1. Introducción

En este cuaderno se emplean dos métodos de interpolación espacial, con el objetivo de estimar el contenido de carbono orgánico del suelo (SOC) en el departamento de Boyacá a una profundidad de 15-30 cm, esto partiendo de datos obtenidos de SoilGrids 250m. El primer método, determinista, que se usará es el Inverse Distance Weighting (IDW) y el segundo, probabilístico, es el Ordinary Kriging (OK).

2. Configuración

Primero limpiamos la zona de trabajo:

rm(list=ls())

Luego, Instalamos y cargamos las librerías requeridas:

library(sp)
library(terra)
## terra 1.8.54
library(sf)
## Linking to GEOS 3.13.1, GDAL 3.11.0, PROJ 9.6.0; sf_use_s2() is TRUE
library(stars)
## Cargando paquete requerido: abind
library(gstat)
library(automap)
library(leaflet)
library(leafem)
library(ggplot2)
library(dplyr)
## 
## Adjuntando el paquete: 'dplyr'
## The following objects are masked from 'package:terra':
## 
##     intersect, union
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(curl)
## Using libcurl 8.14.1 with Schannel
h <- new_handle()
handle_setopt(h, http_version = 2)

3. Leer datos de entrada

Con ayuda de la librería terra, leemos nuestro archivo ráster que contiene los datos de SOC del departamento, para tener referencia del mundo real:

archivo <- ("Datos/soc_igh_15_30.tif")
(soc <- rast(archivo))
## class       : SpatRaster 
## size        : 1069, 1173, 1  (nrow, ncol, nlyr)
## resolution  : 250, 250  (x, y)
## extent      : -8326000, -8032750, 518250, 785500  (xmin, xmax, ymin, ymax)
## coord. ref. : Interrupted_Goode_Homolosine 
## source      : soc_igh_15_30.tif 
## name        : soc_igh_15_30 
## min value   :            89 
## max value   :          1344

Ahora, para convertir los datos a porcentaje, se debe tener presente que las unidades de SOC son g/kg, es decir, gramos de carbono orgánico por kilogramo de suelo seco:

soc.perc <-  soc/10

Luego, transformamos el Sistema de Referencia de Coordenadas (CRS) del archivo, ahora llamado soc.perc, para pasar de la proyección Homolosine al CRS WGS84:

geog ="+proj=longlat +datum=WGS84"
(geog.soc = project(soc.perc, geog))
## class       : SpatRaster 
## size        : 1091, 1256, 1  (nrow, ncol, nlyr)
## resolution  : 0.002199529, 0.002199529  (x, y)
## extent      : -74.71029, -71.94768, 4.65658, 7.056267  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : soc_igh_15_30 
## min value   :      8.874964 
## max value   :    293.181030

Después, procedemos a convertir la capa SpatRaster a un objeto stars:

stars.soc = st_as_stars(geog.soc)

Una vez hecho eso, es momento de realizar un gráfico interactivo para representar los datos de SOC. Esto se ejecuta usando las librerías leaflet y leafem:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      stars.soc,
      opacity = 0.8,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 8:130)
    ) 
m

4. Muestreo de datos del mundo

Para este apartado vamos a realizar un muestreo de 500 sitios de manera aleatoria a partir de los datos de la capa ráster geog.soc:

set.seed(123456)

(samples <- spatSample(geog.soc, 500, "random", as.points=TRUE))
##  class       : SpatVector 
##  geometry    : points 
##  dimensions  : 500, 1  (geometries, attributes)
##  extent      : -74.6938, -71.95758, 4.65768, 7.050768  (xmin, xmax, ymin, ymax)
##  coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
##  names       : soc_igh_15_30
##  type        :         <num>
##  values      :         20.35
##                        18.83
##                        32.55

En el resultado se observa que se generó un conjunto de 500 puntos aleatorios (SpatVector), sobre la capa ráster de SOC. Los puntos están distribuidos entre las longitudes -74.69° y -71.95°, y entre las latitudes 4.65° y 7.05°. Su sistema de coordenadas es WGS84, que corresponde al sistema geodésico de coordenadas geográficas usado mundialmente.

A cada punto se le extrajo el valor de SOC (soc_igh_15_30), cuyos datos numéricos representan la concentración de carbono orgánico del suelo. El data frame muestra que los primeros valores fueron 20.35, 18.83 y 32.55 g/kg.

