Climate-Streamflow Relationship and Consequences of Its Instability in Large Rivers of Pakistan: An Elasticity Perspective

Abstract

Precipitation elasticity provides a basic estimate of the sensitivity of long-term streamflow to changes in long-term precipitation, and it is especially useful as the first assessment of climate change impact in land and water resource projects. This study estimated and compared the precipitation elasticity (ε_p) of streamflow in 86 catchments within Pakistan over 50 major rivers using three widely used analytical models: bivariate nonparametric (NP) estimator, multivariate NP analysis, and multivariate double logarithm (DL) model. All the three models gave similar values of elasticity in the range of 0.1–3.5 for over 70–75% of the catchments. This signifies that a 1% change in the annual mean precipitation compared to the long-term historic mean annual precipitation will amplify the streamflow by 0.1–3.5%. In addition, the results suggested that elasticity estimation of streamflow sensitivity using the multivariate DL model is more reliable and realistic. Precipitation elasticity of streamflow is observed high at altitudes ranging between 250 m and 1000 m while the longitudinal and latitudinal pattern of ε_p shows higher values in the range of 70–75 and 32–36 decimal degrees, respectively. The ε_p values were found to have a direct relationship with the mean annual precipitation and an inverse relationship with the catchment areas. Likewise, high ε_p values were noticed in areas where the mean annual temperature ranges between 15 and 24 °C.

1. Introduction

According to the Intergovernmental Panel on Climate Change (IPCC) (2018), the magnitude of the global mean surface temperature has increased by 1.0 °C, and the increase is expected to reach 1.5 °C by year 2030–2052 if human activities responsible for global warming continue at the current rate [1]. Global warming is noticed at the global scale and has caused increasing vulnerability to human settlements worldwide; this could be due to an increase in the frequency and intensity of meteorological events, high temperature, or rising sea levels [2]. Global warming is responsible for intensifying the hydrological cycle, which consequently causes more frequent and intense drought and flood events in response to drier soil conditions and higher humidity [3].

Climate change studies allude that variability in hydrological systems will affect important sectors, including hydropower generation, water supplies of households, and irrigation, as well as industrial demands [4,5,6]. Streamflow alteration and subsequent change in long-term averages, seasonality, and extremes (e.g., floods and droughts) may affect water security, which is a major concern in many watersheds across the globe [7]. Similarly, a lot of studies confirm that South Asia is suffering from climate change which will cause severe threats to natural environments and water resources of South Asia [7,8,9,10,11]. The Indus basin, which starts in the Hindukush–Karakorum–Himalayan (HKH) territory, is highly prone to aggressive climate events and is reported to suffer from huge losses in terms of infrastructure, economics, and human lives [12]. The average surface temperature increase in the HKH territory as projected by 2100 is predicted to exceed the global average surface temperature, which will change the weather pattern and the hydrological cycle of the territory [13].

Climate change, rising temperatures, shifting precipitation patterns, and the increase in the frequency of extreme weather events have a negative impact on food and livelihood security, resulting in land degradation and increasing displacement [1,14,15]. Fifteen percent of people globally feel climate hazards constitute the greatest risk to their safety [16]. Women, the young, the old, and the impoverished are the most disadvantaged and vulnerable to the effects of climate change in the least developed countries [1,14]. Pakistan is one of the top nine countries most vulnerable to climate change [16]. Between 1999 and 2018, Pakistan was classified as the world’s fifth most afflicted country by extreme weather events [17]. Pakistan’s economy relies heavily on agriculture, and any changes in temperature and disruptions in water availability and monsoon patterns can wreak havoc on the livelihoods of millions of people [18]. Climate change and extreme weather events worsen the country’s already serious poverty and food security challenges. From 1998 to 2018, Pakistan witnessed 152 extreme weather events, lost 9989 lives, and suffered economic losses worth $3.8 billion [19].

High uncertainty and vulnerability of water resources in the context of climate change have become a popular research area and are considered as a burning issue. Many hydrological studies are available that assessed streamflow sensitivity in response to climate variables, particularly precipitation and evapotranspiration [7,20,21,22,23,24]. A large share of these studies utilized suitable hydrological models by calibrating input parameters against historical streamflow data to foresee the resulting changes in water assets and the future streamflow of the region [25,26,27,28,29]. Many scholars worked on the quantification of water assets of Pakistan with the primary purpose of seeking the impact of shifting climatic conditions upon its water resources [5,12,13,30,31,32,33,34]. Overall, the above studies were mostly conducted for the Upper Indus basin (UIB), utilizing a suitable modelling technique, e.g., the snow runoff model (SRM), Soil and Water Assessment Tool (SWAT), Hydrologiska Byrans Vattenbalansavdelning (HBV) model, and water and energy budget-based distributed hydrological model (WEB-DHM). The choice of the modelling technique is relatively more reliable in giving estimates of streamflow sensitivity subject to proper calibration of a suitable model [35]. Hydrologic modelling requires accurate precipitation data at a high spatial resolution, which is often limited in many regions of the globe [7,36]. Moreover, the main problem with the modelling approach is the presence of outliers and the requirement for a continuous and comprehensive historical record of different climatic and non-climatic parameters [37]. Schaake (1990) was the first to introduce to the scholarly world the concept of elasticity in the estimation of streamflow sensitivity [25]. He reported a 20% increase in the annual streamflow of the Animas River at Durango, Colorado, by keeping temperature and potential evapotranspiration constant. The concept of elasticity is very simple and can be described with a ratio between the proportional changes occurring in the streamflow (Q) to the corresponding proportional changes occurring in any climate variables, i.e., precipitation (P), temperature (T), evapotranspiration (ET), etc. Schaake (1990) represented precipitation elasticity as follows:

$${\mathsf{\epsilon }}_{\mathrm{p}}\left(\mathrm{P},\mathrm{Q}\right)=\frac{\frac{\mathrm{dQ}}{\mathrm{Q}}}{\frac{\mathrm{dP}}{\mathrm{P}}}=\frac{\mathrm{dQ}}{\mathrm{dP}}\frac{\mathrm{P}}{\mathrm{Q}}$$

(1)

Sankarasubramanian (2001) highlighted that elasticity values are often estimated using a suitable model, and it is always difficult because the model structure in the majority of cases is unknown, and validation is always a basic problem in such models [38]. This uncertainty can be reduced by directly using historical climate and discharge data by employing an NP estimator [38].

