The difficulties in validating aerosol optical depth from spaceborne lidars such as the Geoscience Laser Altimeter System (GLAS) on board IceSat and the lidar on board the Cloud Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) satellite with ground-based instruments are discussed. Because observations are often not collocated, matching errors, which increase with separation distance, confound the validation of instrumental errors. These matching errors can be assessed by comparing the aerosol total-column optical depth measured in a swath by the Moderate Resolution Imaging Spectroradiometer (MODIS) instrument on board the Terra satellite with point Aerosol Robotic Network (AERONET) Sun photometer measurements at different separation distances. In particular, the relationship of the correlation of the two sensors' aerosol total-column optical depth observations with increasing spatial separation is determined. The use of back trajectories to reduce the loss in correlation with increasing spatial separation is then evaluated. Matching errors are found to increase faster over land than over water sites, with the correlation dropping by 20% in 200 km over the land sites and 500 km over the ocean sites. Constraining the area over which the MODIS aerosol optical depth is calculated to within 30° azimuth of the average back trajectory only improved the correlation for a site where long-range transport of aerosols often occurs.
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 Accurate measurements of aerosol optical properties are extremely important to determine observationally based direct and indirect aerosol climate forcing and for input into climate prediction models. Aerosol and cloud radiative effects remain the largest uncertainties in our understanding of climate change [Intergovernmental Panel on Climate Change, 2001]. A number of satellite remote sensors are currently in orbit or soon to be in orbit to gather global aerosol optical property data and reduce these uncertainties.
 Combined with the Moderate Resolution Imaging Spectroradiometer (MODIS) and Clouds and the Earth Radiant Energy system (CERES) instruments on the Aqua satellite, the Cloud Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission will provide global measurements from which observationally based estimates of aerosol direct and indirect radiative forcing of climate can be made [Winker et al., 2003]. CALIPSO and the Geoscience Laser Altimeter System (GLAS) on board ICESat will be the first lidars in Earth orbit to observe aerosols long-term for climate studies. In order for these satellites to provide useful accurate data for climate studies, they must be properly validated. Methods for validating the observations from these sensors have to be developed to evaluate uncertainties in measured aerosol optical properties for characterization of climate forcing and for data assimilation to climate models [Diner et al., 2004; Anderson et al., 2005]. A validation plan has been developed for CALIPSO to organize a network of existing ground-based sensors to obtain correlative data [Kovacs et al., 2004].
 Satellite remote sensors are often validated by comparing measurements to other calibrated instruments with well-characterized and smaller uncertainties. For satellite remote sensors that measure a swath (e.g., the MultiAngle Imaging Spectroradiometer (MISR) [Abdou et al., 2005; Kahn et al., 2005] and MODIS [Abdou et al., 2005; Ichoku et al., 2005; Remer et al., 2002, 2005; Chu et al., 2002]) direct comparisons of another measurement to a spaceborne measurement can be made. For instruments measuring a single profile, comparisons with another measurement within some predetermined distance are performed often with some technique for trajectory mapping particularly in the more stable stratosphere (e.g., the Stratospheric Aerosol and Gas Experiment II (SAGE II), and the HALogen Occultation Experiment (HALOE)) [Morris et al., 2002]. However, for these point comparisons that are not coincident in space or time, uncertainties from variations in the parameter of interest over space and time add to the measurement uncertainties. Often predetermined acceptable separation distances are used with little assessment of these uncertainties, which can be particularly significant for short-lived species in the troposphere often making validation with ground-based instruments impractical [Cuomo et al., 1998].
 Comparing measurements that are separated in space or time incur a matching error defined as an uncertainty due to spatial or temporal inhomogeneities of the parameter of interest when the two measurements are not collocated. For total-column aerosol optical depth, τa (hereafter τa refers to total-column optical depth), the larger the horizontal gradient, the larger the uncertainty due to matching errors. Techniques for assessing the spatial and temporal variations of aerosols using MODIS [Ichoku et al., 2002] and for developing optimal spatial-temporal matching windows for averaging the satellite and ground aerosol retrievals using Advanced Very High Resolution Radiometer (AVHRR) and Aerosol Robotic Network (AERONET) [Zhao et al., 2002] have been identified. Anderson et al.  documented the variability of aerosols at different spatial and temporal scales using autocorrelations of several instruments and found autocorrelations decreased to 0.8 over a distance of around 200 km or a time span of 10 hours.
