The generalized STAR modelling with three-dimensional of spatial weight matrix in predicting the Indonesia peatland’s water level

Abstract

The release rate of CO₂ gas can be influenced by peatlands’ physical properties, such as water level and soil moisture, and rainfall. To anticipate the unstable condition which is when the peatland emit more carbon, we developed the Generalized Space Time Autoregressive (GSTAR) model in predicting these physical properties for the following weeks. As the innovation in modelling, the spatial weight matrix was based on three-dimensional coordinates with a modification on the height factor. The data we used are real-time data of water level on the peatlands in Pulang Pisau Regency, Central Kalimantan Province from 20 February 2021 to 18 March 2023. We then used Ordinary Kriging interpolation on the prediction results to create contour maps on different dates. There were empty data on several dates, especially from 24 March until 3 August 2022. To fill the empty data, we used linear interpolation and then we added white noise to the interpolation results. From the data, the water level has a downward trend pattern from around November to September and an upward trend pattern from October to November. Furthermore, we found that the best model for water level was GSTAR (2;0.1) with a modified matrix $a=0.1$ and $b=1.1$. Based on the predicted water level, there is a risk of changes in the properties of the peatlands in several areas in Pulang Pisau Regency.

Introduction

One of the most important roles of peatlands on the environment is their function as the largest carbon storage. On a global scale, peatlands can store twice as much carbon as other forests in total. Peatlands accumulate vast amounts of carbon in the long term over thousands of years, because the absorbed carbon through plant photosynthesis is not all released back into the atmosphere as a consequence of incomplete decomposition. However, the condition changes dramatically for peatlands in unstable conditions, because they could release huge amounts of carbon that has been stored. The unstable condition can be caused by drainage and fires, which commonly occur on degraded peatlands [6]. This condition could affect the global carbon cycle and contribute positively to global warming.

The rate of carbon input and release from the peatland is significantly affected by the water level position [5, 13, 15, 16, 35]. A shallow water level position leads to the shorter residence time of plant litter or organic matter in an unsaturated zone that supports carbon accumulation. Contrastingly, a deeper water level increases the thickness of the unsaturated zone, which results in a more significant carbon release from the peatland [17]. Furthermore, a deep water level position could produce peatland fire [33] and leads to the unstable condition of the peatland. Therefore, modelling and predicting the water level are crucial to anticipate the unstable condition of the peatland.

The water level can be modelled by deterministic or stochastic approaches. The deterministic models [5, 17, 35] do not capture the uncertainties of the system that could influence the model outputs. Contrastingly, the stochastic models involve uncertainties that might improve the model capability and accuracy, particularly for complex systems including peatland water level. Tamea et al. [32] developed a stochastic model of the water level for the wetlands that can describe a long-term probability distribution of water level depth in temporarily inundated sites. However, this model ignores the influence of spatial interactions on the water level position, which becomes a significant limitation. The field observation from Lewis et al. [14] showed that the water level is located in a deeper position at the margin compared to the centre on the peatland due to the variations in hydraulic conductivity and bulk density. This condition is also supported by the simulation from Mahdiyasa et al. [15, 16], who found the spatial variability of the peatland water level through the fully coupled mechanical-ecohydrological model.

To overcome these problems, we propose a novel stochastic approach through the Generalized Space Time Autoregressive (GSTAR) combined with Kriging to provide the future value and spatial interpolation of water level position. The GSTAR model can be employed to predict the future outcome of random variables that are affected not only by past observations but also by observations in neighbouring regions [4, 8, 22]. The GSTAR is the extension of the Space Time Autoregressive (STAR) model [29], because it allows the autoregressive parameters to vary across locations. The specific characteristic of GSTAR is the inclusion of a weight matrix that captures the spatial relationship between different locations [27].

