Interpolación espacial
Daniel Diaz Vargas
2025-11-13
Interpolación espacial
1. Introducción
Este cuaderno ilustra dos técnicas de interpolación espacial: la ponderación por distancia inversa (IDW) y el kriging ordinario (OK). IDW es una técnica determinista, mientras que OK es probabilística. Ambas técnicas se utilizan aquí para obtener una superficie continua de carbono orgánico del suelo (COS) a 15-30 cm de profundidad a partir de muestras obtenidas de SoilGrids de 250 m.
Cabe mencionar que se proporciona una explicación básica del código para facilitar su comprensión y replicación. Sin embargo, los estudiantes de Geomática Básica deben: (i) ampliar la documentación actual; y (ii) interpretar y analizar los resultados del código. Esto les ayudará a mejorar sus habilidades de programación y a demostrar que no se limitan a copiar y pegar.
2. Configuración
Limpiar el espacio de trabajo:
# Uncomment at first-time run
rm(list=ls())Para leer y escribir datos geoespaciales, usaremos terra (para ráster) y sf (para vectores). Para la interpolación espacial, usaremos las bibliotecas gstat y automap. Para la visualización, usaremos ggplot, leaflet y tmap.
Primero, debemos instalar las bibliotecas necesarias. Hágalo desde la consola de R, no desde este cuaderno.
library(terra)## Warning: package 'terra' was built under R version 4.5.2## terra 1.8.80library(sf)## Warning: package 'sf' was built under R version 4.5.2## Linking to GEOS 3.13.1, GDAL 3.11.4, PROJ 9.7.0; sf_use_s2() is TRUElibrary(sp)## Warning: package 'sp' was built under R version 4.5.2library(stars)## Warning: package 'stars' was built under R version 4.5.2## Cargando paquete requerido: abind## Warning: package 'abind' was built under R version 4.5.2library(gstat)## Warning: package 'gstat' was built under R version 4.5.2library(automap)## Warning: package 'automap' was built under R version 4.5.2library(RColorBrewer)## Warning: package 'RColorBrewer' was built under R version 4.5.2library(leaflet)## Warning: package 'leaflet' was built under R version 4.5.2library(leafem)## Warning: package 'leafem' was built under R version 4.5.23. Leer los datos de entrada
Como se indicó en la introducción, nuestras muestras son valores de carbono orgánico del suelo (COS) a una profundidad de 15-30 cm. Se obtuvieron de SoilGrids 250 m utilizando este cuaderno. Guardamos dichas muestras en formato geopackage, en el directorio de GB022/Proyecto05, con el nombre de archivo soc_huila.gpkg.
Primero, verifiquemos que los datos estén disponibles para su uso:
list.files(path="/GB022/Proyecto05", pattern = "soc_huila.gpkg")## [1] "soc_huila.gpkg"Luego, leemos las muestras usando sf:
samples <- sf::st_read("/GB022/Proyecto05/soc_huila.gpkg")## Reading layer `soc_huila' from data source `D:\GB022\Proyecto05\soc_huila.gpkg' using driver `GPKG'
## Simple feature collection with 1937 features and 1 field
## Geometry type: POINT
## Dimension: XY
## Bounding box: xmin: -76.64869 ymin: 1.495825 xmax: -74.50588 ymax: 3.807744
## Geodetic CRS: +proj=longlat +datum=WGS84 +no_defsDado que nuestra área de estudio es Huila, tiene sentido leer también el geopaquete de los municipios correspondientes:
munic <- sf::st_read("/GB022/Proyecto05/munic_huila.gpkg")## Reading layer `col_adm2' from data source `D:\GB022\Proyecto05\munic_huila.gpkg' using driver `GPKG'
## Simple feature collection with 37 features and 11 fields
## Geometry type: MULTIPOLYGON
## Dimension: XY
## Bounding box: xmin: -76.6402 ymin: 1.493471 xmax: -74.5054 ymax: 3.807901
## Geodetic CRS: WGS 844. Explorar los datos de entrada
Obtengamos un resumen de las muestras:
summary(samples)## soc geom
## Min. : 6.981 POINT :1937
## 1st Qu.:15.654 epsg:NA : 0
## Median :23.864 +proj=long...: 0
## Mean :27.322
## 3rd Qu.:37.829
## Max. :75.587generaremos un resumen estadístico básico de los datos contenidos en el objeto samples. Esta función proporciona una visión general de la distribución de los valores para cada variable en el dataset, mostrando para variables numéricas como soc (carbono orgánico del suelo) los valores mínimo, máximo, media, mediana y los cuartiles (1º y 3º).
