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04_interpolation.qmd
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04_interpolation.qmd
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# Interpolation
Products which enter hydrological analysis like temperatures or precipitation are measured **locally at point locations**. Here we will show one well-known method to interpolate across fairly homogeneous space and one method for interpolation across more tedious and complex domain like a river network.
## Inverse distance weighting (IDW)
$$
Z_p = \dfrac{\sum\limits_{i=1}^{n}\dfrac{z_i}{d_i^P}}{\sum\limits_{i=1}^{n}\dfrac{1}{d_i^P}}
$$
Let's create an arbitrary spatial domain
```{r, fig.align='center'}
library(sf) # Spatial Feature library for spatial data
library(scico) # Scientific pallette collection called "scico"
n <- 25
dom <- data.frame(x = runif(n, 0, 50),
y = runif(n, 0, 50),
value = rnorm(n)) |>
sf::st_as_sf(coords = c("x", "y"), crs = 4326)
plot(dom,
nbreaks = 26,
pal = scico(n = 25,
palette = "davos",
end = 0.9,
direction = -1),
main = "Precipitation [mm]",
pch = 20, # Point character selection
graticule = TRUE, # Display graticules
axes = TRUE, # Display plot axes
bgc = "#f0f0f033", # Background color
key.pos = 1) # Legend position
```
We need a grid of new points, which will represent the raster on which we want to recalculate
```{r, fig.align='center', fig.cap='Point location of Precipitation measuring stations'}
grid <- st_make_grid(x = dom,
cellsize = 2,
crs = 4326) |>
st_sf()
plot(x = dom,
nbreaks = 26,
pal = scico(n = 25,
palette = "davos",
end = 0.9,
direction = -1),
main = "Precipitation [mm]",
pch = 20, # Point character selection
graticule = TRUE, # Display graticules
axes = TRUE, # Display plot axes
bgc = "#f0f0f033", # Background color
key.pos = 1, reset = FALSE) # Legend position
plot(x = st_geometry(grid), lwd = 0.1, add = TRUE, reset = TRUE)
```
Now we compute the distances between the points
```{r, eval=TRUE, fig.align='center', fig.cap=''}
distances <- outer(
st_coordinates(grid)[, "X"], st_coordinates(dom)[, "X"],
st_coordinates(grid)[, "Y"], st_coordinates(dom)[, "Y"],
FUN = function(x1, x2, y1, y2) sqrt((x1 - x2)^2 + (y1 - y2)^2)
)
```
```{r, eval=TRUE, fig.align='center', fig.ext='svg'}
power <- 2
weighted_sum <- apply(distances, 1, function(d) sum(dom$value / (d^power)))
weights_sum <- apply(distances, 1, function(d) sum(1 / (d^power)))
interpolated_values <- weighted_sum / weights_sum
results <- data.frame(
x = st_coordinates(grid)[, "X"],
y = st_coordinates(grid)[, "Y"],
value = interpolated_values) |>
sf::st_as_sf(coords = c("x", "y"), crs = 4326)
results$value[which(is.nan(results$value))] <- NA
library(ggplot2)
ggplot() +
geom_sf(data = results, aes(color = value)) +
geom_sf(data = grid, fill = NA) +
theme_bw() +
scale_color_scico(palette = "davos")
ggsave("docs/grid.svg", device = "svg")
# plot(x = results,
# nbreaks = 11,
# pal = scico(n = 10,
# palette = "davos",
# end = 0.9,
# direction = -1),
# main = "Precipitation [mm] estimated by IDW",
# pch = 20, # Point character selection
# graticule = TRUE, # Display graticules
# axes = TRUE, # Display plot axes
# bgc = "#f0f0f033", # Background color
# key.pos = 1, reset = FALSE) # Legend position
# plot(x = st_geometry(grid), lwd = 0.1, add = TRUE, reset = TRUE)
#
# interpolated_matrix <- matrix(interpolated_values, ncol = length(st_coordinates(grid)[, "X"]), nrow = length(st_coordinates(grid)[, "Y"]))
#
# nrow <- length(unique(st_coordinates(results)[, "Y"]))
# ncol <- length(unique(st_coordinates(results)[, "X"]))
# dim(results)
# interpolated_matrix <- matrix(results_df$interpolated_value, nrow = nrow, ncol = ncol, byrow = TRUE)
#
# plot(interpolated_matrix[, 1], interpolated_matrix[, 2])
# contour(unique(st_coordinates(results)[, "X"]), unique(st_coordinates(results)[, "Y"]), z = results$value)
# # Plot the interpolated values as an image
# image(
# x = na.omit(st_coordinates(grid)[, "X"]),
# y = na.omit(st_coordinates(grid)[, "Y"]),
# z = interpolated_matrix,
# col = heat.colors(100), # You can choose a color palette
# main = "Interpolated Values"
# )
```
## `gstat`
There are many libraries listed in CRAN **geostatistics** task view. One of these is called gstat, it was developed and is maintained by Edzer Pebesma, who is also behind the `raster` and `terra` packages.
