Interpolación de datos de precipitación para Caqueta

1. Introducción

Este es un cuaderno realizado en RStudio, basando en el cuaderno de Interpolacion informe 3, que tiene como objetivo ayudar a los estudiantes de Geomática Básica de la Universidad Nacional de Colombia a familiarizarse con las funcionalidades geoespaciales proporcionadas por el entorno del software R. En este cuaderno se describen conceptos básicos de interpolación espacial y además se aplican 3 técnicas de interpolación a un conjunto de datos de precipitación que cubren el departamento de Caqueta.

La interpolación sirve para predecir valores para las celdas de un ráster a partir de una cantidad limitada de puntos de datos de muestra. Esta herramienta se puede utilizar para prever valores desconocidos de cualquier dato de un punto geográfico como por ejemplo la elevación y precipitación. (Toro, G. A., & Melo. 2009)

Las herramientas de interpolación de superficie crean una superficie continua (o predicción) a partir de valores de punto de muestra. Lo anterior se hace viable cuando se hace la suposición de que los objetos distribuidos espacialmente están correlacionados espacialmente, o sea, las cosas que están cerca, tienden a parecerce. Así, de los puntos cercanos a los puntos de muestra, tienen más posibilidades de ser similares que los puntos más alejados (Portalés, C., Boronat-Zarceño. 2008)

Para esto, existen diferentes maneras de derivar una predicción para cada ubicación; cada método se denomina modelo. Con cada modelo, se realizan diferentes suposiciones acerca de los datos, y algunos modelos son más aplicables para determinados datos y cada uno produce predicciones usando diferentes cálculos. (Toro, G. A., & Melo. 2009)

En el presente trabajo, se realizarán 3 métodos de interpolación los cuales se describen a continuación:

Tabla 1. Herramientas de interpolación que se aplicarán a datos de precipitación

Descripción	Herramienta de Interpolación
Interpola una superficie de ráster a partir de puntos	Kriggin
Interpola una superficie de ráster a partir de puntos utilizando una técnica de distancia inversa ponderada	IDW
Se basa en considerar que los puntos de una zona coinciden con el valor medido más próximo y generan, como consecuencia, zonas de influencia siendo el resultado la unión de polígonos que generan distintas zonas temáticas	Polígonos de Thiessen

Las siguientes figuras fueron elaboradas en QGis. La figura 1 enseña la ubicación del departamento en el territorio nacional y la figura 2 muestra los diferentes municipios del departamento:

2. Descripción de la zona de estudio

El Departamento de Caqueta está ubicado al suroccidente del país y al nor-occidente de la región amazónica entre los 00º42’17’’ de latitud sur y 02º04’13" de latitud norte y los 74º18’39’’ y 79º19’35’’ de longitud occidente, representando el 7.79% del territorio colombiano. Tiene una superficie de 88 965 km². formando parte de las regiones Andina y Amazonia, limitando al norte con Meta y Guaviare, al noreste con Vaupés, al sur con Amazonas y Putumayo, y al oeste con Cauca y Huila. Con 88 965 km² es el tercer departamento más extenso —por detrás de Amazonas y Vichada (Gobernación Caqueta. 2018). Está conformado política y administrativamente por en 16 municipios: Florencia, ciudad capital, Albania, Belén de los Andaquíes, Cartagena del Chairá, Curillo, El Doncello, El Paujil, Milán, Montañita, Morelia, Puerto Rico, San José del Fragua, San Vicente del Caguán, Solano, Solita y Valparaíso; al igual que 31 caseríos, 42 inspecciones de policía y 33 centros poblados. Para el año 2016 registra 106.672 predios urbanos y 55.917 rurales (Gobernación Caqueta. 2018).

Caqueta es reconocido económicamente por ser un territorio rico en maderas, resinas, plantas medicinales y gomas. Los suelos, especialmente los que lindan con la cordillera, son fértiles, y se aprovechan con cultivos de arroz, plátano, yuca, cacao y caña de azúcar. En el pie de monte, existen yacimientos de estaño y carbón y en las riberas del Río Caquetá existen vetas de oro (DANE. 2020). Pero la principal actividad del departamento es la ganadera con cerca de millón y medio de cabezas de ganado ocupando la cuarta población ganadera del país (Gobernacion Caqueta. 2018).