Subsequently, many researchers utilized the climate elasticity concept for measuring precipitation elasticity of streamflows (both via bivariate and multivariate approaches) using an NP estimator or regression coefficients for the quantification of water resources in a given country/region and successfully analysed the climate change impacts in a given country/region using precipitation elasticity [6,29,35,38,39,40,41,42,43,44,45]. Numerous studies made a comparison of climate elasticity with other popular available models for streamflow sensitivity and found a robust coherence between them [40,44,46,47]. Similarly, Fu et al. (2007) studied the impacts of climate variability upon the streamflow in the Spokane River basin in the United States of America and the Yellow River basin in China by using two parameters, i.e., precipitation (P) and temperature (T) [48].

It is well-understood that hydrological data suffer from various sources of uncertainty even under the most rigorous measurement settings. The absence of a complete understanding of the hydrological phenomena and processes involved causes hydrological uncertainty. The hydrological cycle is primarily driven by precipitation, and the hunt for consistent and precise worldwide precipitation estimates is, for the most part, a story of compromise [49]. Every dataset has strengths and weaknesses that are inextricably linked [50]. Ground-based precipitation measurements, such as rain gauge and radar networks, are either few or non-existent in many parts of the world, including in the developing countries, owing to the high costs of constructing and maintaining the infrastructure. This problem is worsened in areas with complex topography, where precipitation has a high degree of spatiotemporal unpredictability [51]. Thus, in complex terrain regions, precipitation estimates can be associated with significant errors due to variability and uncertainty introduced by orographic effects [51,52,53]. Precipitation over various types of terrain has long been recognized as having a significant impact on local weather [54,55,56], as well as on the interaction between land surface and atmosphere, which influences large-scale atmospheric circulation and even global climate [57,58,59,60].

In this study, an effort was made to utilize ground-based observation stations for climate data instead of satellite-based stations because the latter ones are more susceptible to errors and need proper calibration and correction factors before use in climate change research [61,62,63]. Since Pakistan is a developing country where datasets for many meteorological factors for conducting the streamflow sensitivity analysis using a hydrological model at the country level are not available, which forced the authors to use the elasticity approach to bring streamflow sensitivity at the country level to the forefront. Moreover, based on the available literature [5,12,13,30,31,32,33,34], it is believed that there had been no research to gauge the potential climate change effects upon the water resources of Pakistan on a large scale using analytical models, i.e., climate elasticity models. This research study aimed to suggest that naïve utilization of precipitation elasticity of the streamflow without wise consideration of the precipitation–streamflow relationship yields false, deceptive, improbable, and impractical results. Additionally, our purpose of carrying out this study was to devise a robust and low-biased estimator for gauging stream sensitivity to climate change that can provide reliable results of streamflow sensitivity.

2. Materials and Methods

2.1. Study Area

This research was carried out on 86 catchments with a streamflow monitoring station at their outlets (Table A1 of Appendix A), 48 precipitation and 34 temperature monitoring stations (Table A2 of Appendix A) covering 50 major and minor rivers of Pakistan and their main tributaries (Figure 1). The sub-basins of the study area are shown in Figure A1 of Appendix A.

Figure 1. Study area map showing location of the streamflow and meteorological gauging stations.

The Köppen–Geiger climate classification system can be used to better understand the climatic conditions of the study area. S. Sarfaraz et al. (2014) successfully produced Köppen–Geiger climatic zones of Pakistan by using the 30-year monthly normal area-weighted precipitation and temperature dataset of 59 meteorological sites well-spread across Pakistan. The climatic variables used in the Köppen–Geiger system were calculated at each of the 59 meteorological stations. The result clearly manifests that the climate of more than three-fourths of Pakistan is arid or semiarid (central and southern Pakistan). It is characterized by high temperatures and low rainfall. About 17% of the meteorological stations used in the study are in the temperate climate (submountain areas in the north), and just over 5% fall under the cold-type climate (in northeastern Pakistan, three GB stations are in the D type climate). S. Sarfaraz et al. (2014) concluded that, in total, the calculated Köppen climate classes across Pakistan come out to be 12 classes as shown in Figure 2 [64].

Figure 2. Pakistan climate classification map based on the Köppen climate classification system showing the spatial distribution of 12 Köppen climate classes with the dominant one being BWhw, followed by the rest [64].

The primary focus of this study is the Indus River basin in Pakistan. The Indus River basin, which is ranked as one of the mightiest basins of the world, covers areas of Afghanistan, China, Pakistan, and India. Pakistan contributes 56% of the total area of the Indus basin, which is the largest amongst all the other neighbouring countries [65]. The Indus River basin in Pakistan covers 520,000 km², which is 65% of the total area of Pakistan [66]. The climatic conditions of the Indus basin features high variability, from subtropical arid and partial arid to moderate subhumid over the plain areas of two provinces, Sindh and Punjab [67]. The historical record shows an annual precipitation in the range of 100–500 mm in the plain areas compared to the highest value of 2000 mm on alpine slopes [67]. Snowfall is the major source of river runoff at higher elevations of almost 2500 m [68].