```{r, eval=FALSE}
library(gstat)
gstat::idw(precipitation ~ x + y, locations = ~x+y)
```
```{r, eval=FALSE}
power <- 2 # You can adjust the power parameter
weighted_sum <- apply(distances, 1, function(d) sum(df$value / (d^power)))
weighted_sum
```
## Krigging
Another interpolation technique is called **Krigging**.
### Data
Let's use some artificial data set for this example as well.
$$
\mathrm{Distances} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}\qquad \mathrm{for}\:i, j = 1,2, \ldots,n
$$
$$
\mathrm{Differences} = \Delta Z_{i,j} = Z_i - Z_j\qquad \mathrm{for}
$$ \### Semivariogram calculation
$$
\gamma(h) = \dfrac{1}{2N(h)}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\Delta Z_{ij}(h)^2
$$ where $\gamma(h)$ represents the semivariance lag at distance $h$.
```{r, eval=FALSE}
df <- data.frame(
x = runif(10, 0, 50),
y = runif(10, 0, 50),
value = rnorm(10)
)
n <- nrow(df)
dist_matrix <- as.matrix(dist(cbind(df$x, df$y)))
differences_matrix <- matrix(0, n, n)
for (i in 1:n) {
differences_matrix[i, ] <- df$value - df$value[i]
}
lag_tol <- 1 # Lag tolerance, you can adjust this
max_lag <- 10 # Maximum lag distance, you can adjust this
semivariance <- rep(0, max_lag / lag_tol)
num_pairs <- rep(0, max_lag / lag_tol)
for (i in 1:(n - 1)) {
for (j in (i + 1):n) {
distance_ij <- sqrt((df$x[i] - df$x[j])^2 + (df$y[i] - df$y[j])^2)
lag <- round(distance_ij / lag_tol)
if (lag <= max_lag / lag_tol) {
semivariance[lag] <- semivariance[lag] + (df$value[i] - df$value[j])^2
num_pairs[lag] <- num_pairs[lag] + 1
}
}
}
semivariance <- semivariance / (2 * num_pairs)
prediction_grid <- expand.grid(
x = seq(0, 50, by = 1),
y = seq(0, 50, by = 1)
)
n_grid <- nrow(prediction_grid)
semivariance_at_grid <- rep(0, n_grid)
for (i in 1:n_grid) {
differences_to_data <- df$value - prediction_grid[i, ]
distances_to_data <- sqrt((df$x - prediction_grid[i, "x"])^2 + (df$y - prediction_grid[i, "y"])^2)
lag_values <- round(distances_to_data / lag_tol)
semivariance_at_grid[i] <- sum((differences_to_data)^2 / (2 * lag_values))
}
n_lags <- length(semivariance)
kriged_values <- rep(0, n_grid)
for (i in 1:n_grid) {
kriging_weights <- rep(0, n)
distances_to_data <- sqrt((df$x - prediction_grid[i, "x"])^2 + (df$y - prediction_grid[i, "y"])^2)
lag_values <- round(distances_to_data / lag_tol)
for (j in 1:n) {
if (lag_values[j] <= n_lags) {
kriging_weights[j] <- (semivariance[lag_values[j]] - semivariance_at_grid[i]) / semivariance[lag_values[j]]
}
}
# Check if there are valid weights to avoid replacement with length zero
if (any(kriging_weights != 0)) {
kriged_values[i] <- sum(kriging_weights * df$value)
}
}
```
$$
\left[
\begin{matrix}
\mathbf{G} & \mathbf{j}\\
\mathbf{j^\prime} & \mathbf{0}\\
\end{matrix}
\right]
\left[
\begin{matrix}
\lambda\\
\mu_L
\end{matrix}
\right] =
\left[
\begin{matrix}
\mathbf{g}\\
1
\end{matrix}
\right]
$$ Where $\mathbf{G}$ is a rectangular matrix containing the values of variograph for all doubles of measured points, $\mathbf{1}$ is a vector of weights s with values of
### Now we have to state the varigram model. We will calculate the semivariogram
to model the **spatial covariance** structure. Then calculate the pairwise distances and differences between data points.
```{r, eval=FALSE}
n <- nrow(df)
dist_matrix <- as.matrix(dist(cbind(df$x, df$y)))
differences_matrix <- matrix(0, n, n)
for (i in 1:n) {
differences_matrix[i, ] <- df$value - df$value[i]
}
```