for application in computer tomography. In the case of noise-free observations, the original distribution can be completely recovered except for spatial frequencies where the spectrum of the aperture function (the pixel rectangle) exhibits nulls. For noisy observations 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 observations of tropospheric air pollution have become available at spatial resolutions suitable for studying air quality on the regional scale. Pollution patterns of short-lived species such as NO2 are more or less immobile as they are tied to stationary emission sources like cities, motorways, power plants, etc. By combining the views from multiple satellite overpasses it is therefore possible to resolve details beyond the resolution of the individual satellite footprint. Based on an idealized test case, we here present an analysis of the capabilities and limitations of enhanced resolution imaging from simulated (OMI) satellite NO2 observations using a suite of iterative methods for image reconstruction. 2. IMAGE RECONSTRUCTION TECHNIQUES Let a(x,y) represent the true NO2 column at point (x,y). A satellite measurement sk can then be modelled by

∫∫ += noise ),(),( dxdyyxayxhs kk (1)

where hk(x,y) is the aperture response of the kth observation. In our case this is a simple rectangle with height and width equal to the satellite pixel dimensions (e.g. 13 x 24 km2 for OMI at nadir). More generally, we can write equation (1) as noise+= Has , (2) where H is a linear operator projecting the NO2 pollution field onto the observations. H is therefore frequently referred to as the observation operator. We are interested in the inverse problem sHa 1ˆˆ −= (3) where â is an estimate of a given the measurements. The inverse of the operator H, 1ˆ −H is exact only if H is invertible and the measurements are noise free, in which case, â = a. However, the problem is often ill-posed as there may be fewer knowns (observations) than unknowns (number of cells of the reconstruction grid). In addition, the measurements contain noise which turns equation (2) into a minimization problem (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 observations s. The first method (Fig. 1a) may be termed “direct mapping” or “pixel area mapping”. This method only inverts the sampling step but not the smoothing effect of the aperture function. The second method (b) is similar to (a), but instead of spreading an observation over all grid cells covered by the satellite pixel the observed value is only assigned to the grid cell in the centre of the pixel. Method (c) represents one of the iterative methods presented below, which not only invert the sampling step but, in addition, de-convolve the aperture function. A number of iterative image reconstruction techniques have been presented in [1]. These methods include algebraic reconstruction techniques (ART) [2] such as additive AART and multiplicative MART which have been developed for application in computer tomography. We applied an additional algorithm (termed SIMPLE in the following) motivated by the iterative trajectory statistics method presented in [3].

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Proc. ‘Envisat Symposium 2007’, Montreux, Switzerland 23–27 April 2007 (ESA SP-636, July 2007)

Figure 1: Techniques for mapping satellite pixels onto a high resolution grid. (a) “Pixel area mapping”. âi is the value attributed to cell i (i=1,..,N) of the fine grid, sk is the value of satellite pixel k (k=1,..,K), and δik is 1

if pixel k (fully) covers cell i and 0 otherwise. (b) “Pixel center mapping”. The satellite pixel values are

only attributed 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 distribution aij

from a previous iteration is used in some way to calculate the distribution ai

j+1 in step j+1.

Instead of assigning a constant value sk to all grid cells covered by a satellite pixel, this method uses the field âj from the previous iteration step as a-priori information for redistributing sk inside a given satellite pixel. A description of the AART and MART algorithms can be found in [1]. 3. RECONSTRUCTION IN 1-D In this section, the method is illustrated based on a one-dimensional problem. The true distribution a to be reconstructed is a simple superposition of three Gaussian peaks (black solid lines in Fig. 2). Randomly sampled observations of an imaginary instrument with a wide aperture window (indicated by the horizontal bar in Fig. 2a) are overlaid as black diamonds. The root mean square (RMS) difference between the reconstructed âj 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 but does not improve any further.

Figure 2: Reconstruction of a simple function

(superposition of three Gaussian, thick black line) from a random set of observations (black diamonds) with an aperture window [x-2,x+2]. (a) Reconstruction using

SIMPLE method. (b) MART algorithm. The reconstructed functions âj are shown as coloured lines

for different iteration steps j.

Figure 3: Quality of reconstruction (in terms of RMS

difference from original function) as a function of iteration step for AART (black), SIMPLE (red) and

MART (blue).

a

b

aperture

MART reaches the highest score after about 300 iterations but then “overshoots” (cf. Fig. 2b). 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 observations by randomly placing 5000 OMI pixels over a high-resolution NO2 emission inventory for Switzerland [4] as shown in Fig. 4. Each OMI pixel was assigned a “NO2 concentration” obtained by averaging over all grid cells of the inventory covered by the pixel. The original emission inventory representing our target function a to be reconstructed is shown in Fig. 4a. The artificial set of OMI observations placed over Switzerland is shown in Fig. 4b. Only the pixel frames are plotted with colour representing the pixel mean concentration.

