Bayesian object-based approach

4.3.1.1 Bayesian optimization of a winter wheat object

The optimization procedure is described in detail for the example of a winter wheat field (object id = 6) for which ground measurements were available. The left side of Figure 4.1 shows the variation of the total, radiometric and a priori costs, the variation of the six free variables and the variation of the damping factor through the iterations of the optimization algorithm. Iteration 0 represents the starting point of the optimization, and the algorithm stopped at iteration 5 because the changes for all variables became smaller than 1% of their a priori variation range. The values at iteration 5 therefore are the estimates. The comparison with reference data (see Table 4.3) shows that good estimates were obtained for LAI and Cab, whereas Cw and Cdm were overestimated. Through the iterations, the total cost decreased from about 1600 to close to 900. The total cost consisted almost entirely of the radiometric cost, because the a priori cost was very low (< 8). This situation of very high radiometric cost and very small a priori cost was similar for all objects in the study area. The effect of constraining the variables in their a priori range cannot be seen in this example. After iteration 2, the changes in the variable values became small, until the damping factor (µ) increased between iterations 4 and 5, leading to the termination of the optimization.

The right side of Figure 4.1 shows the variations of the simulated radiance through the optimization in visible (VIS), near-infrared (NIR) and shortwave infrared (SWIR) feature spaces. Iteration 1 brought the simulation closer to the APEX measurements in the three spectral domains. Iteration 2 provided an improvement in the SWIR, but not in the VIS and NIR. The small changes in the parameter values in iteration 3 to 5 caused small changes in the simulated radiance. No changes were visible in the VIS and SWIR, but the simulated radiance at 847 nm drifted slightly away from the APEX measurement. The variable changes in iterations 3 to 5, however, decreased the cost value, so that they must have provided improvements in spectral bands not shown in Figure 4.1.

The final match between APEX measurement and model simulation at the end of the optimization is presented in Figure 4.2. The overall agreement was very good, with slightly higher simulated radiances in the VIS and SWIR spectral domains. The

Figure 4.1 Variation of the costs, variables, damping factor µ (left) and radiance values (right) for a winter wheat object (id = 6) through the optimization iterations. The allowed variation range for the variables and damping factors were used as limits for the y axes of the corresponding plots. In the right part of the figure, the numbers refer to iterations 0, 1, 2 and 5, and VIS, NIR and SWIR refer to the visible, near-infrared and shortwave infrared domains.

Figure 4.2 Radiometric match and absolute radiometric error at the end of the Bayesian optimization of a winter wheat object (id = 6).

absolute error was less than 1 mW/(m2 sr nm) in the SWIR, and less than 5 mW/(m2 sr nm) in the VIS. The obvious spikes in the error curve in the NIR domain have no apparent trend to over- or under-simulation.

4.3.1.2 Object-level results

The results obtained from the Bayesian optimization at object level are presented in Figure 4.3. The object-level crop classification had an overall accuracy of 79 %. The misclassifications concerned mostly the fields having low canopy cover, such as short grass, bean and pea. For all variables, the estimated values covered the entire a priori ranges, except for very high values of Cab (Cabmax was 100 μg/cm2) and fB (fBmax was 0.5). A visual check with the true colour APEX image (not shown) revealed that the pattern of fields having low estimated Cv is realistic. Further, very high LAI values are mostly found in winter wheat objects, but some are found in objects having low Cv values and may therefore be erroneous. The leaf variables show a lot of variation from object to object, but it is not possible to visually evaluate the plausibility of the results. The cost values are a good proxy to judge the estimation performance. Low costs indicate a good radiometric match and a reliable estimate and vice versa. More than half of the study area had costs lower than 2839 and only a few objects had very high costs indicating unrealistic estimates. These were mainly fields having low canopy cover (short grass, bean and pea).

The object estimates of LAI, Cab, Cdm and Cw were compared with the average of the four point measurements taken in the corresponding field. Table 4.3 shows that

LAI was estimated most accurately, followed by Cw, and Cdm. Cab was poorly estimated. A positive bias was observed for LAI and Cab.

4.3.1.3 Pixel-level results: LAI and Cab estimates

The LAI, Cab and cost maps obtained at pixel level from the object-based LUTs (Figure 4.4) present similar spatial patterns to those observed at the object level (Figure 4.3), but intra-object variability appears. This intra-object variability was

Figure 4.3 Object crop map, maps of the estimates of the seven variables at the object level, and map of the total cost, obtained from step 1 of the Bayesian object-based approach (raw image geometry, north towards the top).

Table 4.3 Comparison of the object-level estimates (step 1 of the Bayesian object-based approach) with the ground measurements (averages of the four point measurements taken in each field). For explanation of crop types see Table 4.1.

Crop Object

id

LAI (-) Cab (μg/cm2_{) Cdm (g/cm}2_{) Cw (cm)}

Measured Estimated Measured Estimated Measured Estimated Measured Estimated

W 6 3.4 3.3 36 50 0.005 0.009 0.01 0.004 Co 36 1.3 1.9 25 32 0.003 0.003 0.01 0.02 S 37 2.3 2.1 34 48 0.005 0.004 0.03 0.03 P 50 1.5 2 35 39 0.003 0.004 0.02 0.02 Cl 58 4.5 3.8 36 27 0.004 0.007 0.01 0.01 R2 _{0.95 0.11 0.36 0.68} RMSE 1.1 23 0.005 0.012 Bias 1.0 16 -0.0006 0.002

expected and is related to small scale variations due to management practices of the farmers or due to local variation in the soil conditions. Another spatial pattern appearing on the pixel-level Cab map was a very narrow vertical striping. This effect is typical for pushbroom sensors like APEX and can be explained by small deviations of the response of detector elements across track. In the objects, the cost values are higher in areas with less vegetation, indicating a higher estimation uncertainty. A closer look at this effect reveals that objects presenting a wide range of Cab values are typically characterized by a low LAI, whereas objects with higher LAI have a lower variability in Cab. This reflects the smaller sensitivity to Cab in canopies having a smaller LAI.

The estimates were compared with the 20 available ground measurements using 3 by 3 averaging windows. The comparison (left plots in Figure 4.5) shows that the quality of the estimates was rather poor and that LAI (R2 _{= 0.45) was estimated more} accurately than Cab (R2 = 0.18).

Figure 4.4 Maps of the LAI and Cab estimates and of the total cost obtained from step 2 of the Bayesian object- based approach (raw image geometry, north towards the top).

Figure 4.5 Comparison of the LAI and Cab estimates obtained from the two approaches. The plots on the left side present the comparison of the estimates with the ground measurements (the line is the 1:1 line), and the maps on the right side present the absolute difference between the estimates obtained from the two approaches, red indicates that the estimate from the Bayesian object-based approach had a higher value than that of the LUT-BCF approach (raw image geometry, north towards the top). The LAI values are unitless, and the Cab values are in

μg/cm2.