Tabla 2. Actividades económicas en Caqueta

Actividad económica	Porcentaje de participación a nivel nacional (%)
Agricultura	25,37
Ganaderia	36
Industria	13,81
Turismo	12,52
Comercio	12,74

El departamento de Caquetá se encuentra en mayor proporción en el piso térmico cálido húmedo (91,3%), el resto del territorio se localiza en los pisos templado (5,6%), frío (2,8%) y muy frío a extremadamente frío (0,3%). Los vientos alisios del sureste traen la humedad de la cuenca amazónica entre junio y agosto y dan lugar a un régimen de lluvias monomodal, es decir, a una sola estación pluviosa durante todo el año (Gobernacion Caqueta. 2018).

Las principales fuentes hídricas del departamento, son los ríos Orteguaza (110 kilómetros), el Fragua Chorroso, Fragua Grande, Caguán (250 kilómetros), Guayas, Yarí, Pescado y el Apaporis. El río Caquetá tiene cerca de 2.200 kilómetros de longitud de los cuales 1.200 kilómetros transcurren por territorio Colombiano. En el Departamento del Caquetá, abarca desde la desembocadura del río Fragua Grande, en límite con el departamento del Cauca, hasta la desembocadura del río Caguán, en el extremo oriental, abarcando una longitud aproximada de 400 kilómetros.(IDEAM. 2018) El Río Caquetá constituye uno de los afluentes fluviales más importantes no solo del departamento sino también del país (Gobernacion Caqueta. 2018).

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## 
## Attaching package: 'geojson'

## The following object is masked from 'package:graphics':
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## Success! File is at ./chirps/ppoints.geojson

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Preparación adicional

En caso de que se pierda la conexión, podemos volver a leer el shapefile con el área de interés:

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## 3  BELÃ‰N DE LOS ANDAQUÃ\215ES  18094 POLYGON ((-75.78705 1.74982...
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## although coordinates are longitude/latitude, st_intersection assumes that they are planar

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montes.sf.magna <- st_transform(montes.sf, crs=3116)

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## 3. Descripción de datos y métodos

3.1 Datos

Para este reporte técnico se manejarán datos de precipitación obtenidos de la página de CHIRPS, el cual es un fondo que ha desarrollado técnicas para producir mapas de lluvia, principalmente en aquellas áreas donde los datos de superficie son escasos. La resolución de los datos es de 0,05°, estos se obtubieron con imágenes de satélite combinadas con datos de estaciones. CHIRPS tiene convenios con todos los paises y reciben diariamente información climatológica. De este modo, crean un formato raster tifs de precipitación que logran por medio de algoritmos que integran esta información, el cual es util para monitoriar precipitación o predecir tiempos de sequía (Portalés, C., Boronat-Zarceño. 2008).

Se descargó un conjunto de datos de precipitación de los últimos días de abril (del 23 al 30), en donde se tienen 2000 filas, 7200 columnas y 14400000 celdas y se encuentran en coordenadas geográficas.

Por otro lado, se necesita tambíen un shapefile que representa el departamento de Caqueta que se encuentra en coordenadas WGS84 y tiene 16 municipios, con el objetivo de asociar los datos de precipitación al área de estudio.

3.2 Métodos

Se utilizarán puntos que expresan precipitación para realizar las técnicas de interpolación, aplicados en Caqueta los cuales se muestran en la siguiente gráfica:

A continuación se abordarán las técnicas de interpolación que se utilizarán en el trabajo:

Polígonos de Thiessen

La interpolación de áreas utiliza el concepto de polígonos de Thiessen para dibujar límites en torno a las áreas que tienen el mismo valor. Este tipo de interpolacion se llama interpolacion proximal o euclidia. Esta técnica construye polígonos mediante la asignación por cada punto de un área de interés un valor del punto de muestra que sea más cercano. Crea superficies escalonadas, donde los valores pueden cambiar drásticamente en distancias cortas, es decir, un valor estimado está representado dentro del área de cada polígono. (Jimenez. j. 2018).

