Introduction

Modeling the challenges of environment is tough and several researchers have successfully contributed to this domain in an excellent manner. For example, Antunes da Silva et al. (Antunes da Silva et al. 2024) utilized high-resolution remote sensing data for hydraulic modeling to create inundation maps in the Acre River Basin, Brazil, showcasing the critical integration of advanced technology in flood management. Naganur et al. (Naganur et al. 2024) evaluated best management practices (BMPs) and their environmental flow impacts using the SWAT+ model, highlighting the importance of sustainable practices in watershed management. Anand and Pradhan (Anand and Pradhan 2024) focused on in-situ channel migration and bank stability of the Brahmani River in Odisha, providing vital insights into riverine dynamics and geomorphology. Sambou et al. (Sambou, et al. 2024) studied the impact of climate change on the habitat range and distribution of key plant species in Senegal, emphasizing the significance of ecological modeling in understanding and mitigating climate change effects.

Similarly, Nayak et al. (Nayak et al. 2024) employed a transformer-based deep learning architecture to model monthly rainfall in India, illustrating the application of cutting-edge artificial intelligence techniques in meteorology. Jamaa et al. (Jamaa et al. 2024) utilized the DRASTIC model and GIS functions to assess groundwater vulnerability to contamination in the Doukkala Plain, Morocco, demonstrating the crucial role of geospatial analysis in groundwater management. Paul and Roy (Paul and Roy 2024) conducted a geospatial modeling and analysis of groundwater stress-prone areas in Murshidabad district, India, using GIS-based TOPSIS, VIKOR, and EDAS techniques, highlighting innovative multi-criteria decision-making approaches in hydrology. Bint-e Mehmood et al. (Bint-e Mehmood et al. 2024) modeled temperature and precipitation variabilities over the semi-arid region of Pakistan under RCP 4.5 and 8.5 emission scenarios, providing vital insights into climate change impacts in arid regions. Several other studies have been reported where modelling has helped to address the climate challenges (Mansour 2024; Rocha et al. 2024; Rios-Romero et al. 2024; Ochieng 2024; Swetha et al. 2024).

These studies demonstrate the diverse and innovative approaches researchers are taking to address complex environmental challenges.

Climate Change and Carbon Emissions

The increase in global greenhouse gas emissions and their profound impacts on the human life and the environment heightened the natural threats that may occur and impact the human life such as global warming, floods, increased temperature, droughts etc. The main cause of these serious threats is the combustion of fossil fuels. Due to rapid urbanization and development, the demand for fossil fuels has increased significantly and therefore the greenhouse gas emissions also increased showing a direct relation with the demand for fossil fuels (Sikder et al. 2022). Due to bad economic conditions, few countries are not able to cope with these key concerning issues and some countries have failed in providing effective policies in this regard. The rapid increase in the greenhouse gas emissions including methane (CH4), carbon dioxide (CO2), and nitrous oxide (N2O) has raised significant impacts and concerns globally. An increase of almost 30% of overall carbon dioxide emissions level is observed in transition from nineteenth to twentieth century (Akram et al., 2023). According to the report published by World Meteorological Organization (WMO) Greenhouse Gas Bulletin, the overall level of atmospheric carbon dioxide in 2021 is 149% which is higher than pre-industrial level. The key cause of such high levels of carbon emission has been the burning of fossil fuels and production of cement. Increased occurrence of droughts, wildfires on the land, and rising sea surface temperature in the ocean has an inverse impact on the environmental capacity of land and ocean to work as carbon sinks. As a result, an increasingly larger amount of emitted CO2 remains in the atmosphere. The Greenhouse Gas Bulletin report issues that out of the total amount of carbon dioxide emitted during 2011–2020, 48% is stored in the atmosphere. Oceans absorb approximately 26%, while the land ecosystem absorbs 29%. Hence, around 50% of the total carbon emission accumulates in the atmosphere which intensifies the threat of global warming (Pradhan et al. 2024).

The increasing pattern of carbon emissions in the atmosphere is leading towards intense environmental degradation. This issue has led to the establishment of the Intergovernmental Panel on Climatic Change (IPCC) and the United Nations Framework Convention on Climatic Change (UNFCCC). Various agreements such as Kyoto Protocol and the Paris Agreement, have been acknowledged and signed by numerous countries to save the planet, Earth. An agreement was supported by various countries to confine the increase in global temperature by 1.5 °C above pre-industrial levels by the year 2050 in the UN Climate Change Conference held in Glasgow (COP26) in 2021 (Yu et al. 2021). In spite of all such events, the increased levels of environmental threats connected with achieving highest levels of economic growth shows a great conflict in the actual efforts to curb the environmental concerns.