The GSTAR model has rapidly developed both in theory and application. On the theoretical side, the development of weight matrices has been carried out using various approaches, including the kernel approach [37] and minimum spanning tree [24]. In application, the GSTAR model has been explored for economic [26], oil palm production [21], commodity prices [10], tea production [36], criminal cases [18], coffee berry borer attack [30], Dengue Fever cases [22], layer peat soil [12], rainfall [34], copper and gold grades [28], toll gates density [25], COVID-19 cases [8], and climate data [31]. Those applications considered the spatial dependency among time series observations in many locations and used the two-dimensional weight matrices. Here, by considering that the current water level can be seen as the impact of the previous water level in same locations as well its neighbour locations, then the GSTAR modelling be applied. Furthermore, since the peatlands have the thickness variable, it is deemed necessary to use a three-dimensional weight matrix.

The location for this study is the Pulang Pisau Regency, Central Kalimantan Province, Indonesia. The region features a mix of lowlands and wetlands, with a significant portion of its land covered by peatlands. Similar to the other area of Central Kalimantan, Pulang Pisau experiences a tropical rainforest climate, with monsoon season. One of the most significant issues in the Central Kalimantan area is the peatland fires. For example, around 20324 peatland fires were detected in the Central Kalimantan in 2015, which becomes the highest density of peatland fires in South East Asia [7, 19]. The distribution of peatland fires indicate the maximum number of fire appear in August until October, which is related to the long dry season and strong El Nino phenomenon [9].

This paper, therefore, sets out to (1) develop a three-dimensional spatial weight matrix for Generalized Space Time Autoregressive (GSTAR) that involves the height factor, (2) estimate the future value of water level position through GSTAR model, and (3) produce water level map of the peatland in Pulang Pisau Regency, Central Kalimantan Province, Indonesia as the case study.

The spatial weight matrix

Suppose $\{{Y}_{i,t}\}$ denotes a process with zero mean at the $i$-th location. Suppose $\left\{{{\varvec{Y}}}_{t}\right\}$ denotes a process vector with zero mean and ${{\varvec{\Phi}}}_{k{\ell}}=diag\left({\phi }_{k{\ell}}^{\left(1\right)},\dots ,{\phi }_{k{\ell}}^{\left(N\right)}\right)$ where ${\phi }_{k{\ell}}^{\left(i\right)}$ denotes the autoregressive parameter at time lag $k$ and spatial lag ${\ell}$ for location $i$. If $\left\{{{\varvec{Y}}}_{t}\right\}$ follows the GSTAR($p;{\lambda }_{1},\dots ,{\lambda }_{p}$) model, we get

$${{\varvec{Y}}}_{t}={\sum }_{k=1}^{p}{\sum }_{{\ell}=0}^{{\lambda }_{k}}{{\varvec{\Phi}}}_{k{\ell}}{{\varvec{W}}}^{({\ell})}{{\varvec{Y}}}_{t-k}+{{\varvec{\varepsilon}}}_{t},$$

where the vector ${{\varvec{Y}}}_{t}$ represents the observation vector at $N$ locations with ${{\varvec{Y}}}_{t}={\left({Y}_{1,t},{Y}_{2,t},\dots ,{Y}_{N,t}\right)}^{\prime}$; matrix ${{\varvec{W}}}^{\left({\ell}\right)}$ represents the spatial weight matrix at spatial lag ${\ell}$ with ${{\varvec{W}}}^{\left({\ell}\right)}=\left({w}_{ij}^{\left({\ell}\right)}\right)$; the notation ${{\varvec{\varepsilon}}}_{t}$ states the error vector at the time $t$ which is a white noise with ${\varepsilon }_{t}={\left({\varepsilon }_{1t},{\varepsilon }_{2t},\dots ,{\varepsilon }_{Nt}\right)}^{\prime}$ [37].

The spatial weight matrix $\left({{\varvec{W}}}^{\left({\ell}\right)}=\left\{{w}_{ij}^{\left({\ell}\right)}\right\}\right)$ serves to describe the spatial relationship between locations. The elements of the spatial weight matrix ${w}_{ij}^{\left({\ell}\right)}$ represent the weight of the spatial relationship between location $j$ and location $i$. Determining the value of ${w}_{ij}^{\left({\ell}\right)}$ is strongly influenced by the spatial lag order definition.