se representa por medio de esta ecuacion, donde un sitio “pi” está construida a partir de las intersecciones de los semiplanos formados al trazar los bisectores de “pi” hacia los sitios “pj,j≠i”. Tomando el caso donde únicamente hay dos sitios “p” y “q”, se traza el segmento de recta “pq¯” y posteriormente se traza el bisector de “pq¯”. Este bisector parte el plano en dos semiplanos, donde el semiplano que contiene a “p” (representado como “h(p,q))” representa el lugar geométrico de todos los puntos más cercanos a “p” que a “q”; y el semiplano que contiene a “q (h(q,p))” alberga a todos los puntos más próximos a “q” que a “p”. De acuerdo con esto, entonces se puede establecer de forma general cómo se define la región de Voronói para un sitio “pi”. (Jimenez. j. 2018).

\[\mathcal{V} (p_i) = \bigcap^n_{j=1,j \neq i} h(p_i,p_j)\]

IDW

La interpolación mediante distancia inversa ponderada es útil para determiar los valores de celda mediante una combinación ponderada linealmente de un conjunto de puntos de muestra. La ponderación es una función de la distancia inversa. La superficie que se interpola debe ser la de una variable dependiente de la ubicación. Así mismo, este método presupone que la variable que se representa cartográficamente disminuye su influencia a mayor distancia desde su ubicación de muestra, es decir, asume que los valores más cercanos están más relacionados que otros con su función.(Jimenez. j. 2018). IDW es un método de interpolación espacial muy flexible ya que se puede configurar la interpolación de diferentes maneras, pues se puede espicificar el radio de búsqueda y la interpolación sólo utilizará el número de puntos conocidos dentro del radio de búsqueda. También se pueden establecer barreras. Si hay crestas en un perfil de elevación, o barreras de ruido, entonces estos son ejemplos apropiados para usar una barrera. Esta barrera de polilínea impide buscar los puntos de entrada de la muestra. A continuacion el modelo matematico que lo representa (Toro, G. A., & Melo. 2009).

\[ u(x) = \begin{cases}{\frac{\sum^N_{i=1} w_i(x)u_i}{\sum^N_{i-1} w_i(x)}}, &{\text{si} \: d(x,x_i)\neq 0 \: \text{para} \: \text{todo} \: i} \\u_i, & \text{si} \: d(x,x_i)= 0 \: \text{para} \: \text{algún} \: i \end{cases}\] \[w_i(x) = \frac {1}{d(x,x_i)^p} \]

Kriging

Es una herramienta de interpolación que basicamente interpola una superficie de ráster a partir de puntos. Este es un proceso intensivo del procesador, por ende, la velocidad de ejecución depende de la calidad de puntos en el conjunto de datos o dataset (Toro, G. A., & Melo. 2009), es deicr, datos que corresponden a los contenidos de una única tabla de base de datos o una única matriz de datos de estadística, donde cada columna de la tabla representa una variable en particular, y cada fila representa a un miembro determinado del conjunto de datos que se están tratando. En un conjunto de datos o dataset tenemos todos los valores que puede tener cada una de las variables, como por ejemplo la altura y el peso de un objeto, que corresponden a cada miembro del conjunto de datos. Cada uno de estos valores se conoce con el nombre de dato. (Balagueró T. 2018).

Kriging presupone que la distancia o la dirección entre los puntos de muestra reflejan una correlación espacial que puede utilizarse para explicar la variación en la superficie. Además ajusta una función matemática a una cantidad especificada de puntos o a todos los puntos dentro de un radio específico para determinar el valor de salida para cada ubicación. (Toro, G. A., & Melo. 2009)

Pondera los valores medidos circundantes para calcular una predicción de una ubicación sin mediciones, estas ponderaciones están basadas no solo en la distancia entre los puntos medidos y la ubicación de la predicción, sino también en la disposición espacial general de los puntos medidos. Para utilizar la disposición espacial en las ponderaciones, la correlación espacial debe estar cuantificada. Por lo tanto, en un kriging ordinario, la ponderación depende de un modelo ajustado a los puntos medidos, la distancia a la ubicación de la predicción y las relaciones espaciales entre los valores medidos alrededor de la ubicación de la predicción. (Balagueró T. 2018). A continuacion el modelo matematico que lo representa.