South Asia comprises of approximately 21% of global population, accounts 4% of global GDP, shares 6% of global energy consumption and contributes 3% to total world merchandise exports (Rehman and Rehman 2022). According to the stats, Asia is considered to be the largest greenhouse gas emitter in the entire world, accounting for approximately 53% of the global emissions. Recently, China has been on the top of the list in the world in terms of carbon emissions, emitting about 10 billion tons per year, accounting for more than one-quarter of worldwide emissions. In 2008, total carbon emissions increased by 85% in Afghanistan showing a drastic increase in global warming. Likewise, a steady increase in carbon emissions in India and Bangladesh was seen due to massive increase in industrialization and overall increase in GDP annually. Initially, low level of carbon emissions was observed in Pakistan due to low increase in GDP, but after the initiation of Chinese program of the China-Pakistan Economic Corridor (CPEC) and One Belt One Road (OBOR), the highest level of carbon emissions was observed with a 22% increase in 2016. In the last few years, Srilanka faced many economic crises, with an observed decrease in carbon emissions, but after OBOR, it also has a same trend as compared to Pakistan, with higher growth rate in 2014 at 21% due to the increase in GDP. Nepal is showing positive economic growth with time and this is also increasing level of carbon emissions, which is evident from its last 5 years’ development i.e. it has increased by 20% for almost every year (Aamir et al. 2022). In recent years, because of rapid increase in carbon emissions this region has faced repeated natural disasters such as South Asian tsunami affected 7seven countries and more than two hundred thousand people were killed in 2004, the 2008 earthquake in Pakistan followed by two floods in 2010 and 2011 making 10 million people homeless, and recent earthquake in Nepal killed around 9000 people (Ahmed et al. 2017). It Is evident from the level of carbon emissions, that they are showing increasing trend with increase in time.

Pakistan is one of the emerging economies with rapid economic growth in Southern Asia. This is set to be the story for a while longer. In any case, regardless of a developing industrial sector, agriculture is the primary predominant segment in Pakistan’s economic development. Yet, this industrial push, along with rapid growth in population rate is the main cause of deforestation which lead towards higher rates of carbon dioxide concentration in the atmosphere. The UNEP has defined Pakistan as one of the countries with top GHG emissions in the world. This accounts for 1.02% of global emissions Pakistan saw an estimated 504.59 million tons of GHG emissions in 2018. It is one of the world’s most exposed nations to the harmful effects of climate change and air pollution.

Pakistan is the first in Asia experiencing massive deforestation due to excessive exploitation of Natural energy from the environment. This leads to environment pollution on a larger scale. The pollution in this circumstance, along with deforestation has generated global warming which has caused climate change. Pakistan’s industrial revolution’s core element is the shift of Pakistan from organic energy economies to inorganic fossil fuel-based economies. Factually, the most greenhouse gas produced by fossil fuels as a by-product is CO2, presumably adding to the CO2 in the atmosphere. Certainly, the pursuit of becoming an industrialized powerhouse comes with significant costs. Pakistan is on a verge of its own complete industrial revolution promoting the use of fossil fuels to accelerate the industrial growth in the economy.

Stochastic Nature of the Climate Change Datasets

The impact of stochastic processes in climate action has been the subject of numerous studies. As the unforeseen calamities of climate change often exhibit stochastic behavior, researchers have focused on understanding these processes to improve climate models and predictions. For instance, Santos Gutiérrez (Santos Gutiérrez 2022) explored operator methods and responses in climate dynamics, shedding light on the complex interactions within the climate system. Montero et al. (Montero et al. 2022) provided details of the distant future under stochastic resettings, high-lighting the importance of considering stochastic elements in discounting future impacts of climate change. Additionally, seminal work by Hasselmann (Hasselmann 1976) laid the groundwork for stochastic climate models, providing a theoretical framework for understanding the stochastic nature of climate processes. Lindgren et al. (Lindgren et al. 2011) further advanced this field by establishing a link between Gaussian fields and Gaussian Markov random fields, contributing to the development of more accurate and comprehensive climate models. These studies collectively demonstrate the crucial role of O-U processes in enhancing our understanding of climate dynamics and guiding effective climate action strategies.