The main source of revenue generation in the economy of Pakistan is reliant on agriculture, which depends upon the water resources of the Indus River [33]. The increase in population and industrial growth has caused a drop in water availability from 5600 cubic meters in 1947 to 1017 cubic meters per capita in 2015, which is anticipated to further decrease under the existing infrastructure and organizational conditions [69]. The majority of the water demand of Pakistan is fulfilled through the Indus River and its contributing tributaries, for which the primary source of feeding are precipitation and snowmelt in the HKH mountainous region [70].

2.2. Datasets Collection

The research objectives were achieved with the help of the river’s mean annual streamflow data, mean annual precipitation data, mean annual temperature data, and geospatial datasets. Geospatial datasets of the digital elevation model (DEM) were obtained from the USGS website [71]. The Shuttle Radar Topography Mission (SRTM) DEM was downloaded in 30 × 30 m resolution. Similarly, hydroclimatic datasets include data on streamflow, precipitation, air temperature, etc. The annual streamflow data are made available from the Surface Water Hydrology Project of the Water and Power Development Authority (WAPDA) and the Global Runoff Data Centre (GRDC). A summary of the data that provides resolution (temporal and spatial) and sources of the data is given in Table 1. Based on the available record, the mean annual streamflow data was acquired for different durations at different stations from 1963 to 2009. In this study, 86 stations were chosen across different rivers, keeping in mind the maximum data availability at a particular flow station. The flow station locations are the outlets of catchments. The details of all these catchments along the different rivers of the study area are given in Appendix A of the manuscript. The datasets of annual precipitation and temperature were acquired from the Pakistan Meteorological Department (PMD). Precipitation data were obtained from 47 meteorological stations, while temperature data—from 34 stations within the study period, i.e., in 1963–2009. For precipitation, every catchment was to have at least one precipitation station within its boundaries contributing to Thiessen weighting at a distance of not more than 200 km in plain areas and 150 km in hilly areas [72], although for the majority of the catchments, the distance is less than 100 km from the precipitation station. For temperature data, in plain areas, every catchment was to have at least one temperature gauging station at a distance of 300 km in the vicinity of the catchment boundary contributing to Thiessen weighting [72]. Again, here, for the majority of the catchments, the distance is less than 100 km from the temperature station because the data of all the 34 temperature stations were acquired at the same weather stations at which the precipitation data were acquired.

Table 1. Summary table indicating data resolution (temporal and spatial) and sources of the data.

S. No.	Data Type	Resolution (Temporal/Spatial)	Source
1	Precipitation data	Annual data	Pakistan Meteorological Department (PMD) [73]
2	Temperature data	Annual data	Pakistan Meteorological Department (PMD) [73]
3	Discharge data	Annual data	1. Water and Power Development Authority (WAPDA) [74] 2. Global Runoff Data Centre’s (GRDC) website [75]
4	Spatial data (digital elevation model (DEM) data)	30 × 30 m	USGS Website [71]

2.3. Data Preparation

The annual mean values of streamflow for the available record at each catchment outlet were computed. Similarly, the annual mean values of precipitation and temperature were calculated for all the selected stations and are given in Table A2 of Appendix A.

These values were arranged in a proper format and set ready for the application of a suitable interpolation technique in ArcGIS. The ArcGIS 10.2 platform provides several interpolation techniques that can be used for interpolating climate variables. Many researchers utilized different interpolation techniques for different climate parameters [23,40,48]. For this study, we estimated the basin-averaged precipitation by applying the Thiessen polygon method to the subbasin [71,72,73,74,75,76], whereas an inverse distance-weighted (IDW) model was adopted for the interpolation of both precipitation and temperature elasticity data in ArcGIS 10.2. The interpolated annual mean time series values of precipitation and temperature were extracted for the all the 86 catchments within the study period of 1963–2009.

2.4. Data Uncertainty

The streamflow, precipitation, and temperature data were checked for data quality (missing values), which is indicated by −1 or −100 in the available data record for the streamflow, precipitation, and minimum and maximum temperature values. For this study, the historic record showed that the found missing values in each month of an individual year were fewer than 15 at all the stations. These missing values were linearly interpolated to all such stations [23]. Thus, for this study, it is believed that the influence of time series inhomogeneity on the results was very meagre.

2.5. Methods

In this study, precipitation elasticity of the streamflow was calculated analytically by using long-term hydroclimatic datasets of streamflow, air temperature, and precipitation. Here, we applied the NP bivariate elasticity model, the multivariate NP analysis model, and the multivariate DL analysis model for the estimation of elasticity through NP estimator ɛ_p.

2.5.1. NP (NP) Bivariate Model

The NP bivariate elasticity model of Sankarasubramanian et al. (2001) [28] was used for the determination of precipitation elasticity in all the 86 catchments. The NP bivariate model for streamflow elasticity is given below.

$${\mathsf{\epsilon }}_{\mathrm{p}}=\mathrm{median}(\frac{{\mathrm{Q}}_{\mathrm{t}}-\overline{\mathrm{Q}}}{{\mathrm{P}}_{\mathrm{t}}-\overline{\mathrm{P}}})\frac{\overline{\mathrm{P}}}{\overline{\mathrm{Q}}}$$

(2)

In Equation (2), variables P and Q are quite general and can be used as instantaneous, monthly, or annual values [77,78,79]. In this study, the mean annual values of streamflow Q and precipitation P for the estimation of ${\mathsf{\epsilon }}_{\mathrm{p}}$ were utilized. Where $\overline{\mathrm{P}}$ and $\overline{\mathrm{Q}}$ are the long-term historical means of time data series of the annual mean values of precipitation P and streamflow Q, respectively, at a particular catchment outlet. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ is estimated for each set of ${\mathrm{Q}}_{\mathrm{t}}$ and ${\mathrm{P}}_{\mathrm{t}}$ for an individual year in the annual time series data. The median value of all the calculated elasticity values of the available historic record at a particular catchment outlet is NP precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$. The main advantage of this relation is nonparametric and it has low biasness so this is the major advantage of this relationship.