In this paper, we defined a spatial lag order which resembles a layered rhombus. This definition is a development of the square grid system [3] and the geographic location definition [26]. Suppose ${s}_{i}$ represents the $i$th reference location’s square in the grid with its two-dimensional coordinates, namely (${p}_{i}$,${q}_{i}$), for $i=\text{1,2},\dots ,N$. Assume there is only one location in ${s}_{i}$. Neighbours ordered ${\ell}=1$ from the location located in ${s}_{i}$ are locations with squares directly adjacent to ${s}_{i}$. This makes neighbours of order ${\ell}=1$ located in squares with their two-dimensional coordinates, namely $\left({p}_{i}-1,{q}_{i}\right)$, $\left({p}_{i}+1,{q}_{i}\right)$, $\left({p}_{i},{q}_{i}-1\right)$, or $\left({p}_{i},{q}_{i}+1\right)$. For order ${\ell}=2, 3,\dots$, neighbours of order ${\ell}$ are all neighbours of order (${\ell}-1$) from neighbours of order (${\ell}-1$) of the location located in ${s}_{i}$ other than the reference location itself. As an alternative definition, a location located in ${s}_{j}$ is an ${\ell}$th order neighbour of a location located in ${s}_{i}$ if

$$C\left( {\ell - 1} \right) < \left| {p_{j} - p_{i} } \right| + \left| {q_{j} - q_{i} } \right| \le C\ell ,$$

(1)

where $C$ is a positive constant. An illustrative example of this definition is shown in Fig. 1.

Fig. 1

Illustration of the spatial lag order definition resulting from the development for $C=2$

After defining the spatial lag order, we can formulate the spatial weight matrix for each respective spatial lag. To accommodate peatlands that have a thickness factor, a three-dimensional weight matrix was developed. In determining the weight value in three dimensions, we proposed the following three ways.

1) Three-dimensional distance inverse matrix with modifications to the height element

Let ${{{\varvec{W}}}^{{\varvec{M}}{\varvec{o}}{\varvec{d}}}}^{\left({\ell}\right)}=\left\{{w}_{ij}^{Mo{d}^{\left({\ell}\right)}}\right\}$ represents a three-dimensional inverse distance spatial weight matrix with modifications to the height element having dimensions $N\times N$ with ${\ell}$th spatial lag order. The matrix ${{{\varvec{W}}}^{{\varvec{M}}{\varvec{o}}{\varvec{d}}}}^{\left({\ell}\right)}$ is the normalized form of ${{\varvec{W}}}^{{\varvec{S}}{{\varvec{H}}}^{({\ell})}}=\left\{{{w}^{SH}}_{ij}^{\left({\ell}\right)}\right\}$ which is the Hadamard multiplication matrix between matrices ${\mathcal{S}}^{\left({\ell}\right)}$ and $\mathcal{H}$ which has the same dimensions as ${{{\varvec{W}}}^{{\varvec{M}}{\varvec{o}}{\varvec{d}}}}^{\left({\ell}\right)}$.

The matrix ${\mathcal{S}}^{\left({\ell}\right)}$ represents the spatial relationship between locations in two dimensions with ${\ell}$ spatial lag order with where

With ${d}_{ij}$ denotes the distance between the centre of the reference location $i$ and the neighbour $j$ according to its position in the plane.

The matrix $\mathcal{H}$ represents the spatial relationship between locations according to the difference in height with . Thus, the element represents the weight by height of the neighbour $j$ towards the reference location $i$, so it can be defined

where $a,b\ge 0$ is a fixed parameter, ${l}_{i}$ represents the height of the centre of the reference location $i$, ${l}_{j}$ represents the height of the centre of the neighbouring location $j$, ${d}_{ij}$ represents the euclidian distance between the centres of locations i and j on the plane, and $\mathcal{d}$ is a constant. In this paper, we define $\mathcal{d}$ as the average distance from each pair of reference locations and neighbouring locations.