Research Statement

In the light of above mentioned serious climatic concerns, it is clear that the level of carbon emission is rising tremendously over the years. For that reason, it is important to maintain a proper check and balance on carbon emissions.

During this research we aim to utilize the Ornstein-Uhlenbeck (O-U) process, renowned for its ability to model mean-reverting behavior through stochastic differential equations, to serve as a powerful tool for simulating carbon emissions.

An important point to note is that while the O-U process inherently aims to maintain emissions near their mean, the stark reality of contemporary trends reveals a significant divergence from desired targets. This discrepancy underscores the complex interplay of technological advancements and socioeconomic factors driving emissions upwards, challenging the conventional application of mean-reverting models to effectively manage and mitigate the pressing environmental challenges of our time. (a) In the first phase of our modeling approach, we employ smart optimization tools to fit the model to this increasing data. (b) Once the model is calibrated, future predictions can be generated by adjusting the mean-reverting rates. This approach enables users and policymakers to focus on promoting mean reversion, thereby striving to control emissions at levels that facilitate reversibility of the current upward trajectory. A schematic of our research strategy is provided in Fig. 1. 

Fig. 1
figure 1

Steps in Analyzing CO2 Emissions Using the Ornstein–Uhlenbeck Model

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There is a noticeable lack of research on the application of the O-U process, despite the abundance of literature on carbon emissions. This study will fill this vacuum by providing a fresh strategy that blends sound statistical modeling with useful policy recommendations. In the next section, we will outline the modeling process, optimization steps, and present the parametric thresholds in the results. Following this, we will engage in a discussion on how these parameter values can inform and guide the formulation of essential policies for ensuring the sustainable future of highly impacted countries and the global community at large.

Materials and Methods

Mathematical Model

The Ornstein-Uhlenbeck (OU) process is a prototype of a random process characterized by continuous sample paths, similar to the Wiener process denoted by the symbol Wt.

Ornstein-Uhlehnbeck continuous time models were introduced to explore the stochastic processes by Uhlenbeck & Ornstein in 1930 and are now used extensively to address the climate action challenges (Turvey et al. 2023; Lashof 1989; Giebel and Rainer 2011). The evolution of the state variable over time in an Ornstein-Uhlenbeck (O-U) process is characterized by two key components: mean reversion and random fluctuations. The mean reversion aspect of the O-U process means that the variable tends to move towards a central or equilibrium value. However, the process is also influenced by random fluctuations, or noise, which cause the variable to deviate from its mean even when it is close to the equilibrium.

In many basic finance problems, for example those involving stock prices or interest rates, the accumulation over time follows a pattern that is modeled as normally distributed, where the variance scales with the duration and the expected value adjusts accordingly. Some useful examples can be found from the work of Shreve (2005); Wilmott 1998) and (Hull and Basu 2016) (chapter 11) which discusses these results in detail.

Let us derive the Ornstein-Uhlenbeck (O-U) process for carbon emissions, represented by Yt, which incorporates a drift towards an equilibrium state, originating from a Wiener process.

Therefore, our research problem is considered as a Gaussian Markov process defined as Y = {Y (t), t ≥ 0} with a parameter α > 0 and an initial value Y0. This perspective is particularly relevant for modeling the stochastic data of carbon emissions because it allows us to capture the continuous fluctuations in emissions over time.

Mathematically, it is described by the stochastic differential equation:

$${dY}_{t}=\alpha (\mu -{Y}_{t}) dt + {\eta }_{0} {dW}_{t}$$
(1)

where µ represents the equilibrium level, α controls the rate of reversion to µ, η0 is the volatility parameter, and Wt denotes the Wiener process.

Assuming α > 0, Y0 is Gaussian. Under the assumption that the process exhibits a mean-reverting behavior towards an equilibrium point and in the absence of boundaries, the unrestricted solution to equation 1 can be expressed as:

$${Y}_{t}={e}^{-at}{Y}_{0}+\mu \left(1-{e}^{-at}\right)+\eta_0 {\int }_{0}^{t}{e}^{-a(t-s)}{dW}_{s}$$
(2)

where the autocorrelation function of this stationary Gaussian process decays exponentially over time. The moments of mean, variance, and covariance are derived as follows. These moments can be used to estimate parameters α and η0. Note that the process of integration from equation 1 to equation 2 is not straightforward but involves the itoˆ’s Lemma and isometry and interested readers are referred to the pioneering works of Uhlenbeck and Ornstein (1930); Cox et al. 2005; Oksendal 2013). Furthermore, for the purpose of knowledge sharing, we have provided the detailed derivation in the supplimentry file.