\[ \sum^n_{j=1} \lambda_j\gamma(u_i-u_j)+ \mu = \gamma(u_i-u) \: \: \: \: \: i=1,...,n\] con \[ \sum^n_{j=1} \lambda_j = 1\] dónde λj son los pesos ponderados de cada punto con la varianza dada por

\[\sigma^2_K(u) = \sum^n_{j=1} \lambda_j\gamma(u_i-u)+ \mu\] ## 4. Presentación de resultados

4.1- Usando el método de Polígonos de Thiessen

## 
## Attaching package: 'nlme'

## The following object is masked from 'package:dplyr':
## 
##     collapse

## The following object is masked from 'package:raster':
## 
##     getData

## Registered S3 method overwritten by 'spatstat':
##   method     from
##   print.boxx cli

## 
## spatstat 1.64-1       (nickname: 'Help you I can, yes!') 
## For an introduction to spatstat, type 'beginner'

## 
## Note: spatstat version 1.64-1 is out of date by more than 7 months; we recommend upgrading to the latest version.

## 
## Attaching package: 'spatstat'

## The following object is masked from 'package:gstat':
## 
##     idw

## The following objects are masked from 'package:raster':
## 
##     area, rotate, shift

## CRS arguments:
##  +proj=tmerc +lat_0=4.59620041666667 +lon_0=-74.0775079166667 +k=1
## +x_0=1000000 +y_0=1000000 +ellps=GRS80 +units=m +no_defs

## CRS arguments:
##  +proj=tmerc +lat_0=4.59620041666667 +lon_0=-74.0775079166667 +k=1
## +x_0=1000000 +y_0=1000000 +ellps=GRS80 +units=m +no_defs

## 4.2 - Usando el método IDW

# Create an empty grid where n is the total number of cells
grd              <- as.data.frame(spsample(precip2, "regular", n=100000))
# You need to figure out what is the expected size of the output grd
names(grd)       <- c("X", "Y")
coordinates(grd) <- c("X", "Y")
gridded(grd)     <- TRUE  # Create SpatialPixel object
fullgrid(grd)    <- TRUE  # Create SpatialGrid object

# Add P's projection information to the empty grid
proj4string(grd) <- proj4string(precip2)

## Warning in proj4string(precip2): CRS object has comment, which is lost in output

## Warning in showSRID(uprojargs, format = "PROJ", multiline = "NO", prefer_proj =
## prefer_proj): Discarded datum Unknown based on GRS80 ellipsoid in CRS definition

# Interpolate the grid cells using a power value of 2 (idp=2.0)
P.idw <- gstat::idw(rainfall ~ 1, precip2, newdata=grd, idp=2.0)

## Warning in proj4string(d$data): CRS object has comment, which is lost in output

## Warning in proj4string(newdata): CRS object has comment, which is lost in output

## [inverse distance weighted interpolation]

## Warning in proj4string(newdata): CRS object has comment, which is lost in output

## Warning in showSRID(uprojargs, format = "PROJ", multiline = "NO", prefer_proj =
## prefer_proj): Discarded datum Unknown based on GRS80 ellipsoid in CRS definition

# Convert to raster object then clip to AOI
r       <- raster(P.idw)

## class      : RasterLayer 
## dimensions : 267, 375, 100125  (nrow, ncol, ncell)
## resolution : 1471.008, 1471.008  (x, y)
## extent     : 755174.6, 1306803, 417272, 810031.1  (xmin, xmax, ymin, ymax)
## crs        : +proj=tmerc +lat_0=4.59620041666667 +lon_0=-74.0775079166667 +k=1 +x_0=1000000 +y_0=1000000 +ellps=GRS80 +units=m +no_defs 
## source     : memory
## names      : var1.pred 
## values     : 1.458398, 64.48055  (min, max)