Retomando, vamos a convertir el objeto SpatVector en un objeto de simple feature:

(muestras <- sf::st_as_sf(samples))

Ahora eliminamos los datos vacíos o faltantes y los datos completos se guardan en nmuestras:

nmuestras <- na.omit(muestras)

Extraemos los datos de coordenadas (latitud y longitud) y los valores de SOC:

longit <- st_coordinates(muestras)[,1]
latit <- st_coordinates(muestras)[,2]
soc <- muestras$soc_igh_15_30

Se identifica individualmente cada punto:

id <- seq(1,500,1)

Por último, creamos un data frame con los resultados obtenidos de los pasos anteriores:

(sitios <- data.frame(id, longit, latit, soc))

Como se observa, hay datos NA que debemos eliminar:

sitios <- na.omit(sitios)
head(sitios)

Teniendo los datos completos, realizamos una gráfica con Leaflet que muestra la capa raster de carbono orgánico del suelo, y sobre ella los puntos muestreados (sitios):

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      stars.soc,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 8:130)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions())
m

5. Interpolación

Para realizar la interpolación, debemos crear un objeto de la gstat, usando la función del mismo nombre: gstat. Un objeto de esta clase brinda la información necesaria para la interpolación, es decir, la definición del modelo y los datos de calibración.

La función gstat consiste en tres parámetros: fórmula, datos y modelo.

5.1 Interpolación IDW

Para estimar el carbono orgánico del suelo (SOC) en Boyacá, se aplica un método de interpolación espacial conocido como Inverse Distance Weighting (IDW). Con la función gstat vamos a construir un modelo de interpolación a partir de los valores de muestreo para generar una superficie continua de predicción.

Primero creamos el modelo especificando los parámetros del gstat:

g1 = gstat(formula = soc_igh_15_30 ~ 1, data = nmuestras)

El contenido de carbono orgánico depende de sí mismo, lo que se representa como ~1.

Ahora, una vez definido el modelo de interpolación g1, podemos aplicar la función predict() para estimar valores en ubicaciones sin datos. Esta función requiere un modelo (gstat) y un raster que actúe como base para las predicciones. El raster cumple dos funciones principales: define los lugares donde se desea predecir y, en métodos como el Kriging Universal, también proporciona las covariables necesarias.

Agrupamos las celdas en bloques, es decir, por cada 4*4 hay una celda, y luego creamos un ráster con valores de celda igual a 1:

rrr = aggregate(geog.soc, 4)
rrr
## class       : SpatRaster 
## size        : 273, 314, 1  (nrow, ncol, nlyr)
## resolution  : 0.008798118, 0.008798118  (x, y)
## extent      : -74.71029, -71.94768, 4.65438, 7.056267  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : soc_igh_15_30 
## min value   :       9.67114 
## max value   :     147.83141

Definimos los nuevos valores:

values(rrr) <-1

Definimos los nuevos nombres:

names(rrr) <- "valor"

Revisamos nuevamente el objeto rrr:

rrr
## class       : SpatRaster 
## size        : 273, 314, 1  (nrow, ncol, nlyr)
## resolution  : 0.008798118, 0.008798118  (x, y)
## extent      : -74.71029, -71.94768, 4.65438, 7.056267  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : valor 
## min value   :     1 
## max value   :     1

Ahora convertimos el ráster rrr a formato stars, para usarlo más adelante con la librería leaflet:

stars.rrr = st_as_stars(rrr)

La predicción se realiza con la función predict(), que utiliza como entrada el modelo (g1) y un raster (convertido a stars) que define el área de análisis. Este raster se genera con la librería terra, a partir de las coordenadas que delimitan la región de estudio en Boyacá. Creamos Z1 para guardar la interpolación:

z1 = predict(g1, stars.rrr)
## [inverse distance weighted interpolation]

Revisamos el objeto z1:

z1
## stars object with 2 dimensions and 2 attributes
## attribute(s):
##                Min.  1st Qu.   Median     Mean  3rd Qu.    Max.  NA's
## var1.pred  10.81947 31.83749 37.69393 38.04114 44.39646 122.633     0
## var1.var         NA       NA       NA      NaN       NA      NA 85722
## dimension(s):
##   from  to offset     delta                       refsys x/y
## x    1 314 -74.71  0.008798 +proj=longlat +datum=WGS8... [x]
## y    1 273  7.056 -0.008798 +proj=longlat +datum=WGS8... [y]

Extraemos solo la predicción del SOC (var1.pred) del objeto z1 y luego la renombramos como “soc”:

z1 = z1["var1.pred",,]
names(z1) = "soc"

Definimos una nueva paleta de colores:

paleta <- colorNumeric(palette = c("orange", "yellow", "cyan", "green"), domain = 10:100, na.color = "transparent")

A continuación, creamos la gráfica para representar el ráster SOC interpolado, utilizando IDW:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      z1,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 11:55)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions()) %>%
    addLegend("bottomright", pal=paleta, values= z1$soc,
    title = "IDW SOC interpolation [%]"
    )
## Warning in pal(c(r[1], cuts, r[2])): Some values were outside the color scale
## and will be treated as NA
m

5.2 Interpolación OK

En los métodos de kriging, es necesario construir un modelo de variograma. Este modelo permite cuantificar la autocorrelación espacial de los datos y asignar pesos adecuados en las predicciones. Como primer paso, calculamos el variograma empírico mediante la función variogram(), la cual analiza la dependencia espacial en los datos.

