## 1 Introduction  
Because agricultural productivity directly influences food prices, trade and greenhouse-gas emission policies, and human livelihood [1, 2], accurate and timely yield estimation is of primary interest to the scientific community and governments [3]. Compared with traditional ground-based methods, such as visual examination and survey sampling, remote sensing that provides synoptic and repetitive observations of the land surface is well suited for agricultural mapping [4, 5, 6] and monitoring [7, 8] large geographic areas. In particular, satellite-derived vegetation indices, as measures of plant chlorophyll abundance and vegetation radiation absorption [9], have proven to be closely related to crop growth in field studies and theoretical models [10, 11, 12]. Despite tremendous efforts, obtaining accurate yield estimations based on remotely sensed data remains difficult [13, 14]. One approach develops regression models that relate satellite-derived vegetation indices directly to historical yield data [15, 16, 17]. These models are essentially retrospective and are based empirically on indirect inferences, whereby changes in vegetation indices can determine variations in plant productions [14, 18]. Because the regression relationship varies largely on a year-to-year basis due to inter-annual variations in climate, water availability, and management practices, the application of these models is limited to the studied regions and periods and is difficult under extreme conditions (e.g., flooding and drought) beyond historical records. Another approach applies satellite data to calibrate physiology-based crop models [19, 20, 21] that simulate the physical process of crop growth, where energy, water, carbon dioxide, and nutrients are converted into biomass [22, 23]. These models have proven useful at the field scale but possess one limitation for large-scale application: they often require numerous inputs related to soil characteristics, management practices, and local weather conditions [24, 25, 26]. Given the difficulties of the two above-mentioned approaches, improving yield estimation with satellite data based on the Production Efficiency Models (PEMs) is a highly desirable goal. The theory proposed by [PERSON] [27, 28] proposes a basis for estimating the primary productivity of vegetation from remotely sensed data. It suggests that crop yields under non-stressed conditions linearly correlate with the amount of absorbed photosynthetically active radiation. Satellite data at different resolutions have been synthesized in production efficiency models to monitor vegetation productivities [29, 30, 31, 32] and estimate crop yields [33, 34, 35] on regional or global scales. Based on production efficiency models, a routine product (MOD17) has been derived from high temporal and moderate spatial resolution MODIS data, which has proven useful to studies on global carbon cycles [36, 37]. However, validations against flux tower data indicate that MOD17 estimates exhibit large uncertainties in croplands [38]. Crop yield estimates based on the MOD17 products were found to have weak correlation with national agricultural data at the county level for the states of Montana and North Dakota in the US [18]. The MOD17 algorithm prescribes constant values of model parameters such as radiation use efficiency (RUE) for the agricultural biome regardless of crop types [36, 37]. However, crops are known to possess distinctive carbon sequestration abilities and thereby different radiation use efficiencies across species, notably the C\({}^{3}\) and C\({}^{4}\) pathways. Moreover, yield estimation based on remotely sensed data requires acknowledging spatial heterogeneity effects because the pixels are typically mixtures of diverse land-cover types at the MODIS spatial resolution [39, 40]. Remote sensing models that consider the conceptual partition of mixed pixels are likely to improve the yield estimates of specific crop types. This study aims to improve yield estimation with MODIS data based on production efficiency models.