hist(samples$soc)
Redondeemos los valores SOC a dos dígitos:
samples$soc = round(samples$soc,2)Ahora, visualicemos las muestras para conocer su distribución espacial.
pal <- colorNumeric(
c("#E1F5C4", "#EDE574", "#F9D423", "#FC913A", "#FF4E50"),
# colors depend on the count variable
domain = samples$soc,
)visualizaremos un mapa interactivo utilizando la biblioteca Leaflet para graficar la distribución espacial de las muestras de carbono orgánico del suelo (SOC) en el departamento de Huila. Primero, define una paleta de colores secuencial (pal) que asigna diferentes tonos (desde verde claro hasta rojo intenso) a los valores de SOC, creando una escala visual que facilita la interpretación de los datos. Luego, construye el mapa superponiendo varias capas: los polígonos municipiales en color gris con transparencia como base contextual, los puntos de muestreo representados como círculos marcadores coloreados según su valor de SOC, una leyenda explicativa de la escala de colores, y finalmente un fondo de mapa base de OpenStreetMap para referencia geográfica.
# This a simple visualization
# Change the code to suit your preferences
leaflet() %>%
addPolygons(
data = munic,
color = "gray",
# set the opacity of the outline
opacity = 1,
# set the stroke width in pixels
weight = 1,
# set the fill opacity
fillOpacity = 0.2) %>%
addCircleMarkers(
data = samples,
radius= 1.5,
label = ~soc,
color = ~pal(soc),
fillOpacity = 1,
stroke = F
) %>%
addLegend(
data = samples,
pal = pal,
values = ~soc,
position = "bottomleft",
title = "SOC:",
opacity = 0.9) %>%
addProviderTiles("OpenStreetMap")5. Interpolación
##5.1 Creación del objeto gstat
Para interpolar, primero necesitamos crear un objeto de la clase gstat, utilizando la función del mismo nombre: gstat.
Un objeto gstat contiene toda la información necesaria para realizar la interpolación espacial, a saber:
La definición del modelo
Los datos de calibración
Según sus argumentos, la función gstat determina qué tipo de modelo de interpolación queremos usar:
Sin modelo de variograma → Distancia de ponderación inversa (IDW)
Modelo de variograma, sin covariables → Kriging ordinario (OK)
Utilizaremos tres parámetros de la función gstat:
formula: la fórmula de predicción que especifica las variables dependientes e independientes (también conocidas como covariables)
data: los datos de calibración (también conocidos como datos de entrenamiento)
model: el modelo de variograma
EL MODELO DE LA FUNCION
#LA CALIBRACION DE LA INFORMACION
BASADO EN ESTOS ARGUMENTOS LA FUNCIONN gstat NO ENTIENDE QUE TIPO DE INTERPOLACION QUIERE USSAR
SIN MODELO DE VARIOGRAMA ->IVERSO A LA DISTANCIA (IDW) MODELO DE VARIOGRAMA ->ORIDINARY KRIGING (OK)
5.2 EMPEZAMOS CON LA INTERPOLACION IDM
en funcion de la muestra vamos a usar los datos variables.
g1 = gstat(formula = soc ~ 1, data = samples)cuando vamos a interpolar vamos a usar la funcion llamada predict, que le pasa un raster que es donde mete los datos en una cuadricula, si el no tiene cuadricula el no sabe a donde debe ingresarlos.