2.5.2. Multivariate NP Analysis Model

Multivariate NP analysis calculates multiple “factor” elasticities in the form of regression coefficients as a result of the multivariate regression model. The multivariate function describes the mutual relationship within climatic variables (precipitation, temperature, humidity, land use, etc.) and streamflow Q_i (i indicates the mean flow) [39]. This can be expressed for precipitation and temperature elasticity mathematically in the form of the following equation:

$${\mathrm{Q}}_{\mathrm{i}}=\mathrm{f}\left(\mathrm{P},\mathrm{T}\right)$$

(3)

The NP multivariate model is developed by using the chain rule on Equation (3) and supposing that an absolute change in streamflow Q is a linear combination of an absolute change in precipitation P and temperature T.

$$\mathrm{dQ}=\frac{\partial \mathrm{Q}}{\partial \mathrm{P}}\mathrm{dP}+\frac{\partial \mathrm{Q}}{\partial \mathrm{T}}\mathrm{dT}$$

(4)

Inserting the absolute change in every term of Equation (4) for their difference from the mean value, we get the following:

$$\mathrm{Q}-\overline{\mathrm{Q}}=\frac{\partial \mathrm{Q}}{\partial \mathrm{P}} \left(\mathrm{P}-\overline{\mathrm{P}}\right)+\frac{\partial \mathrm{Q}}{\partial \mathrm{T}} \left(\mathrm{T}-\overline{\mathrm{T}}\right)$$

(5)

On rearrangement of Equation (5) we get the following:

$$\frac{\mathrm{Q}-\overline{\mathrm{Q}}}{\mathrm{Q}}=\frac{\partial \mathrm{Q}}{\partial \mathrm{P}}\frac{\mathrm{P}}{\mathrm{Q}} \left(\frac{\mathrm{P}-\overline{\mathrm{P}}}{\mathrm{P}}\right)+\frac{\partial \mathrm{Q}}{\partial \mathrm{T}}\frac{\mathrm{T}}{\mathrm{Q}} \left(\frac{\mathrm{T}-\overline{\mathrm{T}}}{\mathrm{T}}\right)$$

(6)

Applying the definition of elasticity to Equation (6), we can substitute the corresponding elasticity estimator for precipitation and temperature as follows:

$$\frac{{\mathrm{Q}}_{\mathrm{t}}-\overline{\mathrm{Q}}}{{\mathrm{Q}}_{\mathrm{t}}}={\mathsf{\epsilon }}_{\overline{\mathrm{P}}}\left(\frac{{\mathrm{P}}_{\mathrm{t}}-\overline{\mathrm{P}}}{{\mathrm{P}}_{\mathrm{t}}}\right)+{\mathsf{\epsilon }}_{\overline{\mathrm{T}}}\left(\frac{{\mathrm{T}}_{\mathrm{t}}-\overline{\mathrm{T}}}{{\mathrm{T}}_{\mathrm{t}}}\right)$$

(7)

In Equation (7), ${\mathsf{\epsilon }}_{\overline{\mathrm{P}}}$ and ${\mathsf{\epsilon }}_{\overline{\mathrm{T}}}$ gives the mean “factor” elasticities of streamflow Q, where $\overline{\mathrm{P}}$ and $\overline{\mathrm{Q}}$ are the long-term historical means of the time series data of the annual mean values of streamflow Q, precipitation P, and temperature T, respectively. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and temperature elasticity ${\mathsf{\epsilon }}_{\mathrm{T}}$ were obtained as coefficients of the ordinary least squares (OLS) regression. The OLS regression was performed on the values obtained from each set of $\frac{{\mathrm{Q}}_{\mathrm{t}}-\overline{\mathrm{Q}}}{{\mathrm{Q}}_{\mathrm{t}}}$, $\frac{{\mathrm{P}}_{\mathrm{t}}-\overline{\mathrm{P}}}{{\mathrm{P}}_{\mathrm{t}}}$, and $\frac{{\mathrm{T}}_{\mathrm{t}}-\overline{\mathrm{T}}}{{\mathrm{T}}_{\mathrm{t}}}$, for one complete year time t in the time series data. During calculations of regressions, the intercept term was put unadjusted, i.e., the intercept term was taken as zero.

2.5.3. Multivariate DL Analysis Model

The multivariate DL analysis model is also employed in order to get a comparison of the precipitation elasticities obtained through different models and seeks a conclusion as to which model is the most suitable. A more recent study [39] evaluated the impact of the regional factor on streamflow Q by utilizing multivariate regression analysis. It was assumed that the effect of this regional factor on streamflow Q is a dimensionless indicator and so can be marked as factor elasticity of streamflow ${\mathsf{\epsilon }}_{{\mathrm{X}}_{\mathrm{j}}}$ as follows:

$${\mathsf{\epsilon }}_{{\mathrm{X}}_{\mathrm{j}}}=\frac{\left(\frac{\partial \mathrm{Q}}{\mathrm{Q}}\right)}{\left(\frac{\partial {\mathrm{X}}_{\mathrm{j}}}{{\mathrm{X}}_{\mathrm{j}}}\right)}$$

(8)