 CALIPSO and GLAS provide measurements of aerosol optical properties such as aerosol extinction and τa at 0.532 μm and 1.064 μm [Winker et al., 2003; Spinhirne et al., 2005]. CALIPSO also provides aerosol depolarization ratio with an additional channel that measures the backscatter perpendicular to the polarized plane of the transmitted beam. Measurements are made in the nadir and have high resolution along the orbital path (333 m nominal horizontal resolution for CALIPSO). However, no observations are made cross-track and subsequent orbits are spaced 24.75° longitude or approximately 2750 km at the equator. Because of the mesoscale variation and short lifetime of aerosols in the troposphere, trajectory mapping, used to fill in this gap, would significantly add to the uncertainty of the direct observations. This paper presents an assessment of comparing τa from a spaceborne instrument with a network of ground-based instruments made at a different temporal and varying spatial separations and an assessment of using back trajectories to improve these comparisons. The assessment applies only to τa and is not meant to apply to other CALIPSO products such as the vertical profile of extinction.
 In this paper, τa observed at AERONET sites, τAER, will be compared with τa retrieved from MODIS data, τMOD, at different distances from the AERONET site. The objective is to determine how the comparisons degrade with distance. The results will be compared to the results of the assessments of Anderson et al. , which are based on instrument autocorrelations. Then, back trajectories will be calculated and the information will be used to attempt to increase the separation distance over which the two measurements are highly correlated. The purpose here is to define the direction of the source of air at the ground-based site on the basis of back trajectories and constrain the azimuth angle over which comparisons are made. This constraint is hypothesized to increase the separation distance over which the two measurements are highly correlated and thus increase the number of correlative measurements with minimal reduction in the quality of the correlative measurements. Back trajectories are often considered useful for this purpose though they do not account for the fact that aerosols evolve as they advect and therefore these calculations have additional uncertainties that also increase spatially and temporally. Single back trajectories based only on dynamics are also not very useful in the planetary boundary layer where turbulent mixing distorts the air parcel such that it quickly loses its identity [Stohl, 1998]. Unfortunately, because most aerosol sources are at the Earth's surface, most aerosols build up within this layer. Nevertheless, back trajectories are often discussed as useful to help validate CALIPSO data products in the troposphere. This paper will assess if these back trajectory calculations are useful when applied in a few cases.
 The methodology for comparing τAER and τMOD at different distances is described in section 2. Initially, comparisons are made between τAER and τMOD without the use of back trajectories so that all τa a certain distance from an AERONET site are averaged. Results of these comparisons and how they change with distance are presented in section 3. Back trajectories are then used to determine an average direction of airflow into a site so that τMOD within several distance bins and at a specified azimuth angle around this average direction is averaged. This methodology is applied to four sites and the effect it has on measurement correlations with distance is demonstrated in section 4. Summary and concluding remarks are given in section 5.
 AERONET data from a number of sites scattered around the world and representative of a number of aerosol source and transport regions will form the basis of comparison to the MODIS satellite retrievals of τMOD. AERONET is a federation of ground-based remote sensing aerosol networks, largely Sun photometers, around the world. AERONET data are freely available at http://aeronet.gsfc.nasa.gov. AERONET uses CIMEL Sun photometers that obtain τAER on the basis of the attenuation of radiation directly from the solar disk as it transfers through the atmosphere. These instruments are calibrated using the Langley method to obtain an exoatmospheric solar irradiance. The instruments obtain τAER with uncertainties of approximately 0.01 [Holben et al., 2001] such that they are often considered ground truth for validating other remote sensors that measure or infer τa [Zhao et al., 2002]. The Sun photometers automatically track the sun and make direct sun measurements with a 1.2° field-of-view every 15 min at nominal wavelengths centered at 0.340, 0.380, 0.440, 0.500, 0.675, 0.870, 0.940, and 1.020 μm [Holben et al., 2001]. In this study Level 2.0 data is used, which is cloud screened and quality assured by applying a final calibration to the data [Smirnov et al., 2000].
