What is the difference between kriging and idw

The purpose of this chapter is to demonstrate a comprehensive approach to evaluate spatial interpolation, including: common quantitative assessment, 2D visualization, and 3D visualization.

This chapter also presents a special case, a closed system consisting of three variables. Spatial interpolation techniques were applied to the three variables separately and independently to create surfaces. This chapter is organized as follows: Section 2—study area and data, Section 3—spatial interpolation methods, Section 4—quantitative assessments, Section 5—2D and 3D visualization, Section 6—special case of a closed system, and Section 7—conclusions. The study area is a Soils within the field were mapped as mollisols.

Soil samples were collected in January using five soils sampling schemes outlined in a previous study [ 15 ]: 0. The soil pH value of the sample points from the 0. The five soil cores were composited to form the sample for each respective grid sample location. Soil pH was as determined using the standard laboratory method of the United States Department of Agriculture [ 16 ].

Study area: R. Spatial interpolation, or spatial prediction, is a process to estimate values of locations that were not surveyed based on a network of points with known values [ 1 , 2 , 10 , 11 ].

In most cases, the input data is a network of points, while the output is a surface that divides the study area into small cells with a data value for each cell. There are two basic assumptions for spatial interpolation. The second assumption is that values are smooth and continuous over space. Many spatial interpolation techniques were developed based on these two assumptions.

Commercial GIS or statistical software provides several spatial interpolation functions, such as inverse distance weighted IDW , kriging, spline, and others. Even with the same technique and same input point data, different parameters may result in different surfaces. Potentially, a given set of points and a given spatial interpolation technique can generate many different surfaces [ 10 , 14 ]. Therefore, it is important to evaluate and understand the accuracy and reliability of surface data generated from spatial interpolation.

In this study, IDW, kriging, and spline will be used to demonstrate the process to evaluate and visualize spatial interpolation surfaces. Inverse distance weighted is a deterministic estimation method where values at unmeasured points are determined by a linear combination of values at nearby measured points.

Among available parameters, the power parameter can significantly affect the results. As the power parameter increases, IDW acts similarly to the nearest neighbor interpolation method in which the interpolated value is close to the value of the nearest measured value. The advantages of IDW are that it is simple, easy to understand, and efficient.

Disadvantages are that it is sensitive to outliers and there is no indication of error [ 1 ]. Schloeder et al. They concluded that IDW and kriging performed similarly and that both are more accurate than the spline interpolation method. Mueller et al. Though individual performance differed greatly depending on the existence of spatial structure and sampling density, they concluded little difference between the overall performances between IDW and kriging.

Kravchenko [ 20 ] conducted another study to compare IDW and kriging on soil properties. He reported that spatial structure significantly affected the accuracy of interpolation performance. He also reported that known variograms can greatly improve kriging performance, which may result in a better performance than IDW. Lu and Wong [ 21 ] developed a new form of IDW, which estimated data values at an unsampled location based on spatial pattern found in its neighborhood.

As already reported in Refs. Their new form of IDW may perform better than kriging without variograms. Kriging is a stochastic method similar to IDW in that it also uses a linear combination of weights at known locations to estimate the data value of an unknown location. Variogram is an important input in kriging interpolation.

It is a measure of spatial correlation between two points. With known variograms, weights can change according to the spatial arrangement of the samples. A major advantage of kriging is that, in addition to the estimated surface, kriging also provides a measure of error or uncertainty of the estimated surface.

A disadvantage is that it requires substantially more computing time and more input from users, compared to IDW and spline [ 1 ]. Bekele et al. They found that kriging generally performed better than IDW. Laslett et al. Gotway et al.

Bishop and McBratney [ 25 ] conducted a study to explore the effect of having secondary data such as color aerial photos in the interpolation process. They reported an improved kriging performance. It can be imagined as fitting a flexible surface through a set of known points using a mathematical function. A major advantage of spline is that it can create fairly accurate and visually appealing surfaces based on only a few sample points.

Disadvantages of spline are that the resultant surface may have different minimum and maximum values from the input data set, it is sensitive to outliers, and there is no indication of errors [ 1 ]. They reported though each method may perform better than others under certain situations, overall spline and kriging performed relatively better than IDW.

