## 3 Results
### Pixel Size
#### 3.1.1 Square Pixels with Biases  
The XCO\({}_{2}\) observations relative to the constant 400 ppm background for the baseline experiment with no external biases are shown in Figure 1. The quantitative emission estimates are shown in Figure 5, which shows little difference between the accuracy or precision of the emission estimates for different pixel sizes. The high accuracy emission estimates resulting from all pixel sizes were a result of including no biases in the simulation. The emission estimates from the 2 \(\times\) 2, 4 \(\times\) 4, and 7 \(\times\) 7 km\({}^{2}\) pixels all had approximately 2.5 Mt yr\({}^{-1}\) precision for all emission rates. This was equivalent to 42%, 19%, and 10% precision for the 6, 13, and 25 Mt yr\({}^{-1}\) sources, respectively. The 10 \(\times\) 10 km\({}^{2}\) km pixels have 3 Mt yr\({}^{-1}\) precision, or 50%, 23%, and 12% precision for each source emission rate, respectively. The consistent precision across the pixel sizes is expected for this baseline experiment with square pixels. The SNR and maximum plume enhancements are both inversely proportional to the pixel linear side length. Since no external biases are included, the ability to estimate the emissions is primarily controlled by the ratio of the SNR and the maximum enhancements in the plume. Thus, we expect similar results across the pixel sizes. The slightly worse precision of the 10 km pixel illustrates that with decreased relative enhancements the emission estimation method is more sensitive to slight changes in the background noise. The ensemble is also run with a combination of observational biases and a methodological bias for a more realistic scenario. The observational biases are a topography related bias (max XCO\({}_{2}\) anomaly 1.2 ppm), a footprint bias (max XCO\({}_{2}\) anomaly 0.4 ppm), and an albedo bias (max XCO\({}_{2}\) anomaly 0.4 ppm). The methodological error is a mismatched atmospheric stability parameter between the modelled plume (\(a=130.0\) m) and the simulated data (\(a_{sim}=156.0\) m). Biases are described in detail in the supplementary materials. The topography bias was related to the gradient of the surface elevation, similar to the bias that was corrected in OCO-2 V9 data [9]. The simulated data with this combination of biases is shown in Figure 6, and the emission estimates are shown in Figure 7.  
Figure 7 shows large systematic errors due to the included biases when compared to the baseline simulation (Figure 5). The 2, 4, and 10 km pixel sizes have similar accuracy, while the 7 km pixel size has noticeable larger systematic biases. For the 6 Mt yr\({}^{-1}\) source, the 2, 4, and 10 km pixels systematically underestimate the emissions by approximately 3 Mt yr\({}^{-1}\) (50%), while the 7 km pixel systematically underestimates the emissions by approximately 4.5 Mt yr\({}^{-1}\) (75%). For larger sources the absolute bias is larger, but the percent bias decreases. For the 25 Mt yr\({}^{-1}\) source, the systematic biases are approximately 16% for the 2, 4, and 10 km pixels, and 22% for the 7 km pixels. The precision of the estimates is similar to the baseline experiment (Figure 5). However, with the systematic errors induced by the biases, the outside of the range of estimates for the 7 \(\times\) 7 km\({}^{2}\) pixel extends into negative emission rates. Figure 5: Emission rate estimates for all 12 combinations of pixel sizes and emission rates for an ensemble of 30 simulations. The green line is the true simulated emission rate, the orange line in each box is the median emission estimate, the black boxes show the range between the lower and upper quartiles (the interquartile range, IQR), and the whiskers (capped black lines) extend from the smallest datum within 1.5 times the IQR below the box to the largest datum within 1.5 times the IQR above the box. Outliers outside this range are shown as open circles. (**a**) Emission rate estimates for a 6 Mt yr\({}^{-1}\) source. (**b**) Emission rate estimates for a 12 Mt yr\({}^{-1}\) source.