Figure 1.
Location of South Esk Hydrological model, northeast of Tasmania, Australia. The exact coordinate locations of the red bounded region are: 42°00S, 147°00E (bottom, left) and 41°00S, 148°30E (top, right), which covers approximately 18,588 km (image adapted from Google Maps).
Figure 1.
Location of South Esk Hydrological model, northeast of Tasmania, Australia. The exact coordinate locations of the red bounded region are: 42°00S, 147°00E (bottom, left) and 41°00S, 148°30E (top, right), which covers approximately 18,588 km (image adapted from Google Maps).
Figure 2.
An illustration of the TIN interpolation technique at point x. In this case, point w lies within a triangle formed by points , and . The weight of each point is calculated based on the corresponding area. For example, has the weight of , and so on.
Figure 2.
An illustration of the TIN interpolation technique at point x. In this case, point w lies within a triangle formed by points , and . The weight of each point is calculated based on the corresponding area. For example, has the weight of , and so on.
Figure 3.
A demonstration of the improved IDW. The dots are the sample data points, and the x is the point location to be interpolated. R is a user-defined radius parameter that indicates the farthest distance to be included from the points x. In this case, only a total of six sample points will have an influence on the interpolation process. However, no empirical approach has been developed to obtain the optimal value for the parameter R.
Figure 3.
A demonstration of the improved IDW. The dots are the sample data points, and the x is the point location to be interpolated. R is a user-defined radius parameter that indicates the farthest distance to be included from the points x. In this case, only a total of six sample points will have an influence on the interpolation process. However, no empirical approach has been developed to obtain the optimal value for the parameter R.
Figure 4.
Distribution methods that are proposed in this work for comparisons: (a) Gaussian; (b) Lorentzian and (c) Laplacian distributions. Within the proposed DDW, the horizontal axis represents the distance, and the vertical axis is the weight values.
Figure 4.
Distribution methods that are proposed in this work for comparisons: (a) Gaussian; (b) Lorentzian and (c) Laplacian distributions. Within the proposed DDW, the horizontal axis represents the distance, and the vertical axis is the weight values.
Figure 5.
Demonstration of the entire simulation setup for each time slice using a single scheme. The example above interpolates Time Index 5, using scheme (2, 3). Map 1 (top) is the “estimated” map generated using the STI method; Map 2 (bottom) is the “observed” map based on the original data. The objective performance of the algorithm is measured using RMSE.
Figure 5.
Demonstration of the entire simulation setup for each time slice using a single scheme. The example above interpolates Time Index 5, using scheme (2, 3). Map 1 (top) is the “estimated” map generated using the STI method; Map 2 (bottom) is the “observed” map based on the original data. The objective performance of the algorithm is measured using RMSE.
Figure 6.
Visualisation of the South Esk hydrological modelled dataset used throughout the simulation: (a) the elevation map, in meters; and (b) the air temperature data, in Kelvin.
Figure 6.
Visualisation of the South Esk hydrological modelled dataset used throughout the simulation: (a) the elevation map, in meters; and (b) the air temperature data, in Kelvin.
Figure 7.
Visualisation output for: the selected sample point locations (a) using the optimisation technique (DE); and three widely-used different spatial interpolation techniques (a) ordinary kriging, (b) triangular irregular network and (c) inverse distance weighting], compared to different numbers of sample points ().
Figure 7.
Visualisation output for: the selected sample point locations (a) using the optimisation technique (DE); and three widely-used different spatial interpolation techniques (a) ordinary kriging, (b) triangular irregular network and (c) inverse distance weighting], compared to different numbers of sample points ().
Figure 8.
This graph demonstrates a comparison of computational efficiency for different 2D spatial interpolation techniques, with the data based on
Table 3.
Figure 8.
This graph demonstrates a comparison of computational efficiency for different 2D spatial interpolation techniques, with the data based on
Table 3.
Figure 9.
This graph demonstrates the quantitative comparisons (RMSE) between the different 2D spatial interpolation techniques, using the data as shown in
Table 4.
Figure 9.
This graph demonstrates the quantitative comparisons (RMSE) between the different 2D spatial interpolation techniques, using the data as shown in
Table 4.
Figure 10.
This graph demonstrates the MAE with the corresponding SE between the different 2D spatial interpolation techniques, using the data as shown in
Table 4.
Figure 10.
This graph demonstrates the MAE with the corresponding SE between the different 2D spatial interpolation techniques, using the data as shown in
Table 4.
Figure 11.
Performance comparisons between different temporal interpolation approaches using the improved IDW: extension (existing) and reduction (proposed) approaches. The vertical axis is the RMSE of the respective (previous and next) windows’ length.
Figure 11.