Voltz and Webster [ 27 ] compared kriging and spline on soil properties, and concluded that kriging performed overall better than spline. Robinson and Metternicht [ 28 ] compared spline, kriging, and IDW interpolations methods on soil properties.

They reported that no single method was suitable for all situations. Simpson and Wu [ 29 ] compared IDW, kriging, and spline on interpolating lake depth, and reported that spline produced the most accurate results with less than the ideal amount of sampled points. Based on a previous study [ 14 ], six interpolated surfaces were chosen for demonstration purposes.

They are IDW parameters: power 2, 10 neighbors , spline parameters: tension, 10 neighbors , kriging parameters: circular, 10 neighbors , IDW parameters: power 4, 20 neighbors , spline parameters: thin plate, 20 neighbors , and kriging parameters: exponential, 20 neighbors. Each surface was evaluated by cross validation Jackkniffing by the points from the 0. This validation process will go through iterations till all points were processed and validated.

In each iteration, one sample point with known data value was discarded, and the remaining sample points were used to predict the value at the location of the discarded point. The known data values were compared to their counterpart predicted values and a measure of prediction accuracy was calculated.

Four error measures were used as accuracy index [ 14 ]. They are 1 mean absolute error MAE , see Eq. Readings from the accuracy index, the lower values mean less errors, and therefore, higher accuracies and better performances. Table 1 summarizes these four error measures for these six interpolated surfaces. At first glance, they are quite compatible with each, meaning a similar performance. With closer examinations, one may notice that spline parameter: thin plate, 20 neighbors seems to have higher error measures, meaning more errors, and therefore worse performance.

This particular interpolation has 0. Among these four error measures, spline parameter: thin plate, 20 neighbors interpolation has considerably higher values than the other surfaces in three measures. On the other hand, IDW and kriging seem to perform similarly with compatible error measures.

Figure 2 shows these six interpolated surfaces in a flat 2D visualization environment. With visual inspection, one may notice that among these three surfaces with 10 neighbors, kriging parameter: circular, 10 neighbors appears differently.

One may describe it as smoother with less extreme values because of less red colors and blue colors. On the other hand, IDW parameter: power 2, 10 neighbors and spline parameter: tension, 10 neighbors seem to appear similarly.

The same observation can be made in the group of three surfaces with 20 neighbors. Kriging parameter: exponential, 20 neighbors appears smoother than other two surfaces.

IDW parameter: power 4, 20 neighbors and spline parameter: thin plate, 20 neighbors seem to appear similarly. Comparison between the group of 10 neighbors and the group of 20 neighbors, one may observe another interesting trend that the group of 20 neighbors generally appears to have more extreme values, with more red colors and blue colors, than the group of 10 neighbors.

One of the specificity of kriging methods is that they do not only consider the distance between observations but they also intend to capture the spatial structure in the data by comparing observations separated by specific spatial distances two at a time.

The objective is to understand the relationships between observations separated by different lag distances. All this knowledge is accounted for in the variogram. Kriging methods then derive spatial weights for the observations based on this variogram.

It must be noted that kriging techniques will preserve the values of the initial samples in the interpolated map. Kriging methods consider that the process that originated the data can be divided into two major components : a deterministic trend large scale variations and an autocorrelated error the residuals.

Where Z s is the attribute value at the spatial position s , m is the deterministic trend that does not depend on the location s of the observations and e s is the autocorrelated error term which depends on the spatial position s. Note that the trend is deterministic! The variogram is only computed on the residuals which are supposed to exhibit a spatial correlation! Basically, when someone intends to fit a variogram model to the data, it actually tries to fit this model to the residuals of the data, after the trend is removed.

Be aware that, depending on the shape of the variogram, the resulting interpolated map can be heavily smoothed. In fact, if the nugget effect of the variogram is strong, kriging techniques will tend to smooth the data to blur the noise within the fields.

It is sometimes surprising because the range of values after kriging can be smaller than that of the samples. If the nugget component is effectively very high regarding the partial sill, the spatial structure might be relatively poor and it would be better not to interpolate.