In Equation (8), ${\mathrm{X}}_{\mathrm{j}}$ shows the j factor (climate variable, i.e., precipitation and temperature in our case) that influences streamflow Q, where ${\mathsf{\epsilon }}_{{\mathrm{X}}_{\mathrm{j}}}$ represents a ratio of proportional change in streamflow to proportional change in ${\mathrm{X}}_{\mathrm{j}}$. Considering the functional form of Equation (3), we modified the equation introduced by Tsai [39] for evaluating the precipitation elasticity of streamflow as follows:

$$\mathrm{Q}={\mathrm{P}}^{{\mathsf{\beta }}_{\mathrm{P}}}{\mathrm{T}}^{{\mathsf{\beta }}_{\mathrm{T}}}$$

(9)

Taking logarithm of both sides of Equation (9), we get the following:

$$\mathrm{LogQ}={\mathsf{\beta }}_{\mathrm{P}}{\mathrm{LogP}}_{\mathrm{t}}+{\mathsf{\beta }}_{\mathrm{T}}{\mathrm{LogT}}_{\mathrm{t}}$$

(10)

where ${\mathsf{\beta }}_{\mathrm{P}}$ in Equation (8) is equal to the precipitation elasticity of streamflow ${\mathsf{\epsilon }}_{\mathrm{p}}$ and ${\mathsf{\beta }}_{\mathrm{T}}$ is equal to temperature elasticity ${\mathsf{\epsilon }}_{\mathrm{T}}$. The values of ${\mathsf{\epsilon }}_{\mathrm{p}}$ and ${\mathsf{\epsilon }}_{\mathrm{T}}$ were estimated as coefficients of the ordinary least squares (OLS) regression analysis that is performed on the values obtained for each set of Log${\mathrm{Q}}_{\mathrm{t}}$, Log${\mathrm{P}}_{\mathrm{t}}$, and Log${\mathrm{T}}_{\mathrm{t}}$ for one complete year time t in the time series data. During calculations of regressions, the intercept term was put unadjusted, i.e., the intercept term was taken as zero.

3. Results

3.1. Precipitation Elasticity ${\epsilon }_p$ and Different Models

Precipitation elasticity was calculated using all the three models as mentioned in the methodology section. It was observed that for Sankarasubramanian’s NP bivariate elasticity model, the ${\mathsf{\epsilon }}_{\mathrm{p}}$ values were observed in the range from −1.8 to +3.5 with a positive ${\mathsf{\epsilon }}_{\mathrm{p}}$ value for 77 catchments and a negative ${\mathsf{\epsilon }}_{\mathrm{p}}$ value for nine catchments. The multivariate NP analysis model resulted in ${\mathsf{\epsilon }}_{\mathrm{p}}$ values within the range of −2.8 to +3.7 with 74 positive ${\mathsf{\epsilon }}_{\mathrm{p}}$ catchments and 12 negative ${\mathsf{\epsilon }}_{\mathrm{p}}$ catchments. Similarly, the multivariate DL analysis model estimated the ${\mathsf{\epsilon }}_{\mathrm{p}}$ values within the range from −2.7 to +3.9, with 76 positive ${\mathsf{\epsilon }}_{\mathrm{p}}$ catchments and 10 negative ${\mathsf{\epsilon }}_{\mathrm{p}}$ catchments.

Two-dimensional (2D) line plots were also produced for the models stated above so as to give us a better understanding of the different precipitation elasticity models (Figure 3). It can be seen from Figure 3 that precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ of all the 86 catchments (at their outlets) are almost the same for the three employed models, i.e., the ${\mathsf{\epsilon }}_{\mathrm{p}}$ values closely matched one another at the majority of the stations.

Figure 3. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ using the three different models.

Furthermore, nearly all the three employed models showed homogeneity in estimating positive and negative elasticity values in the majority of the catchments. For all the three models, the estimated elasticity values are in the range of 0.1–3.5 for over 70–75% of the catchments. It means that 10% change causes 10–35% change in streamflow for over 70–75% of the catchments. The elasticity estimates of our study were in line with other recent studies that had been conducted on streamflow sensitivity analysis in response to precipitation elasticity for a few subbasins situated within our study area [74,75]. The results of our study are very similar to their findings, for example, Shah et al. (2021) found that 10% variation in precipitation produces 12–20% change in streamflow in six major rivers situated in Khyber Pakhtunkhwa province of Pakistan [74], while in our case, 10% change caused 10–35% change in streamflow for over 75% of the catchments. The spatial spot variation of precipitation elasticity in all the 86 catchments at their outlets is presented in Figure 4 which further clarifies the scenarios through specifying ranges for precipitation elasticity for all the three employed models.

Figure 4. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ of the streamflow: (a) NP bivariate model, (b) multivariate NP model, (c) multivariate DL model.

Pakistan is a country with complex topography where precipitation has a high degree of spatiotemporal unpredictability and precipitation elasticity estimates are of variable nature, lacking a clear trend. In general, catchments in the UIB are less sensitive to precipitation elasticity (${\mathsf{\epsilon }}_{\mathrm{p}}$ ≤ 0.5) because the precipitation in this area is usually in the form of snow, and so the proportion of rainfall contribution to the streamflow within this area is too meagre. On the other hand, elasticity values are relatively higher (${\mathsf{\epsilon }}_{\mathrm{p}}$ = 0.1–3) near the federal capital territory and the boundary between Khyber Pakhtunkhwa and Punjab provinces as these areas usually receive more rainfall annually. Similarly, the southern part of Pakistan has moderate elasticity (${\mathsf{\epsilon }}_{\mathrm{p}}$ = 0.1–1) with a few exceptions of high-elasticity catchments.

In order to get an idea of data spread and further elaborate the comparison of the three analytical models for the estimation of precipitation elasticity, we applied statistical tools, i.e., the mean (Figure 5a) and the median (Figure 5b) to the data values of ${\mathsf{\epsilon }}_{\mathrm{p}}$. The mean and median values suggest that all the three models almost equally estimated the precipitation elasticity values.