Esta función requiere una fórmula que defina la variable a modelar y de ser el caso, sus covariables; así como una capa de puntos que contenga la capa de puntos con la variable dependiente y las covariables como atributos de puntos.

v_emp_ok = variogram(soc_igh_15_30 ~ 1, data=nmuestras)

Ahora vamos a graficar el variograma:

plot(v_emp_ok)

#### Existen diferentes formas de ajustar un modelo de variograma al variograma empírico. En este caso, emplearemos la opción más simple: el ajuste automático utilizando la función autofitVariogram del paquete automap, el cual permite escoger el modelo de variograma que mejor se ajuste a los datos.

v_mod_ok = autofitVariogram(soc_igh_15_30 ~ 1, as(nmuestras, "Spatial"))

Graficamos el resultado con el modelo seleccionado:

plot(v_mod_ok)

#### El objeto que devuelve autofitVariogram() es una lista con varios elementos. El componente $var_model contiene el modelo ajustado que se necesita para aplicar kriging posteriormente.

v_mod_ok$var_model

- Nugget: Tiene un valor de 0, lo que indica que no hay variabilidad a escala muy pequeña o no hay error de medición significativo en los datos.

- Ste: Hace referencia al tipo de modelo utilizado para ajustar la curva.

- psill del modelo principal (Ste) es 326.9771, lo que representa la varianza espacial explicada por el modelo.

- Range es 101.73, lo que significa que hasta una distancia de ~101.7 unidades (probablemente km o metros) los datos muestran autocorrelación. Más allá de esa distancia, el valor en un punto no influye sobre otro.

- Kappa con valor 0.3 solo es relevante si el modelo es del tipo Matérn o similar. Si “Ste” representa un modelo esférico (Sph), entonces kappa no tiene un efecto directo.

Ahora, Creamos el modelo de OK usando la variable de interés (soc_igh_15_30), el modelo de variograma ajustado (v_mod_ok$var_model) y los puntos de muestreo (nmuestras) y realiza la predicción espacial sobre un raster definido:

g2 = gstat(formula = soc_igh_15_30 ~ 1, model = v_mod_ok$var_model, data = nmuestras)
z2= predict(g2, stars.rrr)
## [using ordinary kriging]

Luego, seleccionamos solo la capa de valores del resultado del Kriging (“var1.pred”) y renombramos la capa seleccionada como “soc”:

z2 = z2["var1.pred",,]
names(z2) = "soc"

Prosegimos a realizar un mapa que presenta la predicción de carbono orgánico del suelo (SOC) generado mediante Ordinary Kriging:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      z2,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 11:55)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions()) %>%
    addLegend("bottomright", pal = paleta, values= z2$soc,
    title = "OK SOC interpolation [%]"
    )
## Warning in pal(c(r[1], cuts, r[2])): Some values were outside the color scale
## and will be treated as NA
m

6. Evaluación de resultados

6.1 Evaluación cualitativa

A continución, creamos un mapa interactivo con Leaflet que permite comparar visualmente tres capas:

- RealWorld: valores reales medidos de carbono orgánico del suelo (SOC),

- IDW: predicción mediante Interpolación por distancias inversas (IDW),

- OK: predicción mediante Ordinary Kriging (OK).

Para ello, definimos una paleta de colores para representar los valores de SOC y se agregan las capas como grupos superpuestos. Gracias al control de capas, el usuario puede alternar entre las distintas interpolaciones y observar las diferencias:

colores <- colorOptions(palette = c("orange", "yellow", "cyan", "green"), domain = 10:100, na.color = "transparent")
m <- leaflet() %>%
  addTiles() %>%  
  addGeoRaster(stars.soc, opacity = 0.8, colorOptions = colores, group="RealWorld") %>%
  addGeoRaster(z1, colorOptions = colores, opacity = 0.8, group= "IDW")  %>%
  addGeoRaster(z2, colorOptions = colores, opacity = 0.8, group= "OK")  %>%
  # Add layers controls
  addLayersControl(
    overlayGroups = c("RealWorld", "IDW", "OK"),
    options = layersControlOptions(collapsed = FALSE)
  ) %>% 
    addLegend("bottomright", pal = paleta, values= z1$soc,
    title = "Soil organic carbon [%]"
)
## Warning in pal(c(r[1], cuts, r[2])): Some values were outside the color scale
## and will be treated as NA
m