Many variants of kriging depends on the way the trend is characterized. This will be discussed later. Thanks to the computation of the variogram, an error map can be derived on the entire field because the relationships between any two observations separated by a distance h is known.

The error map is a very useful tool because it make access the to the prediction accuracy of the interpolated map. Be aware that the quality of the error map obviously relies on the quality of the model fitting to the variogram. If a variogram model fits poorly to the data, the error map might be questionable. Be aware that the interpolation by any kriging techniques is sensitive to outliers. Here, abnormal observations might mask the data autocorrelation and prevent the spatial structure from being determinated.

The intuitive solution would be to remove these outlying values before kriging. However, it can happen that some values look like they are abnormal because they are much higher or lower than the remaining values in the dataset.

If these observations are clustered, this so-called outliers might be due to a real phenomenom. In order to account for this specificity and not deleting abruptly these outliers, one solution would be to compute the variogram without these observations but perform kriging with them. As such, the spatial structure would still be defined and these extreme observations would be kept inside the dataset.

As the name suggests, simple kriging is the most simple kriging variant. In this case, the deterministic trend, m , is known and considered constant over the whole field under study Fig. Figure 4. Simple kriging and corresponding trend and residuals. This method is global because it does not account for local variations of the trend.

However, if there is no abrupt changes of the variable of interest or if there is no reason there would be one , this assumption can be viable and relevant. As the trend is considered known, the autocorrelated terms are also known which makes the prediction more simple.

It must be understood that, here, predicted values cannot extend beyond the range of the sample values. This technique is the most reported kriging approach. Contrary to simple kriging, this method considers that the trend is constant but only within a local neighbourhood Fig. This assumption is interesting because it ensures to account for the local variations within the field. One could imagine for instance a break in slope within a field that could be interesting to consider.

Here the trend depends on the spatial location of the observation m s. This constant trend is assumed unknown here and has to be derived from the data in the according neighbourhood. Figure 5. Ordinary kriging and corresponding estimated trend and residuals. As such, there is a need to define the extent of the neighbourood : the number of neighbouring observations that will be taken into account for each prediction. As the trend is considered unknown, it may happen that, by construction, the range of the predicted values extend beyond the range of the sample values.

Regression kriging has multiple names : universal kriging or kriging with an external trend. It is similar to ordinary kriging in the way that it considers that the determinitic trend is not constant over the whole field but depends on the spatial location of the observation.

However, here, the trend is modelled by a more complex function, it is not simply considered constant over a neighbourhood Fig. The objective is the same as before : detrend the data so that the autocorrelated residuals can be studied. At the end of the interpolation, the trend is added back to the interpolated residuals. Figure 6. The effect of the inverse distance weights have can often be determined by the user by changing the power that the inverse distance is raised to.

As seen in this diagram you can determine the limits of which data points z values IDW should take into consideration using a search radius. IDW differs from Kriging in that no statistical models are used.

There is no determination of spatial autocorrelation taken into consideration that is to say how correlated variables are at varying distances is not determined. In IDW only known z values and distance weights are used to determine unknown areas. IDW has the advantage that it is easy to define and therefore easy to understand the results. It may be inadvisable to use Kriging if you are unsure of how the results were arrived at. Kriging also suffers when there are outliers see here for an explanation.

ESRI states :. Kriging is most appropriate when you know there is a spatially correlated distance or directional bias in the data. It is often used in soil science and geology. Kriging is a statistical method that makes use of a variograms to calculate the spatial autocorrelation between points at graduated distances A nice introduction can be found here Statios Variogram Introduction and Washington Intro to Variograms.

It uses this calculation of spatial autocorrelation to determine the weights that should be applied at various distances. Spatial autocorrelation is determined by taking squared differences between points. To clarify Kriging is similar to IDW in that:. Like IDW interpolation, kriging forms weights from surrounding measured values to predict unmeasured locations. As with IDW interpolation, the measured values closest to the unmeasured locations have the most influence.

There are various different niche types of Kriging. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn more. Ask Question.