For the mean-reverting O-U process, the moments are defined as follows:

$$E({Y}_{t}) = {Y}_{0}{e}^{-\alpha t}+\mu (1-{e}^{-\alpha t})$$
(3)
$$\text{Var}\left({Y}_{t}\right)=\frac{{\eta }^{2}_0}{2\alpha}(1-{e}^{-2\alpha t})$$
(4)
$$\text{Cov}\left({\text{Y}}_{t}, {\text{Y}}_{s}\right)=\frac{{\eta }^{2}_0}{2\alpha}({e}^{-\alpha \left|t-s\right|}-{e}^{-\alpha |t+s|})$$
(5)

Working with these process moments provides a convenient way to determine parameters such as α, η0, and µ for s < t.

It is worth mentioning that the pioneering work of Gillespie (Gillespie 1996) has the potential to solve basic O-U- processes very effectively. Furthermore, these methods can serve as useful starting point for the qualitative understanding of the processes that follow Brownian trends or beyond that (Blomberg et al. 2020).

Results and Discussion

Fitting of an Ornstein-Uhlenbeck (O-U) process to the given dataset can help to perform future measures more accurately. The OU process is commonly used to model mean-reverting processes in various fields, including finance and biology. The code uses the least squares curve fitting function lsqcurvefit to estimate the parameters of the O-U process that best fit the dataset.

The O-U process of carbon emission data is described by the stochastic differential equation (SDE) 1 and its solution as presented in equation 2. When the state variable is the carbon emissions then the parameters of the model can be depicted as presented in table 1.

Table 1 Parameters for Ornstein–Uhlenbeck Process in Carbon Emissions Modeling
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The parameters were initialized with initial guesses to define the O-U process function using the data. After fitting the O-U process, the code simulates the process with reduced noise and plots the observed and simulated data.

The scaling factor is applied to reduce the noise level in the simulation. As depicted in Figure 2, we observe how effectively the fitted Ornstein-Uhlenbeck (O-U) process captures the underlying dynamics of the dataset.

Fig. 2
figure 2

Carbon emission data and stochastic model approximation. The difference in amplitudes reflects stochasticity and reversion

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In Figure 3, it is evident that after a few iterations, the numerical results converged closely to the observed data, demonstrating the effectiveness of the trained model.

Fig. 3
figure 3

Carbon emission data optimizer performance

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Optimization process helps to refine predictive accuracy and model fidelity as evident from the literature (Sohail 2023; Wang et al. 2024). The optimization algorithm (Levenberg-Marquardt algorithm) aims to minimize the objective function, which is the sum of squared differences between the observed values and the values predicted by the OU process model. This approach ensures that the model aligns closely with observed data, enhancing its predictive power and reliability for real-world applications.

Mathematically, the objective function can be expressed as:

$$\text{minimize }f(x)=\sum_{i=1}^{n}{({y}_{i}-F(x, {t}_{i}))}^{2}$$

where n is the number of data points, yi is the observed value at time ti, F (x, ti) is the predicted value by the OU process model at time ti with parameters x.

The Levenberg-Marquardt algorithm iteratively updates the parameter vector x to minimize the objective function. At each iteration, it computes the Jacobian matrix of the model function with respect to the parameters, ∇F (x, ti), and the Hessian matrix, H, which approximates the local curvature of the objective function. The details of the optimization technique can be found from Nocedal and Wright (1999); Press 2007) and the cross references.

The parameter update at each iteration is given by:

$${x}_{\text{new}}={x}_{\text{old}}-{\left({J}^{T}J+\uplambda I\right)}^{-1}{J}^{T}(F\left({x}_{\text{old}},t\right)-y)$$
(6)

where J is the Jacobian matrix, λ is the damping parameter, I is the identity matrix, y is the vector of observed values.

The damping parameter λ is adjusted at each iteration to balance between the steepest descent direction and the Newton direction. This helps to avoid large updates that may lead to divergence.

By iteratively updating the parameter vector x, the algorithm converges to a local minimum of the objective function, providing the optimal parameters for the O-U process model that best fit the observed data.

Our optimization approach serves a dual purpose. Initially, it aims to fit our model to real-world data on carbon emissions. Subsequently, our focus shifts to incorporate these optimized parameters to predict future emission scenarios. Our future work entails expanding this model using machine learning tools. These tools will enhance our ability to devise more effective control strategies, thereby promoting sustainability. Such advancements are crucial for achieving global health objectives, including initiatives that advocate for renewable energy, energy efficiency, sustainable transportation, reforestation, and climate resilience.