 A large number of observations are necessary for this study to obtain a statistical data set. Therefore 2 years (2002 and 2003) of AERONET data were obtained from each site. Sites were selected if they had at least 183 days (more than half a year) of data for each year. To reduce any seasonal biases, sites were selected that had no more than 1 full month of missing data. A list of the AERONET sites that fit these criteria is located in Table 1. Three of these sites were omitted from the study, Mauna Loa and Venise because of known site-specific problems [Ichoku et al., 2005], and Solar Village because only a few retrievals of τMOD within 100 km of this site were possible because Solar Village is surrounded by bright surfaces that limits the ability to retrieve τMOD.
Table 1. Selected AERONET Stations and Their Location (Latitude and Longitude) Used in This Studya
Number of Comparable Measurements
The last column, number of comparable measurements, represents the number of times MODIS observations occurred within 1 hour and at 20 km or 200 km of an AERONET observation during the 2002 and 2003 calendar years. Three sites (indicated by one asterisk) are omitted from the study because of known site-specific problems. Three other sites (indicated by two asterisks) are known to have a small percentage of observations that fall within the MODIS error bars. Ocean sites are bolded.
MD Science Center
Moscow MSU MO
Rogers Dry Lake**
 MODIS is on board the Terra (launched December 1999) and Aqua (launched May 2002) satellites. Both Terra and Aqua are polar orbiting satellites crossing the equator at 1030 and 1330 local time, respectively. The study in this paper uses only Terra MODIS data, which has a longer data record. Radiances are acquired in 36 spectral bands from 0.405 to 14.385 μm. The MODIS data products are archived and freely available at NASA data centers (e.g., http://modis.gsfc.nasa.gov). MODIS retrieves τMOD separately over land and water at 10 km resolution on the basis of 0.25- and 0.5-km resolution reflectance data, the cloud mask product [Remer et al., 2005; Martins et al., 2002; Ackerman et al., 1998], and meteorological data from the National Center for Environmental Prediction (NCEP). The cloud mask product also identifies the surface as land or water. Over land surfaces, reflectance is determined at 2.13 μm if enough pixels within the 10 km resolution box falls within a threshold value considered dark enough for accurate cloud-free retrievals. Surface reflectance at 0.470 and 0.660 μm are based on an empirical relationship to the reflectance at 2.13 μm. The estimated surface reflectance and the measured mean top of the atmosphere reflectance at 0.470 and 0.660 μm are used to retrieve τMOD at 0.470 and 0.660 μm and the results are also interpolated to 0.550 μm [Kaufman et al., 1997; Remer et al., 2005]. The ocean retrieval is used if all pixels within the 10 km resolution box are covered by ocean surface. Cloudy pixels [Ackerman et al., 1998; Martins et al., 2002; Remer et al., 2005], pixels contaminated with river sediments [Li et al., 2003], and any pixels within a 40° glint angle are removed. The retrieval is based on reflectance data at 0.550, 0.660, 0.870, 1.20, 1.60, and 2.13 μm and an aerosol model lookup table based on Levy et al. . Tanre et al.  and Remer et al.  provide a better description of the ocean algorithm. Aerosol optical depth is retrieved at 0.550, 0.660, 0.870, 1.24, 1.63, and 2.13 μm and extrapolated to 0.470 μm over the ocean. Retrievals of τMOD have been validated over land [Chu et al., 2002] and water [Remer et al., 2002] with uncertainties of ±0.05 ± 0.15τ and ±0.03 ± 0.05τ, respectively [Remer et al., 2005].