By combining the matrices ${\mathcal{S}}^{\left({\ell}\right)}$ and $\mathcal{H}$ through element-by-element multiplication, we get the matrix

With operator ⨀ represents the Hadamard matrix multiplication [20]. After obtaining ${{{\varvec{W}}}^{{\varvec{S}}{\varvec{H}}}}^{\left({\ell}\right)}$, ${{{\varvec{W}}}^{{\varvec{M}}{\varvec{o}}{\varvec{d}}}}^{\left({\ell}\right)}$ can be obtained through this equation for the elements

$$w_{ij}^{{Mod^{\left( \ell \right)} }} = \frac{{w_{ij}^{{SH^{\left( \ell \right)} }} }}{{\mathop \sum \nolimits_{j = 1}^{N} w_{ij}^{{SH^{\left( \ell \right)} }} }}.$$

(2)

To determine the value of ${h}_{ij}$, there are two alternatives in determining the value of $a$ and $b$, and both are

• a. Using $a = 0.1$ and $b = 1.1$ [20].

• b. Using the assumption of the smallest distance value.

Suppose $z=\left|{l}_{i}-{l}_{j}\right|$.

Suppose we define a function

$$f\left(a,b,z\right)=\frac{1}{b{z}^{a}}, a\ge 0, b>0, z>0.$$

Assume the smallest $z$ is $z={10}^{-4}$ and the difference between $1$ and $\frac{1}{b{z}^{a}}$ with $z={10}^{-4}$ is ${10}^{-4}$, we then determine the values of $a$ and $b$, so that

$$\frac{1}{b{\left({10}^{-4}\right)}^{a}}+{10}^{-4}=1.$$

With $a\ge 0$ and $b>1$. Therefore, we obtain

$$\frac{1}{b{\left({10}^{-4}\right)}^{a}}=0.9999.$$

With $a\ge 0$ and $b>1$. From this equation, the value of $b$ can be obtained by the following equation:

$$b=\frac{1}{0.9999{\left(0.0001\right)}^{a}}.$$

Therefore, it can be defined

$$f=\frac{1}{\frac{1}{0.9999{\left(0.0001\right)}^{a}}{z}^{a}}=0.9999{\left(\frac{0.0001}{z}\right)}^{a}.$$

Since $\frac{1}{b{\left|{l}_{i}-{l}_{j}\right|}^{a}}=f$, we obtain

with $a$ a fixed parameter.

From the equation above, for ${l}_{i}\ne {l}_{j}$ and ${d}_{ij}\le \mathcal{d}$, the value of ${\mathcal{h}}_{ij}$ will decrease following the increase in the value of $a$. This states that the greater the value of $a$, the smaller the weight of location $j$ towards the reference location $i$ if locations $i$ and $j$ have different heights. Furthermore, the value $a=0$ causes ${h}_{ij}=0.9999$ for ${l}_{i}\ne {l}_{j}$ and ${d}_{ij}\le d$ so ${{w}_{ij}^{SH}}^{({\ell})}\approx {\mathcal{s}}_{ij}^{\left({\ell}\right)}$. Hence, for $a = 0$, the difference in height has no effect in determining the weight of the neighbouring location towards the reference location.

2) Three-dimensional distance inverse matrix without modification

Let ${{{\varvec{W}}}^{{\varvec{I}}{\varvec{n}}{\varvec{v}}}}^{\left({\ell}\right)}=\left\{{{w}^{Inv}}_{ij}^{\left({\ell}\right)}\right\}$ represents an unmodified three-dimensional inverse distance spatial weight matrix with $N\times N$ dimensions with ${\ell}$th spatial lag order. Then, it can be defined that

$$w_{ij}^{Inv\,\left( \ell \right)} = \frac{{\frac{1}{{d_{ij}^{*} }}}}{{\mathop \sum \nolimits_{j = 1}^{N} \frac{1}{{d_{ij}^{*} }}}}$$

(3)

if location $j$ is the ${\ell}$ th neighbour of location $i$ or ${{w}^{Inv}}_{ij}^{\left({\ell}\right)}=0$ if otherwise where ${d}_{ij}^{*}$ represents the euclidian distance between the centres of location $i$ and $j$ in three-dimensional space. The ${d}_{ij}^{*}$ value can be obtained by the following equation:

$${d}_{ij}^{*}=\sqrt{{\left({p}_{i}-{p}_{j}\right)}^{2}+{\left({q}_{i}-{q}_{j}\right)}^{2}+{\left({r}_{i}-{r}_{j}\right)}^{2},}$$

where $\left({p}_{i},{q}_{i}\right)$ and $\left({p}_{j},{q}_{j}\right)$, respectively, represent the coordinates of locations $i$ and $j$ on the plane, and ${r}_{i}$ and ${r}_{j}$, respectively, represent the heights of locations $i$ and $j$.

