Current methods of spatial prediction are based on either the First Law of Geography or the statistical principle or the combination of these two. The Second Law of Geography contributes to the revision of these methods so they are adaptive to local conditions but at the cost of increasing demand for samples. This paper presents a new thinking about spatial prediction based on the Third Law of Geography which focuses on the similarity of geographic configuration of locations. Under the Third Law of Geography, spatial prediction can be made on the basis of the similarity of geographic configurations between a sample and a prediction point. This allows the representativeness of a single sample to be used in prediction. A case study in predicting spatial variation of soil organic matter content was used to compare the spatial prediction based the Third Law of Geography with those based on the First Law and the statistical principle. It is concluded that spatial prediction based on the Third Law of Geography does not require samples to be over certain size nor to be of a particular spatial distribution to achieve a high quality prediction. The prediction uncertainty associated with spatial prediction based on the Third Law of Geography is more indicative to quality of the prediction, thus more effective in allocating error reduction efforts. These properties make spatial prediction based on the Third Law of Geography more suitable for prediction over large and complex geographic areas.

Most multiple‐flow‐direction algorithms (MFDs) use a flow‐partition coefficient (exponent) to determine the fractions draining to all downslope neighbours. The commonly used MFD often employs a fixed exponent over an entire watershed. The fixed coefficient strategy cannot effectively model the impact of local terrain conditions on the dispersion of local flow. This paper addresses this problem based on the idea that dispersion of local flow varies over space due to the spatial variation of local terrain conditions. Thus, the flow‐partition exponent of an MFD should also vary over space. We present an adaptive approach for determining the flow‐partition exponent based on local topographic attribute which controls local flow partitioning. In our approach, the influence of local terrain on flow partition is modelled by a flow‐partition function which is based on local maximum downslope gradient (we refer to this approach as MFD based on maximum downslope gradient, MFD‐md for short). With this new approach, a steep terrain which induces a convergent flow condition can be modelled using a large value for the flow‐partition exponent. Similarly, a gentle terrain can be modelled using a small value for the flow‐partition exponent. MFD‐md is quantitatively evaluated using four types of mathematical surfaces and their theoretical ‘true’ value of Specific Catchment Area (SCA). The Root Mean Square Error (RMSE) shows that the error of SCA computed by MFD‐md is lower than that of SCA computed by the widely used SFD and MFD algorithms. Application of the new approach using a real DEM of a watershed in Northeast China shows that the flow accumulation computed by MFD‐md is better adapted to terrain conditions based on visual judgement.

Summary: Existing predictive soil mapping ( PSM ) methods often require soil sample data to be sufficient to represent soil–environment relationships throughout the study area. However, in many parts of the world with only a limited quantity of soil sample data to represent the study area, this is still an issue for PSM application. This paper presents a method, named ‘individual predictive soil mapping’ ( iPSM ), which can make use of limited soil sample data for PSM . With the assumption that similar environmental conditions have similar soils, iPSM uses the soil–environment relationship at each individual soil sample location to predict soil properties at unvisited locations and estimate prediction uncertainty. Specifically, the environmental similarities of an unvisited location to a set of soil sample locations are used in a weighted average method to integrate the soil–environment relationships at sample locations for prediction and uncertainty estimation. As a case study, iPSM was applied to map soil organic matter ( SOM ) content (%) in the topsoil layer using two sets of soil samples. Compared with multiple linear regression ( MLR ), iPSM produced a more accurate SOM map (root mean squared error ( RMSE ) 1.43, mean absolute error ( MAE ) 1.16) than MLR ( RMSE 8.54, MAE 7.34) the ability of the sample set to represent the study area is limited and achieved a comparable accuracy ( RMSE 1.10, MAE 0.69) with MLR ( RMSE 1.01, MAE 0.73) when the sample set could represent the study area better. In addition, the prediction uncertainty estimated by iPSM was positively related to prediction residuals in both scenarios. This study demonstrates that iPSM is an effective alternative when existing soil samples are limited in their ability to represent the study area and the prediction uncertainty in iPSM can be used as an indicator of its prediction accuracy.