## class       : SpatialPolygonsDataFrame 
## features    : 16 
## extent      : 751894.5, 1314457, 413692, 819562.9  (xmin, xmax, ymin, ymax)
## crs         : +proj=tmerc +lat_0=4.59620041666667 +lon_0=-74.0775079166667 +k=1 +x_0=1000000 +y_0=1000000 +ellps=GRS80 +units=m +no_defs 
## variables   : 2
## names       :       MUNIC, CODIGO 
## min values  :     ALBANIA,  18001 
## max values  : VALPARAÃSO,  18860

## class      : RasterLayer 
## dimensions : 267, 375, 100125  (nrow, ncol, ncell)
## resolution : 1471.008, 1471.008  (x, y)
## extent     : 755174.6, 1306803, 417272, 810031.1  (xmin, xmax, ymin, ymax)
## crs        : +proj=tmerc +lat_0=4.59620041666667 +lon_0=-74.0775079166667 +k=1 +x_0=1000000 +y_0=1000000 +ellps=GRS80 +units=m +no_defs 
## source     : memory
## names      : var1.pred 
## values     : 1.458398, 64.48055  (min, max)

## class       : SpatialPointsDataFrame 
## features    : 2920 
## extent      : 755367.2, 1306527, 417103.9, 809706.1  (xmin, xmax, ymin, ymax)
## crs         : +proj=tmerc +lat_0=4.59620041666667 +lon_0=-74.0775079166667 +k=1 +x_0=1000000 +y_0=1000000 +ellps=GRS80 +units=m +no_defs 
## variables   : 4
## names       :       MUNIC, CODIGO,             rain, rainfall 
## min values  :     ALBANIA,  18001, 1.01657009124756,        1 
## max values  : VALPARAÃSO,  18860, 67.1218643188477,     67.1

4.2.1 El error cuadrático medio (RMSE) se calcula a partir de IDW.out de la siguiente manera:

## [inverse distance weighted interpolation]
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## [inverse distance weighted interpolation]
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## [1] 4.506279

4.2.2 Polinomio de primer orden

Para ajustar un modelo polinomial de primer orden de la forma precip = intersección + aX + bY

## Variable(s) "NA" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.

4.2.3 Polinomio de segundo orden

Para ajustar un modelo polinomial de segundo orden de la forma precip = intersección + aX + bY + dX2 + eY2 + fXY

4.3 - Usando el método de Interpolacion de Kriging

##   model    psill    range kappa
## 1   Nug   0.0000      0.0   0.0
## 2   Mat 124.6086 103335.3   0.5

## [using universal kriging]

## class      : RasterLayer 
## dimensions : 267, 375, 100125  (nrow, ncol, ncell)
## resolution : 1471.008, 1471.008  (x, y)
## extent     : 755174.6, 1306803, 417272, 810031.1  (xmin, xmax, ymin, ymax)
## crs        : +proj=tmerc +lat_0=4.59620041666667 +lon_0=-74.0775079166667 +k=1 +x_0=1000000 +y_0=1000000 +ellps=GRS80 +units=m +no_defs 
## source     : memory
## names      : var1.pred 
## values     : 1.004153, 66.68696  (min, max)

## 4.3.1 mapas de intervalo de confianza y varianza

Se puede apreciar que al realizar la interpolación con kriging se vuelve a visualizar de una forma más oscura y marcada en la concentración de valores por encima de los 300 a 400 mm en el sector sur del departamento, con algunos valores también en el centro del mismo, asimismo se puede notar en el mapa interactivo que esta área comprende principalmente el sector de curillo, valparaiso, solano y Florencia. Los mapas de varianza e intervalo de confianza nos indican que los puntos ubicados a los bordes del departamento son los de mayor variabilidad y la mayoría de los puntos presentan un alto nivel de confianza.