y el modelo es el objeto gstat
el raster le dice donde debe meter esos datos
DEM <- rast("D:/GB022/Proyecto05/Depa Huila.tif")DEM## class : SpatRaster
## size : 276, 255, 1 (nrow, ncol, nlyr)
## resolution : 0.008333333, 0.008333333 (x, y)
## extent : -76.63333, -74.50833, 1.5, 3.8 (xmin, xmax, ymin, ymax)
## coord. ref. : lon/lat WGS 84 (EPSG:4326)
## source : Depa Huila.tif
## name : Depa HuilaCAMBIAR DATOS POR EL DEM
(rrr <- terra::rast(xmin=-76.63333, xmax=-74.50833, ymin=1.5, ymax=3.8,
resolution = 0.01, vals = 1, crs = "EPSG:4326"))## class : SpatRaster
## size : 230, 213, 1 (nrow, ncol, nlyr)
## resolution : 0.01, 0.01 (x, y)
## extent : -76.63333, -74.50333, 1.5, 3.8 (xmin, xmax, ymin, ymax)
## coord. ref. : lon/lat WGS 84 (EPSG:4326)
## source(s) : memory
## name : lyr.1
## min value : 1
## max value : 1Nota: Si su ordenador tiene recursos limitados (p. ej., poca memoria), utilice una resolución espacial menor (p. ej., 0,008 o 0,01 grados).
Ahora, necesitamos convertir este SpatRaster en un objeto de estrellas:
stars.rrr <- stars::st_as_stars(rrr)ahora interpolamos con IDM
## [inverse distance weighted interpolation]
## running this code takes several minutes
z1 = predict(g1, stars.rrr,nmax = 20)## [inverse distance weighted interpolation]Que es z1?
z1## stars object with 2 dimensions and 2 attributes
## attribute(s):
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## var1.pred 7.090062 19.80855 25.22767 26.97109 33.38493 73.91371 0
## var1.var NA NA NA NaN NA NA 48990
## dimension(s):
## from to offset delta refsys x/y
## x 1 213 -76.63 0.01 WGS 84 [x]
## y 1 230 3.8 -0.01 WGS 84 [y]la varianza es Var1.var es determinista y estima que no tiene error.
realizar una corrección sobre los valores resultantes de la interpolación IDW, específicamente identificando y reemplazando todos los valores NA (no disponibles) en la capa de predicción (z1[[1]]) por el valor 0.0. Esto se logra primero detectando las posiciones de los valores NA usando la función which(is.na(z1[[1]])) y almacenándolas en el vector a, para luego asignar el valor cero a esas ubicaciones específicas en la cuadrícula. Esta acción es necesaria porque algunas celdas de la cuadrícula pueden no tener puntos de muestreo cercanos suficientes para calcular un valor interpolado, generando NAs que podrían causar problemas en análisis posteriores o visualizaciones, asegurando así que toda el área de estudio tenga valores definidos.
# dealing with NA values
a <- which(is.na(z1[[1]]))
z1[[1]][a] = 0.0z1## stars object with 2 dimensions and 2 attributes
## attribute(s):
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## var1.pred 7.090062 19.80855 25.22767 26.97109 33.38493 73.91371 0
## var1.var NA NA NA NaN NA NA 48990
## dimension(s):
## from to offset delta refsys x/y
## x 1 213 -76.63 0.01 WGS 84 [x]
## y 1 230 3.8 -0.01 WGS 84 [y]asignamos un nombre específico a la capa de datos dentro del objeto z1 (que contiene los resultados de la interpolación IDW), cambiando su nombre original (probablemente “Var1.pred”) a “soc”. Esto es importante para mejorar la legibilidad y el contexto de los datos, ya que “soc” (Soil Organic Carbon) identifica claramente el contenido de la capa como los valores interpolados de carbono orgánico del suelo, facilitando su interpretación y uso posterior en análisis, visualizaciones o exportación de resultados.
