The Kriging interpolation technique is being increasingly used in geostatistics these days. But how does Kriging work to create a prediction, after all?

To start with, Kriging is a method where the distance and direction between the sample data points indicate a spatial correlation. This correlation is then used to explain the different variations in the surface.

In cases where the distance and direction give appropriate spatial correlation, Kriging will be able to predict surface variations in the most effective way. As such, we often see Kriging being used in Geology and Soil Sciences.

Kriging generates an optimal output surface for prediction which it estimates based on a scattered set with z-values. The procedure involves investigating the z-values’ spatial behavior in an ‘interactive’ manner where advanced statistical relationships are measured (autocorrelation).

Mathematically speaking, Kriging is somewhat similar to regression analysis and its whole idea is to predict the unknown value of a function at a given point by calculating the weighted average of all known functional values in the neighborhood. To get the output value for a location, we take the weighted sum of already measured values in the surrounding (all the points that we intend to consider around a specific radius), using a formula such as the following:

In a regression equation, *λ***_{i}** would represent the weights of how far the points are from the prediction location. However, in Kriging,

*λ***represent not just the weights of how far the measured points are from prediction location, but also how the measured points are arranged spatially around the prediction location. First, the variograms and covariance functions are generated to create the spatial autocorrelation of data. Then, that data is used to make predictions.**

_{i}Thus, unlike the deterministic interpolation techniques like Inverse Distance Weighted (IDW) and Spline interpolation tools, Kriging goes beyond just estimating a prediction surface. Here, it brings an element of certainty in that prediction surface.

That is why experts rate kriging so highly for a strong prediction. Instead of a weather report forecasting a 2 mm rain on a certain Saturday, Kriging also tells you what is the “probability” of a 2 mm rain on that Saturday.

We hope you enjoy this simple R tutorial on Kriging by Berry Boessenkool.

### Geostatistics: Kriging – spatial interpolation between points, using semivariance

We will be covering following sections in our tutorial with supported illustrations:

- Packages
**read**shapefile- Variogram
- Kriging
- Plotting

#### Kriging: packages

```
install.packages("rgeos")
install.packages("sf")
install.packages("geoR")
library(sf) # for st_read (read shapefiles),
# st_centroid, st_area, st_union
library(geoR) # as.geodata, variog, variofit,
# krige.control, krige.conv, legend.krige
## Warning: package ’sf’ was built under R version 3.4.1
```

#### Kriging: read shapefile / few points for demonstration

```
x <- c(1,1,2,2,3,3,3,4,4,5,6,6,6)
y <- c(4,7,3,6,2,4,6,2,6,5,1,5,7)
z <- c(5,9,2,6,3,5,9,4,8,8,3,6,7)
plot(x,y, pch="+", cex=z/4)
```

#### Kriging: read shapefile II

```
GEODATA <- as.geodata(cbind(x,y,z))
plot(GEODATA)
```

#### Kriging: Variogram I

```
EMP_VARIOGRAM <- variog(GEODATA)
## variog: computing omnidirectional variogram
FIT_VARIOGRAM <- variofit(EMP_VARIOGRAM)
## variofit: covariance model used is matern
## variofit: weights used: npairs
## variofit: minimisation function used: optim
## Warning in variofit(EMP_VARIOGRAM): initial values not provided - running the default search
## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "9.19" "3.65" "0" "0.5"
## status "est" "est" "est" "fix"
## loss value: 401.578968904954
```

#### Kriging: Variogram II

```
plot(EMP_VARIOGRAM)
lines(FIT_VARIOGRAM)
```

#### Kriging: Kriging

```
res <- 0.1
grid <- expand.grid(seq(min(x),max(x),res),
seq(min(y),max(y),res))
krico <- krige.control(type.krige="OK",
obj.model=FIT_VARIOGRAM)
krobj <- krige.conv(GEODATA,
locations=grid, krige=krico)
## krige.conv: model with constant mean
## krige.conv: Kriging performed using global neighbourhood
# KRigingObjekt
```