Performance comparisons between different temporal interpolation approaches using the improved IDW: extension (existing) and reduction (proposed) approaches. The vertical axis is the RMSE of the respective (previous and next) windows’ length.
Figure 12.
Visualisation output for the comparison results between the extension- and reduction-based STI methods. In this output, we used the scheme involving plus-and-minus one time index. This selection is based on the results obtained from
Figure 11, which shows such a configuration produces the best results overall (lowest mean RMSE). (
a) Extension approach; (
b) reduction approach.
Figure 12.
Visualisation output for the comparison results between the extension- and reduction-based STI methods. In this output, we used the scheme involving plus-and-minus one time index. This selection is based on the results obtained from
Figure 11, which shows such a configuration produces the best results overall (lowest mean RMSE). (
a) Extension approach; (
b) reduction approach.
Figure 13.
Visualisation output of IDW (a) and the proposed DDW: (b) Gaussian; (c) Lorentzian; and (d) Laplacian; using the proposed spatiotemporal interpolation technique.
Figure 13.
Visualisation output of IDW (a) and the proposed DDW: (b) Gaussian; (c) Lorentzian; and (d) Laplacian; using the proposed spatiotemporal interpolation technique.
Figure 14.
Objective comparisons between the IDW method and the proposed DDW techniques: using Gaussian, Lorentzian and Laplacian distributions. This graph demonstrates the RMSE comparison between the respective (previous and next) windows’ length.
Figure 14.
Objective comparisons between the IDW method and the proposed DDW techniques: using Gaussian, Lorentzian and Laplacian distributions. This graph demonstrates the RMSE comparison between the respective (previous and next) windows’ length.
Table 1.
List of previous studies that utilised different interpolation techniques for environmental studies, and the findings from each study are also provided. Comparing methods as done here cannot be generalized, as this is limited to the empirical exercises, and no rigorous theoretical demonstration is provided (the table is sorted based on the similarity of interpolation techniques adopted and compared in each study).
Table 1.
List of previous studies that utilised different interpolation techniques for environmental studies, and the findings from each study are also provided. Comparing methods as done here cannot be generalized, as this is limited to the empirical exercises, and no rigorous theoretical demonstration is provided (the table is sorted based on the similarity of interpolation techniques adopted and compared in each study).
Process | Techniques | Findings | Ref. |
---|
Spatial rainfall mapping using IDW in the middle of Taiwan. | IDW | - IDW can be improved by the adjustment of the distance-decay parameter and the search radius. | [4] |
- Does not have significant interpolating ability for extreme values. |
Compared to the developed CAR method with IDW for climate datasets in China. | IDW, CAR | - High-elevation data are an important factor for meteorological studies. | [5] |
- CAR performs slightly better than GIDW in terms of objective comparisons, especially in estimating local neighbouring patterns. |
Comparing different techniques for precipitation and elevation. | IDW, AIDW, Kriging | - Varying the distance-decay parameter of IDW based on the spatial pattern can improve overall performance (AIDW). | [6] |
- AIDW can perform better than kriging in some cases. |
Proposing regression-based IDW and comparing it with IDW and kriging. | IDW, RIDW, Kriging | - Integrating linear regression in IDW provides comparable objective evaluation to kriging and is computationally less demanding. | [7] |
- The confidence interval (CI) of RIDW also surpasses kriging CI. |
Evaluating different interpolation techniques for air temperature data. | Spline, IDW, Kriging | - Kriging performs better overall, followed by IDW and spline. | [8] |
Comparing different techniques for rainfall mapping in Sri Lanka. | IDW, TPS, OK, BK | - Interpolation results are very much dependent on the settings. | [9] |
- Different methods with different settings must be tested to define the suitable technique. |
- Bayesian kriging and splines performed best overall. |
Assessing different interpolation methods to define the most suitable technique for the McArthur Forest Fire Danger Index (FFDI). | IDW, OK, RF, RFOK, RFIDW | - Combination of methods: RFOK and RFIDW shows the most promising results (least error). | [1] |
- Fire danger index is highly related to the behaviour of climate change, and should be considered carefully. |
Investigated several techniques for depth to underground water in northwest China. | IDW, RBF, OK, SK, UK | - Simple kriging is the optimal method in terms of result consistency and the smallest prediction interval of 95%. | [10] |
- Depth of underground water increases significantly over the year because of excessive exploitation. | |
Ranking spatial interpolation techniques using GIS-based DSS. | Spline, IDW, Kriging, TP, TS | - No optimal technique that can accurately predict the rainfall. | [11] |
- Performance of each technique depends on the scale of the input data. |
- Kriging is the recommended technique, as it produces the most consistent results. |
Using an interpolation method to construct a comprehensive archive of Australian climate data. | TPS, OK | - Different climate variables can be more accurately interpolated using different techniques due to the characteristic variability. | [3] |
Table 2.