names(z1) = "soc"Para trazar la superficie interpolada, necesitamos una paleta de colores:
# change this chunk to meet your preferences
paleta <- colorNumeric(palette = c("blue", "yellow", "cyan", "green"), domain = 10:120, na.color = "transparent")Ahora, visualicemos la salida de la interpolación:
m <- leaflet() %>%
addTiles() %>%
leafem:::addGeoRaster(
z1,
opacity = 0.7,
colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"),
domain = 10:100)
) %>%
addCircleMarkers(
data = samples,
radius= 1.5,
label = ~soc,
color = ~paleta(soc),
fillOpacity = 1,
stroke = F
) %>%
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## Warning in paleta(soc): Some values were outside the color scale and will be
## treated as NA
## Warning in paleta(soc): Some values were outside the color scale and will be
## treated as NA## Warning in write_stars.stars(x, dsn = fl): all but first attribute are ignoredm # Print the map5.3 OK Interpolación
Los métodos de kriging requieren un modelo de variograma. Este modelo permite cuantificar objetivamente el patrón de autocorrelación en los datos y asignar ponderaciones al realizar predicciones.
Como primer paso, podemos calcular y examinar el variograma empírico utilizando la función de variograma.
La función requiere dos argumentos:
fórmula: especifica la variable dependiente y las covariables, al igual que en gstat
datos: la capa de puntos con la variable dependiente y las covariables como atributos de punto
La siguiente expresión calcula el variograma empírico de las muestras, sin covariables:
v_emp_ok = variogram(soc ~ 1, data=samples)Grafiquemos el variograma empírico:
El código plot(v_emp_ok) genera una visualización gráfica del variograma empírico (v_emp_ok) que fue calculado previamente para el análisis geoestadístico. Este gráfico muestra la relación entre la distancia espacial entre puntos de muestreo y la semivarianza de los valores de carbono orgánico del suelo, permitiendo evaluar la estructura de autocorrelación espacial de los datos. La visualización ayuda a identificar patrones como el rango (distancia donde la autocorrelación se disipa), la meseta (nivel máximo de variación) y el efecto pepita (variación a distancia cero), lo cual es fundamental para calibrar el modelo de variograma que se utilizará en la interpolación por kriging ordinario.
plot(v_emp_ok)
Existen varias maneras de ajustar un modelo de variograma a un variograma empírico. Aquí, utilizaremos la más sencilla: el ajuste automático mediante la función autofitVariogram del paquete automap.
El código v_mod_ok = automap::autofitVariogram(soc ~ 1, as(samples, “Spatial”)) realiza el ajuste automático de un modelo de variograma teórico al variograma empírico calculado a partir de los datos de muestras de carbono orgánico del suelo. La función autofitVariogram toma como entrada la fórmula soc ~ 1 (indicando que se modela el SOC como variable dependiente sin covariables) y convierte los datos samples a formato Spatial para su procesamiento. Automáticamente prueba múltiples modelos de variograma (como esférico, exponencial o gaussiano) y selecciona el que mejor se ajusta a la estructura espacial observada, estimando parámetros clave como el rango, la meseta y el efecto pepita. Este modelo de variograma ajustado (v_mod_ok) es esencial para realizar la interpolación mediante kriging ordinario, ya que cuantifica matemáticamente la autocorrelación espacial de los datos.
#make sure you understand the parameters needed to run this fitting function
v_mod_ok = automap::autofitVariogram(soc ~ 1, as(samples, "Spatial"))La función selecciona el tipo de modelo que mejor se ajusta y ajusta sus parámetros. Puede usar show.vgms() para mostrar varios tipos de modelos de variograma.
Tenga en cuenta que la función autofitVariogram no funciona con objetos sf, por lo que convertimos el objeto samples en un SpatialPointsDataFrame (paquete sp).
El modelo ajustado se puede representar gráficamente con plot:
plot(v_mod_ok)
El modelo obtenido mediante autofitVariogram es un variograma esférico, el cual representa matemáticamente la estructura de autocorrelación espacial del carbono orgánico del suelo (SOC) en el departamento de Huila. Este modelo esférico se caracteriza por mostrar un aumento progresivo de la semivarianza a medida que aumenta la distancia entre puntos, hasta alcanzar un umbral conocido como “rango”, donde la semivarianza se estabiliza en su valor máximo o “meseta”. El gráfico resultante de plot(v_mod_ok) evidencia este comportamiento mediante puntos que representan el variograma empírico y una línea roja continua que traza el modelo esférico ajustado, permitiendo visualizar claramente la distancia hasta la cual los valores de SOC mantienen correlación espacial y el nivel de variabilidad que presentan a diferentes escalas de distancia.