5. Analisis de resultados

5.1 Resultados de interpolación por IDW

Se usó una potencia de 2.0, debido a que es un grado aceptable en el que disminiuye la interacción espacial en función inversa a la distancia. (Balagueró T. 2018)

Con base a los resultados se puede decir que es un método acertado, pues este coincide con las zonas en las que las precipitaciones son mayores (occidente) y con las zonas en las que las precipitaciones son menores (norte y oriente)

En la siguiente gráfica se muestra cual es el valor de precipitación observada en el eje X y la estimada en interpolación en el eje Y teniendo en cuenta que se usó una potencia=2.0

5.2. Resultados de interpolación por polígonos Thiessen

Este método permite que la precipitación que antes estaba unicamente en los puntos, ahora se encuentra en toda el área de estudio para áreas que son completamente rectangulares. Opera uniendo los puntos a a través de rectas. A las rectas luego se les trazan sus mediatrices hasta que se intersectan entre sí, generando los polígonos que en este caso son cuadrados porque los datos vienen de una grilla regular y tienen la misma distancia tanto en el eje X como en el Y.(Jiménez J.2018). Cabe restaltar que cuando la distribución de puntos no es irregular no crea cuadrados por lo que se hace más apropiado este método para interpolar. La interpolación por polígonos de Thiessen es sencilla pero no brinda una presisión y exactitud confiables a la hora de estudiar alguna variable puesto que se hace uso de poligonos unidos por mediatrices y no superficies continuas a la hora de hacer la interpolación, lo que redunda en el hecho de que se ignoran algunos valores.

5.3 Resultados de interpolación por Kriging

Por medio de este método se obtiene un producto de mejor resolución que tenían los datos, se puede observar que los resultados siguen la tendencia natural de lo que se midio. Este método es más adecuado cuando se sabe que hay una influencia direccional o de la distancia correlacionada espacialmente en los datos, este presupone que la distancia o la dirección entre los puntos de muestra reflejan una correlación espacial que puede utilizarse para explicar la variación en la superficie. En los resultados se puede observar que la zona con mayor precipitación es el occidente del departamento, lo cual tiene bastante sentido puesto que esta zona corresponde a los municipios de currilo y Valparaiso; mucicipios cercanos al departamento del Cesar, es uno departamento que presenta mayores precipitaciones a visperas de fin de año en el país (Jiménez J. 2018).

Conclusiones

La interpolación es útil en múltiples disciplinas, especialmente en la ingeniería agronomómica, pues gracias a esta herramienta se pueden estimar datos importantes relacionados a la producción agraria que, posteriormente ayudarán a la elaboración de planes de producción.
Se pueden utilizar de manera confiable en las herramientas de interpolación IDW y Kriggin para aplicar algún método científico.
Interpolar es una estrategia viable para hallar algún valor del que no se tiene conocimiento, y que además, es necesario conoce para lograr algún cálculo para algún problema de aplicación.
Emplear el método de polígonos de Thiessen para una interpolación no ofrece tanta precisión como IDW y Kriging

Bibliografía

Toro, G. A., & Melo, C. E. (2009). Aplicación de métodos de interpolación geoestadísticos para la predicción de niveles digitales de una imagen satelital con líneas perdidas y efecto sal y pimienta. Tecnura, 12(24), 55-67.
Gobernación Caqueta (2018) Información del departamento de caqueta, recuperado en 20/06/2020 de https://www.caqueta.gov.co/publicaciones/60135/identificacion-del-departamento/
Portalés, C., Boronat-Zarceño, N., Pardo-Pascual, J. E., & Balaguer-Beser, A. (2008). Un nuevo método para el cálculo de precipitaciones medias mediante técnicas de interpolación geoestadística considerando las características geográficas y topográficas del territorio. In Congreso Internacional sobre Ingeniería Geomática y Topográfica, TOP-CART.
Instituto de Hidrología, Meteorología y Estudios Ambientales (IDEAM) http://www.ideam.gov.co/ MAPAS: Atlas climatológico de Colombia - IDEAM
DANE (2020) Necesidades básicas instatisfechas (NBI), recuperado el 20/06/2020 de https://www.dane.gov.co/index.php/estadisticas-por-tema/pobreza-y-condiciones-de-vida/necesidades-basicas-insatisfechas-nbi
Jiménez J. (2018) Interpolación con la Distancia Inversa Ponderada (IDW), recuperado el 20/06/2020 de https://acolita.com/interpolacion-con-la-distancia-inversa-ponderada-idw/
Balagueró T. (2018) ¿Qué son los datasets y los dataframes en el Big Data? recuperado el 20/06/2020 de https://www.deustoformacion.com/blog/programacion-diseno-web/que-son-datasets-dataframes-big-data