Uniform weight matrix

Let ${{{\varvec{W}}}^{{\varvec{U}}{\varvec{n}}{\varvec{i}}{\varvec{f}}}}^{\left({\ell}\right)}=\left\{{{w}^{unif}}_{ij}^{\left({\ell}\right)}\right\}$ denotes a uniform weight matrix with ${\ell}$ th spatial lag order. Then, it can be defined that

$$w^{{unif_{{ij}}^{{\left( \ell \right)}} }} = \left\{ {\begin{array}{*{20}c} {\frac{1}{{n_{i}^{{\left( \ell \right)}} }},} & {{\text{if}}\;{\text{location}}\;j\;{\text{is}}\;{\text{the}}\;\ell {\text{th}}\;{\text{neighbour}}\;{\text{of}}\;{\text{location}}\;i} \\ {0,} & {{\text{if}}\;{\text{location}}\;j\;{\text{is}}\;{\text{not}}\;{\text{the}}\;\ell {\text{th}}\;{\text{neighbour}}\;{\text{of}}\;{\text{location}}\;i} \\ \end{array} } \right.$$

(4)

with ${n}_{i}^{\left({\ell}\right)}$ denotes the number of locations which are the ${\ell}$ th neighbours of location $i$ [26].

The GSTAR-Kriging modelling

In this paper, we analysed the water level using the GSTAR model. The GSTAR modelling consisted of model identification, parameter estimation, and diagnostic tests. Then, we used Ordinary Kriging interpolation to create contour maps of the water level on the observed area. The flowchart of GSTAR-Kriging modelling be presented in Fig. 2 and the detail steps are explained in Algorithm 1.

Fig. 2

Flowchart of GSTAR-Kriging modelling (a) and flowchart of kriging modelling in GSTAR-Kriging modelling (b)

Algorithm 1

GSTAR-Kriging modelling

Data

The data used in this paper are the average daily water level at peatland location points in Pulang Pisau Regency, Central Kalimantan Province from 20 February 2021 to 18 March 2023 with 7 days apart. The data were obtained from the Peatland and Mangrove Restoration Agency of Republic of Indonesia (BRGM Indonesia) via the prims.brg.go.id website. In addition, data on the distribution of peat thickness were also taken from the Indonesian Centre for Agricultural Land Resources Research and Development (BBSDLP) in 2019 from the same site. Information about the location points used is shown in Fig. 3 with the location names decided by the author. For water level, let locations 1, 2, 3, 4, 5, and 6, respectively, represent Jabiren, Jabiren2, Jabiren5, Jabiren7, Kahayan Hilir, and Pandih Batu. The position of the location points used along with their thickness level can also be seen in Fig. 3.

Fig. 3

Map of the observed groundwater level (GWL) measurement location points along with the distribution of peatland thickness according to BBSDLP in 2019

From the data obtained, it turns out that there are several dates with empty observation values. To overcome this problem, a linear interpolation was carried out to fill in the gaps as seen in Fig. 4. After that, the interpolation results were summed with white noise or the simulation results of a normally distributed data with zero mean and variance according to the data.

Fig. 4

Water level graphs in meters of the original data (black) and the interpolated results (red) for Jabiren (a), Jabiren2 (b), Jabiren5 (c), Jabiren7 (d), Kahayan Hilir (e), and Pandih Batu (f). The green arrows show an increasing trend, while the orange arrows show a decreasing trend

The combined numerical summary of the original data and interpolated results for water level is shown in Table 1. In Fig. 4, the water level tends to fall from November 2021 to August 2022 and then rises again as we enter October 2022 at all locations except Jabiren2. At Jabiren2, water level tends to fall from February 2022 to August 2022.

Table 1 Numerical summary of water level (m) at the observed location points

The longitude and latitude coordinates of each location were then converted according to the Universal Transverse Mercator (UTM) coordinate system, specifically in the 50S Zone. In addition, the elevation position of each location was obtained by subtracting the midpoint of the thickness range by 7 m (the largest number in the peat thickness range), so that a height position of 7 m was used as a reference point. Hence, with this definition, the coordinates of each location on the plane and the altitude position used were obtained, as shown in Table 2.