El objeto resultante es en realidad una lista con varios componentes, incluyendo el variograma empírico y el modelo de variograma ajustado. El componente $var_model del objeto resultante contiene el modelo real:
v_mod_ok$var_model## model psill range
## 1 Nug 23.1327 0.00000
## 2 Sph 161.6777 69.87124Ahora, el modelo de variograma se puede pasar a la función gstat y podemos continuar con la interpolación de Kriging ordinario:
## [using ordinary kriging]
## This takes several minutes
## (or even ages depending on your raster cell size and your computer resources)
g2 = gstat(formula = soc ~ 1, model = v_mod_ok$var_model, data = samples)
z2= predict(g2, stars.rrr)## [using ordinary kriging]Nuevamente, subconjunto el atributo de valores predichos y le cambiaremos el nombre:
a <- which(is.na(z2[[1]]))
z2[[1]][a] = 0.0
names(z2) = "soc"Las predicciones OK se muestran en el siguiente gráfico:
m <- leaflet() %>%
addTiles() %>%
leafem:::addGeoRaster(
z2,
opacity = 0.7,
colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"),
domain = 10:100)
) %>%
addCircleMarkers(
data = samples,
radius= 1.5,
label = ~soc,
color = ~paleta(soc),
fillOpacity = 1,
stroke = F
) %>%
addLegend("bottomright", pal = paleta, values= z2$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## Warning in paleta(soc): Some values were outside the color scale and will be
## treated as NA
## Warning in paleta(soc): Some values were outside the color scale and will be
## treated as NA## Warning in write_stars.stars(x, dsn = fl): all but first attribute are ignoredm # Print the map#6. Evaluación de resultados
#6.1 Evaluación cualitativa
colores <- colorOptions(palette = c("grey", "yellow", "cyan", "green"), domain = 10:100, na.color = "transparent")m <- leaflet() %>%
addTiles() %>%
addGeoRaster(z1, colorOptions = colores, opacity = 0.8, group= "IDW") %>%
addGeoRaster(z2, colorOptions = colores, opacity = 0.8, group= "OK") %>%
# Add layers controls
addLayersControl(
overlayGroups = c("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## Warning in write_stars.stars(x, dsn = fl): all but first attribute are ignored
## Warning in write_stars.stars(x, dsn = fl): all but first attribute are ignoredm # Print the map#6.2 Validación cruzada Hemos estimado las superficies SOC utilizando dos métodos diferentes: IDW y Kriging ordinario. Si bien es útil examinar y comparar los resultados gráficamente, también necesitamos una forma objetiva de evaluar la precisión de la interpolación. De esta manera, podemos elegir objetivamente el método más preciso entre los métodos de interpolación disponibles.
En la validación cruzada de “dejar uno fuera”:
Extraemos un punto de los datos de calibración. Hacemos una predicción para ese punto. Repetimos la operación para todos los puntos. Finalmente, obtenemos una tabla con un valor observado y un valor predicho para todos los puntos. Podemos ejecutar la validación cruzada de “dejar uno fuera” utilizando la función gstat.cv, que acepta un objeto gstat.
Al escribir el siguiente fragmento, oculte el mensaje y los resultados.