Discussion

The application of the Ornstein-Uhlenbeck (O-U) model using carbon emission data from 1990 to 2020 in case of Pakistan (source of data is provided) shows deep insights into the trends of carbon emissions over the time period. O-U model, which is famous for its mean reverting characteristics efficiently captures fundamental dynamics of carbon emissions, initially exhibiting increasing trends then following the slight decline in later years.

The early years of Pakistan’s industrialization and demographic expansion have seen a notable rise in carbon emissions, primarily driven by intensified energy consumption and rapid industrial growth. The Pakistan Economic Survey, published annually by the Ministry of Finance, Government of Pakistan, provides comprehensive insights into the country’s energy consumption patterns, industrial development, and demographic trends, all of which influence carbon emissions trajectories. Reports from the World Bank delve into Pakistan’s energy sector and industrial development, highlighting environmental impacts and offering insights into carbon emissions trends. Additionally, the International Energy Agency (IEA) publishes annual reports that include specific analyses of Pakistan’s energy landscape, contributing further clarity on carbon emissions dynamics. Academic journals such as ’Energy Policy,’ ’Environmental Research Letters,’ and ’Journal of Cleaner Production’ feature empirical studies and data analyses that explore carbon emissions trends in Pakistan. Government publications from the Ministry of Climate Change, Ministry of Energy, and Ministry of Industries also provide valuable data and analyses on carbon emissions and related trends. Furthermore, the United Nations Environment Programme (UNEP) offers global and regional perspectives on climate change and environmental issues, including assessments of Pakistan’s carbon emissions and associated factors. some useful recent studies includes (Raza and Dongsheng 2023; Rehman et al. 2021; Uzair Ali et al. 2022).

The early years of Pakistan shows a continuous increasing trend in carbon emissions, basically due to increased energy consumption, rapid shift in industrialization and increase in population. Conversely, the later years show fluctuations i.e. intermittent rise and fall in carbon emissions were observed. The reasons behind the fluctuations are based on several activities done in these years, such as implementation of China Pakistan Economic Corridor causes carbon emissions to increase due to increased industrial activities. However, decrease in carbon emissions may be due to the economic recession or due to the implementation of policies to mitigate climate change impacts. Furthermore, it is evident from these fluctuations that Pakistan is well aware of climate risk and emergency and showing serious efforts to mitigate climate impacts by adopting green technologies to reduce the carbon emissions.

Indeed, we find that O-U model’s state-dependent volatility term does an efficient job of accounting for these large fluctuations, which suggests that while there is an overall increasing trend of carbon emissions, various shocks cause the series to deviate from this trend. This finding is important to our understanding of the issue, because it shows that whether or not policymakers must act to ameliorate the fluctuations. Through identifying the root reasons of emission variation, policymakers can employ more stable and effective strategies to mitigate carbon emissions in a long-term perspective.

Conclusions and Future Work

Discussion pertaining to the application of Ornstein-Uhlenbeck model on carbon emissions data of Pakistan form 1990–2020 allows us to interpret complete emission trends and underlying determinants generating variability in it. As a result, the mean-reverting feature of our model and the fact that volatility is state-dependent, greatly enriches our insights into the dynamics of carbon emissions. They also highlight the urgent requirement of policy interventions to prevent further acceleration in emissions and normalize the variations. The study emphasizes on cleaner technologies, energy efficiency and strict environmental standards for meeting sustainable development objectives. Utilising the lessons extracted from the O-U model, policymakers could construct better policies to control carbon emissions and thereby help in reducing carbon in relation global climate efforts.

Future work should focus on integration of machine learning with the O-U model might increase prediction performance and enable real-time emissions data analysis. Such an approach makes for the design of adaptive policies able to respond successfully to fluctuating environmental and economic conditions. Future studies should also consider the persistence of carbon emissions reduction over time due to mitigation schemes in connection with long term effects of climate change adaptation measures, population growth, urbanization and industrial globalization. Commercial-scale implementation of emerging technologies (e.g. carbon capture and storage) and their incorporation into national energy systems similar to the natural economic order needs to be investigated as an empirical mechanism for reaching substantial emission reductions. Future research can contribute to addressing these areas for combating climate emergency and achieving sustainability.