###### Abstract
To use the estimate as representation for subsequent processing, we collapse the density to bins resemblant of a histogram, with the advantage of inter- and extrapolation over observation gaps and reduced noise in pixel values, especially for parcels of varying size. We also consider Bayesian estimators [22] to account for uncertainty stemming from small pixel counts; in our data the field parcel size varies from tens to hundreds of pixels. We apply the proposed method to a case of off-season soil tillage monitoring in Finland. From the standpoint of environmentally and economically sustainable agriculture, soil erosion and nutrient runoff from crop fields to surface waters is a long-standing challenge, to which soil tillage operations are a contributing factor [23,24]. Large-scale information on annual off-season tillage status of arable land is of interest for agro-environmental monitoring administration, policy makers as well as wide range of academic domains from terrestrial carbon studies to hydrological research. The problem is made challenging by the irregular shapes and small sizes of the parcels in Finland, and the limited amount of labeled training data within a single country. Furthermore, the annual off-season observation time window is limited and must take place during a relatively cloudy time of the year in Finland. The proposed computational pipeline can address all of these challenges. Previous remote sensing studies of soil tillage detection have focused on spectral reflectance characteristics [25, 26, 27] or radar response [28, 29, 30] of soils, green vegetation, and crop residues. SAR signal penetrates cloud cover and is inherently sensitive to target 3D structure affecting backscatter mechanisms and angle. On the other hand, optical, for example. multispectral reflectance from satellite images can be fully exploited only at cloudless moments, but it can characterize a wide range of chemical and physical properties of matter as well as reveal dynamics of organic phenomena. The differences in the physical processes involved in radar backscatter and optical reflectance signals allow them to respond complementarily to the phenomena being interpreted, which results in enhanced accuracy of classification and regression for a range of remote sensing applications, for example, in agricultural or land cover contexts [31, 32, 33, 34, 35], or recently in soil tillage detection [36, 37]. Our contribution to the topic is a practical, effort-reducing SAR-MSI data fusion technique. We use Copernicus Sentinel-1 (S1) SAR data with dimensions of two polarity bands, overlaid with polygon data of crop fields as provided by the Finnish Food Authority. Similarly to SAR bands, we construct density estimates over two spectral indices calculated from a Sentinel-2 (S2) MSI for the same fields during a suitably narrow time window. Both sources are then merged to estimate the soil tillage category. Besides the Sentinel data used here, we expect the approach to be applicable to other SAR and MSI data sources, such as Radarsat-2, Landsat, or Modis. To summarize, as the main contributions of this work we:  
* Present a method for off-season tillage and vegetation cover detection on crop field parcels, which is a challenging task due to the heterogeneous size of the objects and a limited amount of training observations;
* Propose a representation of a free-form raster image object as a non-parametric probability density estimate, to be used for increasing robustness to variability in object size, count and missing pixel data;
* Introduce an easy-to-use framework for multi-sensor raster data fusion for sources of varying spatial resolution, applicable also outside the specific task considered here.