Table 2 Coordinate of each location in meters (m)

Results and discussion

The GSTAR modelling begins by defining the spatial lag order for the lag spatial weight matrix. For that purpose, we defined a grid as be shown in Fig. 5. To guarantee that each location has at least one neighbour location in its second spatial lag order, we use $C=4$ for Eq. (1). It also can reduce the possibility of error in the parameter estimation process. In Fig. 5, each grid has a size of $7500 \text{m}\times 7500 \text{m}$.

Fig. 5

Defining the 15 × 6 grid at the observed location points (a) and spatial lag order resulting from the development for $C=4$ of reference location Kahayan Hilir (b)

The candidate models were identified using the STACF and STPACF [29]. The obtained models were GSTAR(2,0,1) and GSTAR(2;1,1) with modification and conventional weight matrices. After obtaining the candidate models from the model identification stage, parameter estimation and then diagnostic testing can be performed. For the diagnostic test, the Inverse of Autocovariance Matrix (IAcM) approach was used to test stationarity [23], the Ljung–Box test to examine the independent nature of the residuals, and the Kolmogorov–Smirnov test to check for normality with a zero mean of the residuals. For the Ljung–Box test and the Kolmogorov–Smirnov test, a significance level of 0.05 was used. The candidate models which passed the stationarity test and the independence test, be compared according to the Mean Squared Error (MSE) using the training data (in sample) and the testing data (out sample). In this paper, MSE in sample is defined as the average of squared difference between the training data and model estimation of the training data, while MSE out sample is defined as average of squared difference between the testing data and prediction from the training data. The MSE comparison between the GSTAR model candidates is shown in Table 3. The smaller the MSE, the better the model and the more valid the model is.

Table 3 Selection of the best model for water level (no differencing)

From Table 3, GSTAR(2;1,1) with a modified matrix $a=0.1$ and $b=1.1$ has the lowest MSE in sample value, while GSTAR(2;0,1) with a modified matrix $a=0.1$ and $b=1.1$ has the lowest MSE out sample value, indicating that GSTAR(2;1,1) with a modified matrix $a=0.1$ and $b=1.1$ is the best model in estimating the training data, while GSTAR(2;0,1) with a modified matrix $a=0.1$ and $b=1.1$ is the best model in predicting the testing data. Because the main objective is to estimate the future value of water level, GSTAR(2;0,1) with a modified matrix $a=0.1$ and $b=1.1$ was chosen as the best model for water level. This indicates that the water level is affected by the values of that location and the values of the neighbouring locations in the past. However, the difference in the altitude position between the locations only has a significant effect on the weights for water level on the spatial weight matrix. The obtained models for water level in each location, which is marked with the dark blue for Jabiren, green for Jabiren2, brown for Jabiren5, grey for Jabiren7, purple for Kahayan Hilir, and blue for Pandih Batu, can be expressed as follows:

In the above equation, black marks positive coefficients, red marks negative coefficients, and bold marks significant coefficients. The above equation indicates that water level at Jabiren 7 days prior (${Y}_{1,t}$) contributes positively to the present water level at Jabiren (${\widehat{Y}}_{1,t})$ with coeficient 0.640, while the 14 days prior water level at Jabiren (${Y}_{1,t-2}$) and Jabiren2 (${Y}_{2,t-2})$ contributes negatively to the present water level at Jabiren, with respective coefficients are –0.013 and –0.506. Overall, for each location $i=\text{1,2},\dots , 6$, the 7 days prior water level of respective location (${Y}_{i,t-1}$) has the positive and biggest influence to the current water level (${Y}_{i,t})$. As for the neighbour locations have less significant influences to its reference locations, except for Jabiren2 which has negative and positive contribute to Jabiren and Pandih Batu consecutively. As a note, all locations have smaller positive contribution to Pandih Batu. From those models, we suspect that Jabiren2 has important roles to predict the water level in Pulang Pisau Region.