Figure 4: (a) NOx emission inventory. (b) Artificial set of 5000 noise-free OMI “observations”. OMI pixels

are placed randomly over the inventory and each pixel is assigned the mean value of the emissions of all grid

cells covered by the pixel.

Reconstructions based on the MART algorithm are presented in Fig. 5 after the 0th iteration (Fig. 5a) and after the 500th iteration (Fig. 5b). The 0th iteration corresponds to the direct mapping illustrated in Fig. 1a. Fig. 5b demonstrates that in the noise-free case the original field (Fig. 4a) can be almost completely reconstructed given a sufficient number of iterations. Individual polluted areas such as the cities of Zurich, Basel, Berne or Lausanne can be well identified in Fig. 5b. Even the major traffic routes through the Alps as well as medium sized cities embedded in deep valleys like Chur in the Rhine valley and Sion in the Rhone valley can be identified. Note that for a complete reconstruction more observations than only 5000 would be required because the original field contains 12600 cells (unknowns). The density of the observation sample basically determines which (spatial) frequencies of the original image can be reconstructed.

Figure 5: (a) Image reconstructed from noise-free observations after 0th iteration (corresponding to direct

mapping). (b) MART reconstruction after 500 iterations.

a

b

a

b

In the case of irregularly distributed samples more observations are required than in the case of samples on a regular grid for which the highest resolved frequency is given by the Nyquist theorem [5]. As shown in [1], the range of spatial frequencies which can be successfully reconstructed is not only limited by the sampling density but also by nulls in the spectrum of the aperture function. Because the position of these nulls varies with the width of the aperture function, observations with varying pixel sizes (as is usually the case for scanning instruments like OMI) is beneficial as it allows recovering frequencies that otherwise would be lost. 5. THE NOISY REALITY The situation becomes much more complicated in the presence of noise. Probably more important than the actual measurement noise is real variations in NO2 columns due to changing weather conditions and emission strengths which we here consider as noise, too.

Figure 6: (a) Image reconstructed from noisy (40% of signal) observations after 0th iteration (corresponding

to direct mapping). (b) SIMPLE reconstruction after 15 iterations.

Figure 7: Quality of reconstruction (in terms of RMS difference from original distribution) as a function of iteration step for AART (black), SIMPLE (red) and

MART (blue). Fig. 6 presents reconstructions for the same test environment as in Fig. 4 but in the presence of noise (a noise level of 40% of the signal was chosen). Depending on the level of noise and the number of observations available, the iteration has to be stopped at an earlier or later stage because the reconstruction algorithms tend to amplify the (high-frequency) noise. Lowpass filtering of the observation 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 best for our idealized test case. However, further optimization of the different algorithms and application to real satellite observations may alter this conclusion. It should be noted that the level of noise is much smaller in computer tomography applications for which these algorithms have been developed originally. Applying the same methods to “noisy” satellite observations of air pollution is therefore not straight-forward. 5. CONCLUSIONS NO2 is a short-lived species and therefore its distribution is closely connected to geographically fixed emission sources. Variations due to changing weather conditions and emissions may be considered as noisy deviations from a mean picture. If the mean picture is sufficiently well defined, fine scale details of the NO2 pollution distribution can be reconstructed even from coarsely resolved observations. The success of this reconstruction,

however, critically depends on the sampling density and on the level of noise. Methods for image reconstruction and resolution enhancement such as the Algebraic Reconstruction Techniques (ART) are available. Because, different from other applications for which ART algorithms have been developed, the level of “noise” is very large in the case of air pollution observations, a large oversampling of a scene will be necessary for successful reconstruction. The OMI instrument is particularly suited for this task as it not only offers a comparatively high resolution but also a high temporal and spatial coverage. So far, the methods have only been applied in a controlled test environment where the target distribution and the level of noise of the observations are both known. Applicability to real observations yet has to be demonstrated. REFERENCES 1. Early, D. S., and Long, D. G. (2001). Image

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3. Stohl, A. (1996). Trajectory statistics - a new method to establish source-receptor relationships of air pollutants and its application to the transport of particulate sulfate in Europe, Atm. Environ., 30 (4), 579-587.

4. Keller, J., Andreani-Aksoyoglu, S., Tinguely, M., and Prévôt, A. S. H. (2005). Emission Scenarios 1985 – 2010: Their Influence on Ozone in Switzerland, PSI Bericht Nr. 05-07, Paul Scherrer Institut, Villigen, Switzerland.

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