 In order to produce a merged data set the following procedure that is consistent with the procedure outlined by Zhao et al.  is used. MODIS data files are searched for overlapping data with the AERONET sites over the 2 year period. For each AERONET site a matrix is formed for the distance, ri, between the location of every data point in the MODIS data file and the location of the AERONET site. Bins of MODIS aerosol optical depth retrievals with a quality assurance level of “useful” are averaged at every 20 km distance from the AERONET site, MOD (bin), such that the first bin contains an average of all the MODIS data between 0 and 20 km, the second bin contains an average of all the data between 20 and 40 km, and so on up to the last bin that contains an average of all the data between 980 and 1000 km. That is,
where n is the number of MODIS retrievals for a single overpass. This averaging utilizes no directional information provided by back trajectories later in the paper. It is assumed here that τa is horizontally homogeneous. Then, all the level 2.0 AERONET data within 1 hour of the satellite overpass is averaged. For an average synoptic speed of 20 km hr−1 this should provide an integration volume match between the temporally varying AERONET data and the spatially varying MODIS data. One hour is also found by Zhao et al.  to be the optimal temporal averaging time for AERONET in comparison with AVHRR data of similar resolution (8 km). Averages of AERONET and MODIS data must contain at least two measurements in each bin or the data is not considered in the final correlations.
 Once the merged data set is produced, τAER and τMOD are calculated at 0.550 μm. MODIS observes at this wavelength over ocean and reports an interpolated value at this wavelength over land at 10 km horizontal resolution. τAER is linearly interpolated over the natural logarithm of the spectral bands at 0.470, 0.675, and 0.870 μm. No filters are applied to MODIS for excessive spatial variance; however, if τAER varies by more than a factor of two within 1 hour of the MODIS measurement the data are removed. Also, all negative τAER were removed, which only affected one day at the Konza EDC site. This filtering retains as much natural variation as possible, but some of the rare defects in the AERONET data needed to be removed. After all merging and filtering are complete the correlations between the 2 years of temporally coincident τAER and MOD (bin) are computed for each bin and plotted with separation distance of the pair of observations.
 To attempt to increase the amount of data useful for comparing spaceborne and ground-based measurements of τa, back trajectories are computed for each day there are τAER and τMOD that coincided according to the criteria described above. Back trajectories are based on a kinetic theory analysis using NASA Global Modeling and Assimilation Office (GMAO) assimilated gridded data. These back trajectories are freely available through the AERONET website (http://aeronet.gsfc.nasa.gov). Note that this model only utilizes dynamical information and cannot resolve turbulent mixing within the planetary boundary layer. Though these back trajectories are not representative of all back trajectory models it is the model used on the AERONET site and is an example of a model that one might use in studies of τa. The purpose of this paper is not to evaluate the uncertainties of the back trajectory model when applied to aerosol optical properties, merely to assess using back trajectories to constrain the direction of averaging in the azimuth to improve correlations of two spatially separated measurements of τa. Seven-day back trajectories initialized at 0000 UTC and 1200 UTC were calculated starting at 8 different altitudes, though only the lowest 4 altitudes (950, 850, 750, and 500 hPa) and the 1200 UTC back trajectories were used in this study. On occasion, elevated aerosol layers may exist above these elevations, but they are infrequent and should not be a factor in the analysis of 2 years of data. Back trajectories between 23 and 29 December 2002 and 11 August through 2 November 2003, and back trajectories on 18 and 31 May 2003 and 29 July 2003 were not available and not used in this study.