Figure 5. (a) Boxplots of the mean values of precipitation elasticity, (b) boxplots of the median values of precipitation elasticity. Boxplot A: Sankarasubramanian’s bivariate model, boxplot B: multivariate NP analysis model, boxplot C: multivariate DL analysis model.

It was observed for all the models that the ${\mathsf{\epsilon }}_{\mathrm{p}}$ values for the catchments with a consistent and longer historical record in the northern areas of Pakistan, i.e., the UIB, are generally below 0.5 (except a few stations with a shorter record and misleading results). This is because the precipitation in this area is usually in the form of snow, and thus the precipitation elasticity shows less sensitivity of the streamflow as the proportion of rainfall contribution in the streamflow within this area is too meagre. The negative values were observed for stations with a relatively shorter data span (10 or less than 10 years, which is evident from Table A3 of Appendix A) and limited streamflow anomaly ΔQ. Similarly, for the multivariate regression models the corresponding plots were checked individually during calculation for every catchment and was found that in all cases the linear regression does not give significant results. The negative values of climate elasticity and the same shortcoming of regression analysis for shorter span data is evident from climate elasticity literature [45]. Negative elasticity may also be due to the following: (a) there exist storage reservoirs in the catchments or inter-catchment transfer ahead of the catchment gauge outlet; (b) the averaging period is not long enough, i.e., the rainfall has increased but the water has not yet got to the outlet; (c) evaporation exceeding precipitation (might be due to change in land use in the catchment in terms of afforestation or increased vegetation); (d) erroneous measurement of streamflow, climate variables (e.g., precipitation, temperature, evaporation, etc.) or both.

3.2. Comparison of Multivariate NP Analysis Model and Multivariate DL Model

In order to obtain the statistical solution for investigating the best model, the statistics of the two regressions were checked and compared for identifying the best model. A variety of statistical tests are available to test the results for the goodness of fit for regressions. Tsai (2017) applied adjusted R², probability plot correlation coefficients (PPCC), and variance inflation factors (VIFs) to assess regression goodness of fit. The adjusted R² is an indicator of the overall performance of a regression model [39]. In this study, the regression of the two multivariate models, i.e., multivariate NP analysis and multivariate DL models, was tested against their adjusted R² values as shown in Figure 6a.

Figure 6. (a,b) show adjusted R² and R² boxplots for the multivariate NP analysis and multivariate double logarithm models, respectively.

Since the values of precipitation elasticity obtained by the regression of the multivariate double logarithm showed higher adjusted values, i.e., higher explanatory power, we can say that for this study, the multivariate DL results were more reliable than the multivariate NP analysis model. This statement is made more worthy by comparison of the boxplots of the R² values of the multivariate NP analysis and multivariate DL models as shown in Figure 6b. The plot suggests that the values of the multivariate DL model are more concise and are higher, approaching one, which means that it is comparatively more reliable in this case than the multivariate NP model.

3.3. Bivariate Versus Multivariate Analysis

The justification of Sankarasubramanian et al. (2001) [28] regarding the bivariate NP estimator highlights that the median values of precipitation elasticity of the streamflow calculated using an analytical model, i.e., ${\mathsf{\epsilon }}_{\mathrm{p}}=\mathrm{median}(\frac{{\mathrm{Q}}_{\mathrm{t}}-\overline{\mathrm{Q}}}{{\mathrm{P}}_{\mathrm{t}}-\overline{\mathrm{P}}})\frac{\overline{\mathrm{P}}}{\overline{\mathrm{Q}}}$, is more superior compared to a calibrated deterministic hydrological model, though later research on climate elasticity suggested that the result obtained through a single variable does not give true representation of elasticity; rather, it provides misleading information on ${\mathsf{\epsilon }}_{\mathrm{p}}$ [48]. It was mentioned that the ${\mathsf{\epsilon }}_{\mathrm{p}}$ values using a bivariate model on a single variable do not account for certain other important hydroclimatic factors and catchment characteristics like temperature, land use, humidity, slope, etc. Similarly, it was also found that the regression analysis that includes temperature improves the coefficient of determination (R²) [40]. Since all the subsequent research based on multivariate models suggests that multivariate models are more reliable than bivariate elasticity models [39,42,48,73,74,75,76,77,78,79,80,81,82], it is believed that the multivariate elasticity results of our study are more authentic than the bivariate elasticity results, although there is very small difference between the results as discussed in Section 3.1 above.

3.4. Consequences of Instability Precipitation Elasticity ${\epsilon }_p$

This section discusses the correlations of the three different analytical models, i.e., the NP bivariate elasticity model, the multivariate NP analysis model, and the multivariate DL model of precipitation elasticity against the catchment and hydroclimatic characteristics.

3.4.1. Precipitation Elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and Length of the Available Historical Record

Overall, no significant trend was observed; it can be seen from Figure 7 that in all the three models, negative and outlier behaviour of the precipitation elasticity values was obtained where the historical record was equal to or shorter than 10 years.

Figure 7. Precipitation elasticity ε_p and length of study plots: (a) NP bivariate model; (b) multivariate NP analysis model; (c) multivariate DL model.

Although negative elasticity values were also seen for few catchments where the available length of record was quite larger, their values were very small, near zero, and thus were not significant. The possibility of negative values of elasticity in the estimation of precipitation or temperature elasticity indicates that streamflow decreases with an increase in precipitation or temperature [40,41,45].

3.4.2. Precipitation Elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and Catchment Area

The catchment areas of 80% of the catchments (79 out of the 86 catchments) are less than 25,000 square km, which is evident in Figure 8. Moreover, it is depicted in the plots that elasticity shows a strong relationship with the catchment area. In all the three models, the elasticity values were found higher for smaller catchment areas compared to the larger catchment areas. This means that for smaller catchments, runoff and snowmelt water takes less time to reach the catchment outlets and thus results in higher elasticity values. Conversely, the elasticity values of larger catchment areas show relatively smaller values. This might be due to the losses caused in terms of evaporation, local reservoirs (ponds, lakes, etc.), and vegetation.