After testing the significance of the parameters, predictions were made for the next 5 weeks using the best model. Predictions for water level are shown in Fig. 6. From the prediction results in Fig. 6, the water level tends to decrease at Jabiren, Jabiren7, and Pandih Batu, tends to increase at Jabiren2 and Kahayan Hilir, and tends to remain constant at Jabiren5. The results indicate that there is heterogeneity in the movement of the water level at various points of peatland locations in Pulang Pisau Regency. Besides that, the model is adequate in estimating the water level at Jabiren2 and Jabiren7. However, the model overestimates the water level at Jabiren5 and Kahayan Hilir and underestimates the water level at Jabiren and Pandih Batu. From Fig. 3, Jabiren2, Jabiren5, Jabiren7, and Kahayan Hilir are in proximity, with the shortest distance between locations is the distance between Jabiren2 and Jabiren7, while Jabiren and Pandih Batu are far from other locations. There is a possibility that the goodness of the model at each location is related to the closeness of the location with other locations. From Fig. 6, the model estimation is adequate for the locations with the shortest distance to another location, overestimate for the locations with a distance to another location which is slightly further than the shortest distance, and underestimate for the locations with the longest distance to another location.

Fig. 6

Water level graphs in meters of the original data (training, black and testing, orange), the interpolation results (red), GSTAR(2;0,1) model with modified matrices $a=0.1$ and $b=1.1$ estimation results (green), and the prediction results (blue) for Jabiren (a), Jabiren2 (b), Jabiren5 (c), Jabiren7 (d), Kahayan Hilir (e), and Pandih Batu (f). The arrow at the end shows the trend of prediction. The green arrow shows an increasing trend, the orange arrow shows a decreasing trend, while the blue arrow shows no trend

By obtaining the prediction results for each variable, we will see the interpolation of the observed values of location points that were not observed through ordinary kriging on the observation and prediction dates, namely from 4 March 2023 to 22 April 2023. Some set of parameters used in ordinary kriging for water level is shown in Table 4. Through ordinary kriging, the obtained contour maps are shown in Fig. 7.

Table 4 Ordinary Kriging parameters for water level with three neighbours

Fig. 7

Water level contour maps from data and predicted results on 4 March 2023, 11 March 2023, 18 March 2023, 25 March 2023, 1 April 2023, 8 April 2023, April 15, 2023, and 22 April 2023

From the contour maps made using the QGIS application, patterns of the water level can be obtained in March and April 2023. In Fig. 7, from 4 March 2023 to 25 March 2023, the water level in all areas tends to fall, while from 1 April 2023 to 22 April 2023, the water level in the areas around Jabiren and Kahayan Hilir fluctuates, while in other areas, it tends to be stable. Thus, the area in the middle is always the area with the lowest value indicating that the area is drier than the other areas. There is a possibility that this was caused by growing settlement development due to the existence of the Trans Kalimantan Road which connects Banjarmasin City and Palangka Raya City [1, 11]. This development may have caused the soil to harden which can cause drying and backfilling in several areas.

If the water level falls below 0.4 m below the ground surface, there can also be a change in the characteristics of the peatland [2]. In Fig. 7, the peatland area around Kahayan Hilir had a change in the characteristics from 4 March 2023 to 18 March 2023. The peatland around Jabiren2 had a change in the characteristics from 11 March 2023 to 22 April 2023. The peatland around Pandih Batu had a change in the characteristics at 22 April 2023.

Conclusion

In this paper, GSTAR modelling has been used to predict water level from peatlands in Pulang Pisau Regency, Central Kalimantan Province. Some conclusions can be written as follow:

• 1. The water level at almost every location has a downward trend pattern from around November to September and an upward trend pattern from October to November.

• 2. The best model for water level is GSTAR(2;0.1) with a modified matrix $a=0.1$ and $b=1.1$. It has been predicted that in the next 5 weeks, the water level will tend to decrease. In addition, in Pulang Pisau Regency, the area in the middle is the driest area.

• 3. Based on the results of the prediction of the water level, there is a risk of changes in the properties of the peatlands in several areas in Pulang Pisau Regency.

For further research, physical peat variables such as humidity and rainfall factors can be involved to measure the water level. Furthermore, these variables can be used to predict the risk of drought that will occur in the future.