 The difficulty in validating a satellite remote sensor that does not have a swath of observations with a ground observation that measures at a point or column is illustrated in Figure 1. Figure 1 shows an example of a day of τAER retrievals on 2 June 2003, and a swath of τMOD retrievals at 1530 UTC, 2 June 2003. The AERONET site is located at NASA Goddard Space Flight Center (GSFC), and is centered on the range rings in Figure 1b. Notice the gradient of τa temporally in the AERONET data at the time of the MODIS observation (1530 UTC) (Figure 1a) where τa decreases from near 1.1 to under 0.5 in a 5-hour period. The gradient of τa is spatially shown in the MODIS data at the location of the AERONET site (Figure 1b). Within 200 km of the nearest site, τMOD values range from 0.1 to over 1.0. For a satellite sensor, such as a spaceborne lidar, which observes in a column often spatially and temporally separated from a ground observation, errors will be incurred both because of instrumental errors and the possibility that the air masses are different for the two measurements, i.e., matching errors. However, in order to validate these spaceborne sensors, data that is spatially and temporally separated must be used in order to have enough measurements to provide for a statistical comparison. For example, Anderson et al.  suggests that 54 independent measurements with a correlation of at least 0.8 are needed to establish a linear relationship at the 95% confidence level. Fortunately, aerosol properties exhibit enough spatial homogeneity in most instances so that comparisons, in a statistical way, can be made to measurements that are not collocated. The rest of this section will establish the measurement distance for various locations with different aerosol sources that provides a reasonable amount of correlation. The following section will explore the use of back trajectories to improve upon this distance.
 Comparisons of τa between MODIS measurement locations and selected AERONET sites are displayed in Figure 2. Figure 2 shows histograms of the MODIS and AERONET measurements for four sites over the 2-year period that were made within 20 km and 1 hour of observations between the two sensors. The mean and standard deviation for all measurements over that period are also displayed. Figure 2 shows the span of τa at a number of stations and the number of correlative measurements made over the 2 year period. The number of correlative measurements increases with increasing latitude and decreasing persistence of cloud cover at the sites. The number of correlative measurements also increases as the separation distance increases as shown in Table 1. Though comparisons within 20 km and 1 hour of observation between the two sensors are quite good and highly correlated at these sites for most aerosol loadings there are noticeable biases particularly at the BSRN BAO Boulder site. This site is in a mountainous and somewhat arid region with relatively bright surfaces accentuated by frequent snow cover over half the year. High surface reflectance tends to cause overestimation in the MODIS retrievals of τMOD [Ichoku et al., 2005].
 When all the sites are considered, the comparisons tend to be quite good at small separation distances where matching errors approach zero. Figure 3 shows a comparison of τAER and τMOD at 0.55 μm partitioned between land and ocean sites as designated by Ichoku et al. . The MODIS data have been averaged in 20 km bins of distance from the AERONET site and the 20, 40, 200, and 600 km bins are shown. AERONET data within 1 hour of each of the MODIS data are averaged. Note that the ocean sites comprise only three sites and one of them, Capo Verde, is frequently affected by Saharan dust and has a large average τa causing the ocean sites shown here to appear to have similar τa distributions to the land sites. MODIS data have been extensively validated previously [Ichoku et al., 2005; Remer et al., 2002, 2005; Chu et al., 2002], and it is not the intention of this study to repeat those validation studies. Nonetheless the correlation and number of observations within the error bars are similar to those validation studies at 20 km distance over land. Over the ocean there are a large number of MODIS observations outside the error bars at 20 km. The Coconut Island MODIS observations are high and uncorrelated at all separation distances, probably because of the mountainous terrain and persistent cloud cover [Ichoku et al., 2005]. The Ascension Island MODIS observations are biased low and uncorrelated at 20 km probably because of coastal effects such as not being able to distinguish land from water and attenuating effects of wetland areas [Ichoku et al., 2005]. The correlation coefficient for the 0–20 km bin is 0.86 over land and 0.71 over the ocean sites. The correlation over ocean sites improves to 0.9 by 40 km. The effects of a varying surface reflectance are particularly noticeable over land sites where the correlations at small separation distances are reduced. The comparisons are worse at 200 km and 600 km over land and water where a much larger number of measurements fall outside the error bars and the correlation coefficients are much lower. The slope of the linear regression line steadily decreases beyond 40 km as the MODIS measurements separated from AERONET measurements tend to be biased low. This low bias in measurements away from AERONET site may be due to the AERONET sites used in this study being located preferentially near aerosol sources (i.e., caused by real spatial differences in aerosol loading).