Abbreviation list of the interpolation techniques used in
Table 1.
Table 2.
Abbreviation list of the interpolation techniques used in Table 1.
Title | Abbreviations |
---|
IDW | Inverse Distance Weighting |
AIDW | Angular IDW |
GIDW | Gradient IDW |
SK | Simple Kriging |
OK | Ordinary Kriging |
BK | Bayesian Kriging |
UK | Universal Kriging |
CAR | Clustering Assisted Regression |
TPS | Thin Plate Splines |
RBF | Radial Basis Function |
RF | Random Forest |
RFIDW | RF + IDW |
RFOK | RF + OK |
TS | Trend Surface |
TP | Thiessen Polygon |
Table 3.
Total normalised time elapsed for each method (0: fastest, 1: slowest).
Table 3.
Total normalised time elapsed for each method (0: fastest, 1: slowest).
Number of Observed Points | Total Time Elapsed |
---|
Basic IDW | Improved IDW | TIN | Kriging |
---|
10 | 0.0024 | 0.0252 | 0.0289 | 0.3034 |
30 | 0.0096 | 0.0278 | 0.0281 | 0.3852 |
50 | 0.0214 | 0.0268 | 0.0291 | 0.4706 |
70 | 0.0355 | 0.0253 | 0.0291 | 0.6582 |
90 | 0.0447 | 0.0040 | 0.0286 | 0.7808 |
110 | 0.0545 | 0.0283 | 0.0288 | 0.8633 |
130 | 0.0596 | 0.0322 | 0.0283 | 0.9720 |
150 | 0.0596 | 0.0322 | 0.0283 | 0.9720 |
Table 4.
Statistical error (RMSE and MAE with its corresponding SE) for each method. In this graph, we only show one ‘IDW’, because the errors of both versions of IDW (basic and improved, based on
Table 3) are the same; the only difference between the two being the processing time.
Table 4.
Statistical error (RMSE and MAE with its corresponding SE) for each method. In this graph, we only show one ‘IDW’, because the errors of both versions of IDW (basic and improved, based on Table 3) are the same; the only difference between the two being the processing time.
Number of Observed Points | RMSE | MAE ± SE |
---|
IDW | TIN | Kriging | IDW | TIN | Kriging |
---|
10 | 2.2461 | 2.4767 | 2.1257 | 1.83 ± 0.13 | 1.89 ± 0.16 | 1.60 ± 0.14 |
30 | 1.3667 | 1.4503 | 1.2720 | 1.09 ± 0.08 | 1.03 ± 0.10 | 0.95 ± 0.08 |
50 | 1.4897 | 1.6080 | 1.2718 | 1.14 ± 0.10 | 1.11 ± 0.12 | 0.91 ± 0.09 |
70 | 1.2259 | 1.2755 | 0.9727 | 0.97 ± 0.07 | 0.90 ± 0.09 | 0.71 ± 0.07 |
90 | 0.9985 | 1.2243 | 0.9517 | 0.73 ± 0.07 | 0.85 ± 0.09 | 0.68 ± 0.07 |
110 | 1.0565 | 1.2378 | 1.0640 | 0.77 ± 0.07 | 0.85 ± 0.09 | 0.75 ± 0.07 |
130 | 0.9645 | 1.2533 | 0.9136 | 0.70 ± 0.07 | 0.84 ± 0.09 | 0.64 ± 0.07 |
150 | 0.9146 | 1.1537 | 0.8168 | 0.65 ± 0.06 | 0.75 ± 0.09 | 0.58 ± 0.06 |
Table 5.
Summary of the strengths and weaknesses of the spatial interpolation methods evaluated in this paper: ordinary kriging (OK), inverse distance weighting (IDW) and triangle irregular network (TIN). These evaluations are based on three criteria: objective performance (RMSE), computational efficiency and the final visual output smoothness.
Table 5.
Summary of the strengths and weaknesses of the spatial interpolation methods evaluated in this paper: ordinary kriging (OK), inverse distance weighting (IDW) and triangle irregular network (TIN). These evaluations are based on three criteria: objective performance (RMSE), computational efficiency and the final visual output smoothness.
| Advantages | Disadvantages |
---|
OK | - -
Geostatistical method that considers spatial pattern. - -
Best objective comparison overall. - -
Creates very smooth surface.
| - -
Computationally very expensive. - -
Unsuitable for very large datasets.
|
IDW | - -
Computationally inexpensive. - -
Can be improved using kd-tree to increase computational efficiency. - -
Creates smooth surface.
| - -
Produces “bull’s eye effec” for extreme values. - -
Unsuitable for large scale problems without using kd-tree algorithm.
|
TIN | - -
Computationally inexpensive.
| - -
Produces abrupt surfaces.
|