Sixty Years of Fractal Projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Comment: Submitted to proceedings of Fractals and Stochastics

Overview of the O3M SAF GOME-2 operational atmospheric composition and UV radiation data products and data availability

The three Global Ozone Monitoring Experiment-2 instruments will provide unique and long data sets for atmospheric research and applications. The complete time period will be 2007–2022, including the period of ozone depletion as well as the beginning of ozone layer recovery. Besides ozone chemistry, the GOME-2 (Global Ozone Monitoring Experiment-2) products are important e.g. for air quality studies, climate modelling, policy monitoring and hazard warnings. The heritage for GOME-2 is in the ERS/GOME and Envisat/SCIAMACHY instruments. The current Level 2 (L2) data cover a wide range of products such as ozone and minor trace gas columns (NO₂, BrO, HCHO, H₂O, SO₂), vertical ozone profiles in high and low spatial resolution, absorbing aerosol indices, surface Lambertian-equivalent reflectivity database, clear-sky and cloud-corrected UV indices and surface UV fields with different weightings and photolysis rates. The Satellite Application Facility on Ozone and Atmospheric Chemistry Monitoring (O3M SAF) processes and disseminates data 24/7. Data quality is guaranteed by the detailed review processes for the algorithms, validation of the products as well as by a continuous quality monitoring of the products and processing. This paper provides an overview of the O3M SAF project background, current status and future plans for the utilisation of the GOME-2 data. An important focus is the provision of summaries of the GOME-2 products including product principles and validation examples together with sample images. Furthermore, this paper collects references to the detailed product algorithm and validation papers

Validation of tropospheric NO2 column measurements of GOME-2A and OMI using MAX-DOAS and direct sun network observations

Multi-axis differential optical absorption spectroscopy (MAX-DOAS) and direct sun NO2 vertical column network data are used to investigate the accuracy of tropospheric NO2 column measurements of the GOME-2 instrument on the MetOp-A satellite platform and the OMI instrument on Aura. The study is based on 23 MAX-DOAS and 16 direct sun instruments at stations distributed worldwide. A method to quantify and correct for horizontal dilution effects in heterogeneous NO2 field conditions is proposed. After systematic application of this correction to urban sites, satellite measurements are found to present smaller biases compared to ground-based reference data in almost all cases. We investigate the seasonal dependence of the validation results as well as the impact of using different approaches to select satellite ground pixels in coincidence with ground-based data. In optimal comparison conditions (satellite pixels containing the station) the median bias between satellite tropospheric NO2 column measurements and the ensemble of MAX-DOAS and direct sun measurements is found to be significant and equal to -34 % for GOME-2A and -24 % for OMI. These biases are further reduced to -24 % and -18 % respectively, after application of the dilution correction. Comparisons with the QA4ECV satellite product for both GOME-2A and OMI are also performed, showing less scatter but also a slightly larger median tropospheric NO2 column bias with respect to the ensemble of MAX-DOAS and direct sun measurements

Validation of tropospheric NO₂ column measurements of GOME-2A and OMI using MAX-DOAS and direct sun network observations

MAX-DOAS and direct sun NO2 vertical column network data are used to investigate the accuracy of tropospheric NO2 column measurements of the GOME-2 instrument on the MetOP-A satellite platform and the OMI instrument on Aura. The study is based on 23 MAX-DOAS and 16 direct sun instruments at stations distributed worldwide. A method to quantify and correct for horizontal dilution effects in heterogeneous NO2 field conditions is proposed. After systematic application of this correction to urban sites, satellite measurements are found to present smaller biases compared to ground-based reference data in almost all cases. We investigate the seasonal dependence of the validation results, as well as the impact of using different approaches to select satellite ground pixels in coincidence with ground-based data. In optimal comparison conditions (satellite pixels containing the station) the median bias between satellite tropospheric NO2 column measurements and the ensemble of MAX-DOAS and direct sun measurements is found to be significant and equal to −36 % for GOME-2A and −20 % for OMI. These biases are further reduced to −24 % and −8 % respectively, after application of the dilution correction. Comparisons with the QA4ECV satellite product for both GOME-2A and OMI is also performed, showing less scatter but also a slightly larger median tropospheric NO2 column bias with respect to the ensemble of MAX-DOAS and direct sun measurements