#### Kriging: Plotting I

```
image(krobj, col=rainbow2(100))
legend.krige(col=rainbow2(100),
x.leg=c(6.2,6.7), y.leg=c(2,6),
vert=T, off=-0.5,
values=krobj$predict)
contour(krobj, add=T)
colPoints(x,y,z, col=rainbow2(100), legend=F)
points(x,y)
```

#### Kriging: Plotting II

```
library("berryFunctions") # scatterpoints by color
colPoints(x,y,z, add=F, cex=2, legargs=list(y1=0.8,y2=1))
```

#### Kriging: Plotting III

```
colPoints(grid[ ,1], grid[ ,2], krobj$predict, add=F,
cex=2, col2=NA, legargs=list(y1=0.8,y2=1))
```

#### Time for a real dataset

Precipitation from ca 250 gauges in Brandenburg as Thiessen Polygons with steep gradients at edges:

#### Exercise 41: Kriging

- Load and plot the shapefile in PrecBrandenburg.zip with sf::st_read.
- With colPoints in the package berryFunctions, add the precipitation values at the centroids of the polygons.
- Calculate the variogram and fit a semivariance curve.
- Perform kriging on a grid with a useful resolution (keep in mind that computing time rises exponentially with grid size).
- Plot the interpolated values with image or an equivalent (Rclick 4.15)
- and add contour lines.
- What went wrong? (if you used the defaults, the result will be dissatisfying.) How can you fix it?

#### Solution for exercise 41.1-2: Kriging Data

```
# Shapefile:
p <- sf::st_read("data/PrecBrandenburg/niederschlag.shp",
quiet=TRUE)
# Plot prep
pcol <- colorRampPalette(c("red","yellow","blue"))(50)
clss <- berryFunctions::classify(p$P1, breaks=50)$index
# Plot
par(mar = c(0,0,1.2,0))
plot(p, col=pcol[clss], max.plot=1) # P1: Precipitation
# kriging coordinates
cent <- sf::st_centroid(p)
berryFunctions::colPoints(cent$x, cent$y, p$P1, add=T, cex=0.7,
legargs=list(y1=0.8,y2=1), col=pcol)
points(cent$x, cent$y, cex=0.7)
```

#### Solution for exercise 41.3: Variogram

```
library(geoR)
# Semivariance:
geoprec <- as.geodata(cbind(cent$x,cent$y,p$P1))
vario <- variog(geoprec, max.dist=130000)
## variog: computing omnidirectional variogram
fit <-variofit(vario)
## Warning in variofit(vario): initial values not provided - running the default search
## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "1326.72" "19999.93" "0" "0.5"
## status "est" "est" "est" "fix"
## loss value: 107266266.76371
plot(vario) ; lines(fit)
# distance to closest other point:
d <- sapply(1:nrow(cent), function(i) min(berryFunctions::distance(
cent$x[i], cent$y[i], cent$x[-i], cent$y[-i])))
hist(d/1000, breaks=20, main="distance to closest gauge [km]")
mean(d) # 8 km
## [1] 8165.633
```

#### Solution for exercise 41.4-5: Kriging

```
# Kriging:
res <- 1000 # 1 km, since stations are 8 km apart on average
grid <- expand.grid(seq(min(cent$x),max(cent$x),res),
seq(min(cent$y),max(cent$y),res))
krico <- krige.control(type.krige="OK", obj.model=fit)
krobj <- krige.conv(geoprec, locations=grid, krige=krico)
## krige.conv: model with constant mean
## krige.conv: Kriging performed using global neighbourhood
# Set values outside of Brandenburg to NA:
grid_sf <- sf::st_as_sf(grid, coords=1:2, crs=sf::st_crs(p))
isinp <- sapply(sf::st_within(grid_sf, p), length) > 0
krobj2 <- krobj
krobj2$predict[!isinp] <- NA
```

#### Solution for exercise 41.5: Kriging Visualization

```
geoR:::image.kriging(krobj2, col=pcol)
colPoints(cent$x, cent$y, p$P1, col=pcol, zlab="Prec", cex=0.7,
legargs=list(y1=0.1,y2=0.8, x1=0.78, x2=0.87, horiz=F))
plot(p, add=T, col=NA, border=8)#; points(cent$x,cent$y, cex=0.7)
```