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## this code should be run from the console not from here
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## [using ordinary kriging]The result of gstat.cv has the following attributes:
var1.pred—Predicted value
var1.var—Variance (only for Kriging)
observed—Observed value
residual—Observed-Predicted
zscore—Z-score (only for Kriging)
fold—Cross-validation ID
Let’s convert the cv2 object:
# run from the console
cv2 = st_as_sf(cv2)Let’s plot the residuals:
# run from the console
sp::bubble(as(cv2[, "residual"], "Spatial"))
# VERIFICAR NOMBRES DE COLUMNAS - EJECUTA ESTO PRIMERO
names(cv2)## [1] "var1.pred" "var1.var" "observed" "residual" "zscore" "fold"
## [7] "geometry"La comparación objetiva mediante validación cruzada revela que el kriging ordinario (OK) es más confiable que la ponderación por distancia inversa (IDW) para interpolar el carbono orgánico del suelo en Huila, ya que presenta un menor error cuadrático medio (RMSE) debido a que incorpora la estructura de autocorrelación espacial modelada mediante el variograma esférico, optimizando estadísticamente los pesos de predicción. Además, el gráfico de burbujas de residuos del OK muestra una distribución más aleatoria y menos agrupamiento espacial de errores, confirmando que captura mejor la variabilidad espacial de los datos y produce estimaciones más precisas en ubicaciones no muestreadas en comparación con el enfoque determinista de IDW que solo considera la distancia a los puntos conocidos.
7. Guardar las salidas de interpolación
write_stars(
z1, dsn = "/GB022/Proyecto05/IDW_soc_huila.tif", layer = 1
)write_stars(
z2, dsn = "/GB022/Proyecto05/OK_soc_huila.tif", layer = 1
)8. Conclusiones
La ejecución de este cuaderno permitió consolidar el entendimiento de los fundamentos teórico-prácticos de la interpolación espacial, evidenciando la superioridad del kriging ordinario sobre métodos deterministas como IDW. Mediante el análisis comparativo, se verificó que el kriging, al incorporar la estructura de dependencia espacial a través del modelado variográfico, genera superficies predictivas con mayor precisión (menor RMSE) y proporciona además una cuantificación robusta de la incertidumbre asociada. La implementación del flujo de trabajo completo - desde la caracterización exploratoria de datos hasta la validación cruzada - demostró la criticalidad de seleccionar algoritmos que consideren la autocorrelación espacial cuando se trabaja con variables ambientales como el carbono orgánico del suelo. Este ejercicio destacó cómo la geoestadística trasciende la simple interpolación para convertirse en una herramienta de modelado espacial estadísticamente fundamentada.
9. Referencia
Diaz, D. 2025. Interpolación espacial. Disponible en:
sessionInfo()## R version 4.5.1 (2025-06-13 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 10 x64 (build 19045)
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## [1] LC_COLLATE=Spanish_Colombia.utf8 LC_CTYPE=Spanish_Colombia.utf8
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## time zone: America/Bogota
## tzcode source: internal
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## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] leafem_0.2.5 leaflet_2.2.3 RColorBrewer_1.1-3 automap_1.1-20
## [5] gstat_2.1-4 stars_0.6-8 abind_1.4-8 sp_2.2-0
## [9] sf_1.0-21 terra_1.8-80
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## [1] generics_0.1.4 sass_0.4.10 class_7.3-23
## [4] KernSmooth_2.23-26 lattice_0.22-7 digest_0.6.37
## [7] magrittr_2.0.4 evaluate_1.0.5 grid_4.5.1
## [10] fastmap_1.2.0 plyr_1.8.9 jsonlite_2.0.0
## [13] e1071_1.7-16 reshape_0.8.10 DBI_1.2.3
## [16] crosstalk_1.2.2 scales_1.4.0 codetools_0.2-20
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## [22] units_1.0-0 intervals_0.15.5 base64enc_0.1-3
## [25] cachem_1.1.0 yaml_2.3.10 FNN_1.1.4.1
## [28] raster_3.6-32 tools_4.5.1 parallel_4.5.1
## [31] dplyr_1.1.4 ggplot2_4.0.0 spacetime_1.3-3
## [34] 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
## [40] classInt_0.4-11 leaflet.providers_2.0.0 htmlwidgets_1.6.4
## [43] pkgconfig_2.0.3 pillar_1.11.1 bslib_0.9.0
## [46] gtable_0.3.6 glue_1.8.0 Rcpp_1.1.0
## [49] tidyselect_1.2.1 tibble_3.3.0 xfun_0.53
## [52] rstudioapi_0.17.1 knitr_1.50 farver_2.1.2
## [55] htmltools_0.5.8.1 rmarkdown_2.30 xts_0.14.1
## [58] compiler_4.5.1 S7_0.2.0