 The correlations computed in Figure 3 are repeated for all 20 km bins up to 1000 km. Figure 4 shows the correlation of τAER and τMOD plotted versus separation distance for all temporal matches within 1 hour for each AERONET site partitioned by land and ocean sites. Correlations between AERONET and MODIS observations should be quite high for perfectly collocated observations with homogeneous topography and surface brightness. High correlations are seen over ocean sites beyond 40 km where homogeneity keeps the matching errors small. The correlations over land are smaller in the closest bins, but still large enough that we can see, in a statistical sense, how the matching errors increase with distance. Generally, instrumental error does not systematically increase with distance and the decrease in correlation with distance should be largely due to matching errors.
 Over land the correlations drop below 0.8 at approximately 100 km, but over the ocean the correlation remains above 0.8 for 420 km. The correlations over land generally drop faster than over the ocean because of a decreased spatial homogeneity in τa over land; though a varying surface reflectance that affects the MODIS retrievals may also reduce the correlations over land. Correlation can be used to guide the number of measurements needed to establish statistically significant correlation and a linear relationship between the two measurements. Anderson et al.  suggest that for a correlation of 0.9, 5 independent measurements are needed to establish correlation and 20 are needed to establish the linear relationship at the 95% confidence level, while for a correlation of 0.8, 7 and 54 independent measurements respectively are needed. Assuming that the decrease in correlation with distance is due entirely to matching errors we can get a rough idea of the acceptable separation distance that would need to be defined to achieve correlations of 0.9 and 0.8 by looking at the distances where the correlations drop by 10% and 20%, respectively. Over the ocean the correlation drops from its maximum at 60 km of 0.93 to 20% lower or 0.74 at 560 km; a distance of 500 km. Over land the maximum correlation is 0.85 at 20 km and drops 20% to 0.69 at 220 km; a distance of 200 km. Correlation drops 10% from its maximum in 300 km over the ocean and in 140 km over land. Matching errors increase more rapidly over land because of the increased inhomogeneity of aerosol optical depth. Aerosols have more spatial structure in air masses over land probably because of more inhomogeneous aerosol sources and topography, which tends to accumulate more aerosols in the valleys. The reductions in correlation are summarized in Table 2.
Table 2. Summary of the Distance From AERONET Sites Categorized as Land and Water in Table 1 That Corresponding MODIS Averaged Aerosol Optical Depth Loses 10% and 20% Correlationa
Distance Over Land, km
Distance Over Water, km
Number of Independent Measurements to Establish Correlation
Number of Independent Measurements to Establish a Linear Relationship
The last two columns are the number of independent measurements needed to establish a statistically significant correlation and linear relationship between the two measurements (From Anderson et al. ).
10% drop in correlation
20% drop in correlation
Anderson et al.  saw an autocorrelation drop 20% in 120–160 km for a more limited data set from the ACE-Asia field campaign and the Lidar In-space Technology Experiment (LITE) satellite. Measurement noise in the autocorrelation was shown to be small so that the reduction in correlation was due to matching errors. The resulting decrease in correlation is similar to the decrease observed in this paper given that the ACE-Asia and LITE data in the work by Anderson et al.  was for a polluted plume extending over and off the east coast of the United States and Southeast Asia. It should be noted that the coverage of correlative measurements over the total number of possible comparisons in this study is small especially at small separation distances and this could bias the comparisons toward days when MODIS and AERONET were able to measure τa (i.e., mostly sunny conditions). This bias could explain the slightly higher correlations seen in this study compared to the study by Anderson et al. . However, this paper assesses the correlations for comparisons of satellite and ground-based aerosol observations, which generally can only take place during these same (mostly sunny) conditions.
