- freely available
ISPRS Int. J. Geo-Inf. 2020, 9(1), 30; https://doi.org/10.3390/ijgi9010030
2. Materials and Methods
2.1. Description of the Dataset and Study Area
2.5. Verification and Validation of MAF and ICA Algorithms in the Joint Simulation of Soil Health Indicators
- The test for spatial orthogonality assesses how well the methods (i.e., MAF and ICA) orthogonalize the variogram matrices at various lag distances [29,52]:
- The relative deviation from orthogonality can therefore be defined as:
- The spatial diagonalization efficiency is a measure that compares the sum of squares of off-diagonal elements in ΓMAF (h) with those of the attribute of the semi-variogram matrix ΓY (h):
- Reproduction of direct and cross-variograms, using the average of realizations and randomly selected realizations:For both algorithms to be valid, the direct and cross-variograms of the original variables should be reproduced by the back-transformed simulated realizations. A total of 100 realizations would be averaged (in data space). This is defined as E-type, where E is the “conditional expectation” of realizations (i.e., average estimate of realizations) [51,53], and their direct and cross-variograms would be compared with those of the original variables for both MAF and ICA. In addition, randomly selected back-transformed simulated realizations would be compared. The goal is to ensure these realizations reproduce the spatial structure and characteristics of the original variable;
- Reproduction of the original distribution, cross-correlation, and spatial pattern:Cross-correlation between E-type of the back-transformed realizations and original variables would be compared. The reproduction of the histogram of the original variables would also be explored. In addition to the quantitative comparison above, visual inspections in the form of the reproduction of the spatial pattern of the original variable and back-transformed simulated realizations would be considered. This is to ensure consistency and to validate spatial dependency within the variables and also to guarantee the spatial relationship among the variables.
3. Results and Discussion
3.1. MAF and ICA Results
- Performance evaluation if MAF factors and ICA components are orthogonal for all lag distances, using the measure of spatial orthogonality;
- Reproduction of both direct and cross-variograms of original SHIs, using an average of 100 back-transformed simulated realizations for both MAF and ICA;
- Reproduction of direct and cross-variograms of original SHIs, using randomly selected back-transformed simulated realizations for both MAF and ICA; and
- Exploration of the distribution (histogram), cross-correlation, and spatial pattern of original SHIs and back-transformed simulated realizations.
3.2. Performance Evaluation of MAF and ICA Decorrelation, Using the Spatial Orthogonality Measures
3.3. Reproduction of Original Direct Variograms
3.4. Reproduction of Direct and Cross-Variograms, Using E-Type of Simulations
3.5. Reproductions of Original Variograms and Cross-Variograms, Using Randomly Selected Realizations
3.6. Reproduction of Distributions, Cross-Correlation, and Spatial Patterns of Original Variables by E-Type
3.7. Importance of Uncertainty in Simulating Spatially Correlated Variables and Implications for Best Management Practices (BMP) in Sustainable Management
4. Conclusions and Recommendation
- Comparative analysis between the two methods revealed no marked differences in their performance. However, NST is necessary before MAF transformation. In the case of ICA, it was unnecessary to perform NST before transformation. In other words, NST was only used for IC while generating equally probable realizations via SGS. Therefore, IC can be used directly in other applications that do not require SGS simulation;
- Both techniques satisfy the two criteria for spatial orthogonality suggested by Tercan (1999). These are absolute deviation from diagonality () and relative deviation from diagonality (), with ideal values of approximately 0 and 1, respectively. In other words, the MAF and ICA should be spatially orthogonal with a correlation at zero for all distances before they are used in SGS;
- If MAF and ICA are simulated independently, both methods only require one direct variogram for each factor/component. This is in contrast to the three direct and three cross-variograms that would be needed if a traditional approach such as the model of co-regionalization was used. In other words, both MAF and ICA correctly reproduced the direct and cross-variograms of the original variables despite the variograms of MAF factors and ICA being simulated independently;
- The back-transformed simulated variogram realizations were comparable with the original variables of each variable. Moreover, the E-type, which is the average of 100 realizations, compared well with the original variables. The cross-correlation, histograms, and spatial pattern of the back-transformed realizations, using the E-types, were correctly reproduced.
Conflicts of Interest
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|Bulk Density (g/cm3)||Organic Carbon (g/kg)||Total Nitrogen (g/kg)|
|Coefficient of variation (%)||9.10||89.44||95.65|
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