Figure 8. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ vs. catchment area plots: (a) NP bivariate model; (b) multivariate NP analysis model; (c) multivariate DL model.

3.4.3. Precipitation Elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and Mean Annual Temperature

Logically, there exists a dual relationship between precipitation elasticity and temperature. When temperature increases, evaporation increases, which causes a decrease in runoff water to join the streamflow. On the other hand, the situation is opposite in snow and glacier regions where an increase in temperature causes an increase in runoff and snowmelt and thus boosts the streamflow. The scatterplots in Figure 9 show that there exists relatively lower precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ in cold areas where the mean annual temperature is lower because of the existence of glaciers and snowfall as the main source of precipitation [28]. The lower elasticity values in cold areas are also because of less energy available for snow melting [72]. Similarly, an increasing trend of elasticity values was seen from 15 to 22 °C, followed by a decreasing trend (southern part) where the higher temperature causes a reduction in the streamflow due to evaporation.

Figure 9. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ vs. the mean annual temperature (°C) plots: (a) NP bivariate model; (b) multivariate NP analysis model; (c) multivariate DL model.

3.4.4. Precipitation Elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and Mean Annual Precipitation

It is understood that precipitation has a direct impact on streamflow sensitivity and is without any doubt the primary source of river streamflow. The same phenomenon was observed when plots showing the relationship of precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and the mean annual precipitation were produced as shown in Figure 10a–c.

Figure 10. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ vs. the mean annual precipitation (mm) plots: (a) NP bivariate model; (b) multivariate NP analysis model; (c) multivariate DL model.

It is visible from the plots that precipitation elasticity showed a relatively higher sensitivity in an increasing trend with an increase in the mean annual precipitation. Although some of the catchments in northern areas showed smaller elasticity values in spite of having a higher mean annual precipitation, this is because the precipitation usually occurs in the form of snow or accumulated snow which usually retains water and does not directly contribute to the streamflow.

3.4.5. Precipitation Elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and Mean Annual Streamflow

The mean annual streamflow is dependent on several catchment characteristics like catchment’s slope, terrain, size, shape, altitude, vegetation, land use, etc. Similarly, rainfall intensity, frequency, distribution, and air temperature also significantly affect streamflow, and thus precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$. The plots shown in Figure 11 demonstrate a clear understanding of the probable relation of precipitation elasticity and the mean annual streamflow. It is obvious from the plots that relatively higher elasticity values were found in the catchments with lower mean annual flows. The ${\mathsf{\epsilon }}_{\mathrm{p}}$ values are generally lower than 1.0 where the streamflow is higher [72]. Higher ${\mathsf{\epsilon }}_{\mathrm{p}}$ values were mostly found for smaller catchments where runoff water reaches the gauging station faster. As a result, the streamflow sensitivity becomes high due to less time of concentration and smaller losses in the form of infiltration, inundation, interception, evaporation, etc.

Figure 11. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ vs. the mean annual streamflow (cusecs) plots: (a) NP bivariate model; (b) multivariate NP analysis model; (c) multivariate DL model.

3.4.6. Precipitation Elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and Altitude

The altitude is an important factor in precipitation elasticity and sensitivity of the streamflow as precipitation patterns and air temperature substantially vary with the altitude of a given region. The plots presented in Figure 12 reveal that the ${\mathsf{\epsilon }}_{\mathrm{p}}$ values initially increased with altitude and reached the highest level at an altitude of 250–1000 m because precipitation is more likely at higher altitudes due to a higher chance of lower temperature and more condensation [28,72]. With a further increase in altitude, the values follow a declining trend, which is an indicator of snow and glacier zones in the northern parts of Pakistan, particularly the UIB.

Figure 12. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ vs. altitude (m) plots: (a) NP bivariate model; (b) multivariate NP analysis model; (c) multivariate DL model.

3.4.7. Precipitation Elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ and Spatial Trends

It was observed from the plots in Figure 13 that longitude-wise, higher elasticity values were found between 70 and 75 decimal degrees, while latitude-wise, higher elasticity values were found between 32 and 36 decimal degrees.

Figure 13. Precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ vs. longitude and latitude plots: (a,d) NP bivariate model; (b,e) multivariate NP analysis model; (c,f) multivariate DL model.

This spatial trend of precipitation elasticity is further elaborated by interpolating the elasticity values by using the inverse distance weighting (IDW) technique for the NP bivariate model, the multivariate NP analysis model, and the multivariate DL model in Figure 14a–c, respectively.

Figure 14. Spatial trend of ε_p: (a) NP bivariate model, (b) multivariate NP analysis model, (c) multivariate DL model, (d) monsoon precipitation trend (Hanif et al., 2013 [80]).

The streamflow sensitivity obtained using the three employed models in this study is reinforced by the almost matching results of another study for Pakistan [80] with approximately the same study period, i.e., 1951–2010 (Figure 14d).

Almost 60% of the total mean annual water is contributed by headwaters of the Indus basin, out of which approximately 80% of the annual total water joins the system from June to September every year, which is called the monsoon season in Pakistan [70].

By comparing the precipitation elasticity maps in Figure 14a–c with the monsoon rainfall trend map as shown in Figure 14d (Hanif et al. (2013) [80]), a close resemblance was observed among the areas of higher precipitation elasticity and the areas with higher monsoon rainfall. Since rainfall is the most important and governing climate parameter that contributes to river flows, it is more likely that areas receiving more precipitation will possesses higher streamflow sensitivity due to greater runoff generation and might yield high ${\mathsf{\epsilon }}_{\mathrm{p}}$. The results of this study show higher sensitivity in areas where the monsoon rainfall intensity is higher and vice versa which proves the authenticity of this study and elasticity models.