6.2 Evaluación cuantitativa

Para comparar objetivamente los métodos de interpolación (IDW y Kriging Ordinario), se aplica la validación cruzada Leave-One-Out (LOOCV), que consiste en predecir cada punto excluyéndolo del conjunto de datos, generando así una tabla de valores observados vs. predichos. Esto permite calcular errores y evaluar la precisión de cada método:

cv1 = gstat.cv(g1)
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cv1 = na.omit(cv1)
cv1
## class       : SpatialPointsDataFrame 
## features    : 478 
## extent      : -74.6652, -71.99277, 4.65768, 7.050768  (xmin, xmax, ymin, ymax)
## crs         : +proj=longlat +datum=WGS84 +no_defs 
## variables   : 6
## names       :        var1.pred, var1.var,         observed,          residual, zscore, fold 
## min values  : 15.2321562984189,       NA, 10.7943916320801, -22.7095095025061,     NA,    1 
## max values  :  68.784040479395,       NA, 122.860206604004,  81.8311333713511,     NA,  478
cv1 = st_as_sf(cv1)
sp::bubble(as(cv1[, "residual"], "Spatial"))

sqrt(sum((cv1$var1.pred - cv1$observed)^2) / nrow(cv1))
## [1] 10.6635

Evaluamos la calidad de las predicciones del método IDW usando Leave-One-Out Cross Validation. Podemos isualizar los errores espacialmente (residuos) y cuantificamos el error general (RMSE = 10.66).

El siguiente paso es hacer lo mismo con OK y comparar los resultados:

cv2 = gstat.cv(g2)
## [using ordinary kriging]
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cv2 = st_as_sf(cv2)
sqrt(sum((cv2$var1.pred - cv2$observed)^2) / nrow(cv2))
## [1] 9.890342

Según los resultados obtenidos mediante la validación cruzada, el método de OK mostró un mejor desempeño que el método de IDW, al presentar un menor error cuadrático medio (9.89 frente a 10.66). Esto sugiere que el Kriging Ordinario ofrece estimaciones más cercanas a los valores reales. La razón de su mayor precisión radica en que este método tiene en cuenta la dependencia espacial entre los datos, a través del variograma, lo cual mejora la calidad de las predicciones. En consecuencia, el Ordinary Kriging resulta ser el método de interpolación más adecuado para este caso.

7. Referencias

Lizarazo, I., 2023. Spatial interpolation of soil organic carbon. Available at https://rpubs.com/ials2un/soc_interp.

sessionInfo()
## R version 4.5.1 (2025-06-13 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 11 x64 (build 26100)
## 
## Matrix products: default
##   LAPACK version 3.12.1
## 
## locale:
## [1] LC_COLLATE=Spanish_Colombia.utf8  LC_CTYPE=Spanish_Colombia.utf8   
## [3] LC_MONETARY=Spanish_Colombia.utf8 LC_NUMERIC=C                     
## [5] LC_TIME=Spanish_Colombia.utf8    
## 
## time zone: America/Bogota
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] curl_6.4.0     dplyr_1.1.4    ggplot2_3.5.2  leafem_0.2.4   leaflet_2.2.2 
##  [6] automap_1.1-20 gstat_2.1-4    stars_0.6-8    abind_1.4-8    sf_1.0-21     
## [11] terra_1.8-54   sp_2.2-0      
## 
## loaded via a namespace (and not attached):
##  [1] generics_0.1.4     sass_0.4.10        class_7.3-23       KernSmooth_2.23-26
##  [5] lattice_0.22-7     digest_0.6.37      magrittr_2.0.3     evaluate_1.0.4    
##  [9] grid_4.5.1         RColorBrewer_1.1-3 fastmap_1.2.0      plyr_1.8.9        
## [13] jsonlite_2.0.0     e1071_1.7-16       reshape_0.8.10     DBI_1.2.3         
## [17] crosstalk_1.2.1    scales_1.4.0       codetools_0.2-20   jquerylib_0.1.4   
## [21] cli_3.6.5          rlang_1.1.6        units_0.8-7        intervals_0.15.5  
## [25] withr_3.0.2        base64enc_0.1-3    cachem_1.1.0       yaml_2.3.10       
## [29] FNN_1.1.4.1        raster_3.6-32      tools_4.5.1        parallel_4.5.1    
## [33] spacetime_1.3-3    png_0.1-8          vctrs_0.6.5        R6_2.6.1          
## [37] zoo_1.8-14         proxy_0.4-27       lifecycle_1.0.4    classInt_0.4-11   
## [41] htmlwidgets_1.6.4  pkgconfig_2.0.3    pillar_1.11.0      bslib_0.9.0       
## [45] gtable_0.3.6       glue_1.8.0         Rcpp_1.1.0         tidyselect_1.2.1  
## [49] tibble_3.3.0       xfun_0.52          rstudioapi_0.17.1  knitr_1.50        
## [53] farver_2.1.2       htmltools_0.5.8.1  rmarkdown_2.29     xts_0.14.1        
## [57] compiler_4.5.1