4. Using Back Trajectories to Improve Correlation
 To examine the possibility of increasing the number of highly correlated measurements of τa from satellite remote sensing instruments, back trajectories are calculated from four AERONET sites to determine if constraining our comparisons close to the average back trajectories improves the correlation to larger distances. Back trajectory analysis was done on four selected AERONET sites. Back trajectory analyses for AERONET sites are available each day on the AERONET website and the entire year of available back trajectories for 2002 and 2003 were analyzed and are presented in this section. One ocean site (Capo Verde) was chosen because of the predominantly long-range transport of Saharan dust aerosols to this site. Three land sites were chosen for their varying behavior of their correlation with distance. Alta Floresta was chosen for its smooth decrease in correlation with distance. Ispra and BSRN BAO Boulder were chosen because of their rapid decrease in correlation with distance.
Figure 5 shows the MODIS τa for four days selected with good overlap of MODIS observed τa around the AERONET sites. Also shown are the 1 hour average τAER and the 48 hour back trajectories at 950, 850, 700, and 500 hPa. Because of the altitude of the sites the 950 hPa back trajectory for Ispra and the 950 hPa and 850 hPa back trajectory for BSRN BAO Boulder are not valid.
 To test how well back trajectories can be used to increase the correlation, the averages of τMOD defined in (1) were further constrained to be within a certain azimuth angle, θ, of the 24-hour average computed back trajectory, traj. Each back trajectory level is averaged separately. The average τMOD are computed for each bin with an added restriction to (1),
Figure 6 shows how the correlations between τAER and MOD (bin) vary with distance on the basis of taking averages over all azimuth angles such as in Figure 4 and at 30° around the average 24-hour back trajectory at the three lowest back trajectory levels (2 for BSRN BAO Boulder).
 The first site shown (Figure 6), Capo Verde, is located 600 km west of Dakar, Africa. This site often receives Saharan dust during dust outbreaks at altitudes of 2–5 km or approximately 550–800 hPa [Holben et al., 2001]. The case shown is typical of the higher τa over the Saharan Desert extending out into the ocean to the west, often called the Saharan Aerosol Layer. Back trajectories at the higher levels extend back into the Saharan Aerosol Layer while the lowest back trajectory extends to the northeast along it. In this case the AERONET site is on the edge of the optically thick aerosol layer and all back trajectories extend into this layer. Using back trajectory information improves correlations to larger distances. In fact, the 0.8 correlation increases from 400 km to about 560 km separation distance. If the exact height of the key aerosol layers in the column were known, such as with lidars, the models could account for the advection wind speed and direction at these heights, presumably improving the correlation.
 Alta Floresta exists in a fairly homogeneous aerosol layer typical for this site that produces a fairly high correlation of τa at long distances. Because of this horizontal homogeneity back trajectories offer little improvement (Figure 6). Ispra and BSRN BAO Boulder are located at the foothills of large mountain chains. Mountainous areas increase MODIS retrieval uncertainties due to the bright surfaces and varying topography. Mountainous areas also have valleys that trap aerosols and produce large spatial inhomogeneities in aerosol optical depth. These factors probably cause the correlation to decrease with distance rapidly at these sites. Because of the complex topography, the back trajectory is probably also highly uncertain, causing back trajectory models to be of little use for these sites and, in fact, degrades the correlation.
 A sensitivity study was done to determine if the computed average back trajectory for different averaging times or the angle that was used to restrict the τa averages in the azimuth affect the correlations. Using the back trajectory level in Figure 6 that produces the best correlations within 400 km of the site, the averaging times of 12 and 48 hours were used to calculate the average back trajectory and the results are shown in Figure 7. No major consistent differences in the correlations at various distances were observed at any of the sites when different averaging times of the back trajectory were used.
 Next, the azimuth angle used to restrict τMOD averaging at various distances was changed to 22.5° and 45° around the average 24 hour back trajectory and the resulting effect on correlations are shown in Figure 8. Once again the back trajectory altitude that produced the best correlations in Figure 6 was used in each case. The increased azimuth angle at larger distances had a significant consistent improvement beyond 400 km for Capo Verde, a small but consistent improvement for Ispra and BSRN BAO Boulder and an inconsistent change for Alta Floresta. For Capo Verde the lowest back trajectory level produced the best improvement to the correlations within 400 km, but long-range transport becomes more significant beyond 400 km and the higher back trajectory levels produce more improved correlations at these separation distances. When comparing τa at longer distances from this site, the larger acceptance angle must contain more of this transported air from the Saharan Aerosol Layer.