3.5. Temperature Elasticity ${\epsilon }_T$

In addition to precipitation elasticity, temperature elasticity was also evaluated using three models i.e., Sankarasubramanian’s NP bivariate model, the multivariate NP analysis model, and the multivariate DL model to check the response of the streamflow to the mean temperature. Temperature elasticity ${\mathsf{\epsilon }}_{\mathrm{T}}$ estimates for all the three models are shown in Figure 15.

Figure 15. Comparison chart showing temperature elasticity as obtained using Sankarasubramanian’s NP model, the multivariate NP analysis model, and the multivariate NP DL model.

It was observed that ${\mathsf{\epsilon }}_{\mathrm{T}}$ ranged between −17.9 and +16 for Sankarasubramanian’s NP bivariate model and the multivariate NP analysis model, while for the multivariate double logarithm model, the ${\mathsf{\epsilon }}_{\mathrm{T}}$ values were in the range of +2.3–+4.7. In the case of temperature elasticity, there exist large variations of the maximum and minimum values between the multivariate DL model and the other two models, i.e., Sankarasubramanian’s NP bivariate model and the multivariate NP analysis model. The linear trend in the values of the double logarithm was due to the log transformation behaviour which smoothened the variation in regression. Sankarasubramanian’s NP bivariate model and the multivariate NP analysis model showed relatively similar results at the majority of the catchments.

The estimations of Sankarasubramanian’s NP bivariate elasticity model and the multivariate NP analysis model suggest that about 65% of the catchment showed negative values of ${\mathsf{\epsilon }}_{\mathrm{T}}$. This means that the increase in temperature caused a decrease in the streamflow, which is logical as the increase in temperature accelerates the evaporation process and results in a decreased streamflow. Overall, the values obtained using all the three models, which are comprised ofillogical and unrealistic values. Thus, the results of temperature are not reliable and are misleading. Furthermore, the existing literature also suggests that there is no significant impact of residual temperature on the streamflow compared to the direct and much more significant impact of precipitation on the streamflow, all because of the opposite correlation between precipitation and temperature [76,77,78,79,80,81,82].

3.6. Recommendations Regarding Water Management and Policy-Making Based on Elasticity

Although in this study only a relatively straightforward targeting approach was undertaken, the results of the various comparisons made in the study point to the daunting challenges that will exist in the future for developing and implementing watershed management plans that are effective in improving water management practices in stream systems throughout the country. Generally, the elasticity value is an indicator of sensitivity of the streamflow. The higher the elasticity value, the higher the sensitivity, and vice versa. Consequently, catchments having higher elasticity values are prone to aggressive climate events in the form of flash floods, and thus the existing infrastructure needs proper design to protect the inhabitants and flora and fauna of the catchments against the expected flood risks. Similarly, a lower elasticity value is an indicator of drought, and policymakers need to adopt necessary actions for coping with the drought situation through water management techniques. The smaller catchments were found to be more sensitive, with higher elasticity, and so the water supply schemes and cultivable agricultural land are more susceptible to flooding events and calamities; thus, best management practices must be ensured in all such areas.

4. Conclusions

The design, planning, and management of various preliminary hydrological studies require annual runoff volume for watersheds. For such purposes, regional methods that link streamflow to climate characteristics can offer a better solution. This study presents the estimates of precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ of the streamflow in 86 catchments of Pakistan using the NP bivariate model, the multivariate NP analysis model, and the multivariate DL analysis model. Based on the results of statistical tests, it was concluded that the higher explanatory power of the multivariate DL model suggests that it gave more reliable values of precipitation elasticity ${\mathsf{\epsilon }}_{\mathrm{p}}$ compared to Sankarasubramanian’s NP bivariate elasticity model and the multivariate NP analysis model within the study area.

Additionally, all the employed models showed relatively similar results indicating elasticity in the range of 0.1–3.5 (observed in almost 70% of the total catchments using the multivariate NP analysis model and 75% of the catchments for both Sankarasubramanian’s NP bivariate elasticity model and the multivariate double logarithm model). Precipitation elasticity of the streamflow is defined as the percentage of change in the mean annual streamflow for a given percentage change in the mean annual precipitation. This means that a 1% change in precipitation with respect to long-term historic mean annual precipitation will change the streamflow by ${\mathsf{\epsilon }}_{\mathrm{p}}$%, i.e., by 0.1–3.5% in our case. Similarly, if this change is assumed, a 10% change with respect to long-term mean annual precipitation will amplify the streamflow by 1–35%.

The study further revealed that the elasticity estimates of the catchments having a shorter historical record, i.e., usually less than 10 years, yielded misleading values and showed an outlier behaviour, i.e., either overestimating or underestimating the elasticity. Similarly, it was found that ${\mathsf{\epsilon }}_{\mathrm{p}}$ is relatively higher at an altitude ranging between 250 and 1000 m and at the catchments where the mean annual temperature is relatively high, i.e., from 15 °C to 22 °C. The longitudinal and latitudinal pattern of ${\mathsf{\epsilon }}_{\mathrm{p}}$ showed high elasticity in the range from 70 to 75 and from 32 to 36 decimal degrees, respectively. Furthermore, the precipitation elasticity was found to have a direct relationship with the mean annual precipitation and an inverse relationship with the catchment areas. The study also found that the temperature elasticity values in the majority of the catchment areas were not significant and showed outlier or unrealistic behaviour, and thus the results of temperature elasticity cannot be significantly utilized in analysing streamflow sensitivity; however, it improved the results of precipitation elasticity in multivariate approaches.