5. Summary and Conclusions
 Two years of aerosol optical depth, τa, data from 22 AERONET sites and Terra MODIS retrievals were compared to determine how measurements separated spatially (causing matching errors) contribute to reducing correlations of these two measurements. Correlations of AERONET measured total-column aerosol optical depth, τAER, and MODIS retrieved total-column aerosol optical depth averaged in 20 km range bins from each AERONET site, MOD (bin), at 0.55 μm within 1 hour temporal separation were computed at each range bin. Because instrument errors does not change with distance in a consistent manner the change in correlation should be largely due to matching errors. Correlations were high at the closest separation distance. As separation distance increased, the matching errors caused a decreasing correlation. The correlation decreased by 20% over a 500 km distance over the ocean and 200 km over land and the correlation decreased by 10% over a 320 km and 140 km distance over water and land, respectively. Matching errors probably increase more rapidly over land because of the increased inhomogeneity of aerosol optical depth probably due to more inhomogeneous aerosol sources and topography, which tends to accumulate more aerosols in the valleys.
 Back trajectories were also computed for four sites and were used to constrain the data used to compute MOD (bin) at each separation distance. When the average was constrained to 30° azimuth on either side of the 24 hour average back trajectory the correlations improved only for Capo Verde, which is affected by long-range transport more than the other three sites analyzed with back trajectories. Capo Verde is affected by long-range transport of outflow of desert dust from the Saharan desert. The sites near mountainous areas had smaller correlations because of varying surface reflectance and topography upstream of the site. The varying topography should also affect spaceborne lidar comparisons of τa with ground sites because τa is a column integrated value and topography affects the geometrical size of the column.
 Using back trajectories to attempt to improve correlations suffer from a number of limitations that tend to reduce its effectiveness when applied to aerosols in the troposphere. First, aerosols are not conserved species. They have numerous sources and sinks. Their optical properties change because of changes in relative humidity. When an aerosol layer is distributed over a range of altitudes vertical wind shear will cause different layers of aerosol to advect, which affects the column optical depth. Aerosols are generally concentrated in the planetary boundary layer where all the previously mentioned difficulties are at their worst and where single back trajectories for an air parcel become difficult to define because of turbulent mixing [Stohl, 1998]. The results in this paper and the aforementioned limitations lead to the conclusion that back trajectories are primarily useful in extending high correlation to longer distances when long-range transport predominates and the aerosols are outside of the planetary boundary layer. Furthermore, when validating lidars where the heights of the key aerosol layers in the column are known, back trajectory models could account for the elevation-dependent wind speed and could further improve the correlations.
 Other more sophisticated back trajectory models such as Lagrangian particle dispersion models [Stohl, 1998] that simulate the transport and diffusion of a passive scalar tracer by calculating trajectories of many particles may be more useful when turbulent mixing is important. However, these models still require the tracer to be passive, where aerosols interact and evolve. Eulerian transport models [Grell et al., 2005; Mathur et al., 2005; Chin et al., 2002; Rasch et al., 1997] may be useful to account for transport of aerosols, but cannot resolve fine-scale structure, suffer from considerable uncertainties, and are computationally expensive.
 The Terra MODIS data used in this study were acquired as part of the NASA's Earth Science Enterprise. The algorithms were developed by the MODIS Science Teams. The data were processed by the MODIS Adaptive Processing System (MODAPS) and Goddard Distributed Active Archive Center (DAAC) and are archived and distributed by the Goddard DAAC. The author thanks the principal investigators and their staff for establishing and maintaining the 22 AERONET sites used in this investigation. Thanks to Tom L. Kucsera (SSAI) at NASA/Goddard for back trajectories available at the http://aeronet.gsfc.nasa.gov website. The author also wishes to thank Pat McCormick, Dave Winker, and the two anonymous reviewers for reading and providing helpful suggestions for the paper. Support for this paper was provided by NAS1-97042.