It is a common task to partition the synergistic impacts of drivers in the environmental sciences. However, there is no mathematically precise solution to this partition task. Here I present a line-integral-based method, which addresses the sensitivity to the drivers throughout the drivers' evolutionary paths so as to ensure a precise partition. The method reveals that the partition depends on both the change magnitude and pathway (timing of the change) but not on the magnitude alone unless used for a linear system. To illustrate this method, I applied the Budyko framework to partition the effects of climatic and catchment conditions on the temporal change in the runoff for 19 catchments from Australia and China. The proposed method reduces to the decomposition method when assuming a path in which climate change occurs first, followed by an abrupt change in catchment properties. The proposed method re-defines the widely used sensitivity at a point as the path-averaged sensitivity. The total-differential and the complementary methods simply concern the sensitivity at the initial and/or the terminal state, so they cannot give precise results. Although the path-averaged sensitivities varied greatly among the catchments, they can be readily predicted within the Budyko framework. As a mathematically accurate solution, the proposed method provides a generic tool for conducting quantitative attribution analyses.

The impacts of certain drivers on observed changes of interest often require quantification in environmental sciences. In the hydrology community, both climate and human activities have posed global-scale impact on hydrologic cycle and water resources (Barnett et al., 2008; Xu et al., 2014; Wang and Hejazi, 2011). Diagnosing their relative contributions to runoff is of considerable relevance to the researchers and managers. Unfortunately, performing a quantitative attribution analysis of runoff changes remains a challenge (Wang and Hejazi, 2011; Berghuijs and Woods, 2016; Zhang et al., 2016); this is to a considerable degree due to a lack of a mathematically precise method of decoupling synergistic and often confounding impacts of climate change and human activities.

Numerous studies have detected the long-term variability in runoff and attempted to partition the effects of climate change and human activities through various methods (Dey and Mishra, 2017); these include the paired-catchments method and the hydrological modeling method. The paired-catchment method can filter the effect of climatic variability and thus isolate the runoff change induced by vegetation changes (Brown et al., 2005). However, this method is capital intensive; moreover, it generally involves small catchments and experiences difficulties when extrapolating to large catchments (Zhang et al., 2011). Physically based hydrological models often have limitations such as a high data requirement, labor-intensive calibration and validation processes, and inherent uncertainty and interdependence in parameter estimations (Binley et al., 1991; Wang et al., 2013; Liang et al., 2015). Conceptual models such as Budyko-type equations (see Sect. 2.1) have consequently gained interest in recent years.

Within the Budyko framework, studies (Roderick and Farquhar, 2011; Zhang et al.,
2016) have used the total differential of runoff as a proxy for the runoff
change and the partial derivatives as the sensitivities (hereafter called
the total-differential method). The total differential, however, is simply a
first-order approximation of the observed change (Fig. 1a). This
approximation has caused an error in the calculation of climate impact on
runoff, with the deviation ranging from 0 to

For a nonlinear function

The so-called decomposition method developed by Wang and Hejazi (2011) has also been widely used. The method assumes that climate changes cause a shift along a Budyko curve and then human interferences cause a vertical shift from one Budyko curve to another (Fig. 2). Under this assumption, the method extrapolates the Budyko models that are calibrated using observations of the reference period, in which human impacts remain minimal, to determine the human-induced runoff changes that occur during the evaluation period.

A schematic plot illustrating the decomposition method. Point A
denotes the initial state (the reference period), and point C denotes the
terminal state (the evaluation period). The decomposition method assumes that the catchment state first evolves from Point A to B along the Budyko curve for the reference period, then jumps to Point C.

Recently, Zhou et al. (2016) established a Budyko complementary relationship for runoff and further applied it to partitioning the climate and catchment effects. Superior to the total-differential method, the complementary method culminates by yielding a no-residual partition. Nevertheless, this method depends on a given weighted factor that is determined in an empirical but not a precise way. Furthermore, Zhou et al. (2016) argued that the partition is not unique in the Budyko framework because the path of the climate and catchment changes cannot be uniquely identified.

Obtaining a precise partition remains difficult, even when giving a precise
mathematical model. This difficulty can be illustrated by using a precise
hydrology model

The aim of this study is to propose a mathematically precise method to conduct a quantitative attribution to drivers. The method is based on the line integer (called the LI method hereafter) and takes into account the sensitivity throughout the evolutionary path of the drivers rather than at a point as the total-differential method does. To present and evaluate the proposed method, I decomposed the relative influences of climate and catchment conditions on runoff within the Budyko framework using data from 19 catchments from Australia and China.

Budyko (1974) argued that mean annual evapotranspiration (

The partial differentials of

We start by considering an example of a two-variable function

A schematic plot illustrating the LI method.

As shown in Fig. 3, points

Unlike the total-differential method, the sum of

The mathematical derivation above applies to a three-variable function as
well. By doing the line integrals for the MCY equation, we obtain the
desired results:

Determining

A curve can always be approximated as a series of line segments. Hence, we
can first handle the case of a linear integral path. Given two consecutive
periods and assuming that the catchment state has evolved from (

Unfortunately, I could not determine the antiderivatives of

Dividing the evaluation period into a number of subperiods:

I first determined a change point and divided the whole observation period into the reference and evaluation periods. To determine the integral path, the evaluation period was further divided into a number of subperiods. The Budyko framework assumes a steady-state condition of a catchment and therefore requires no change in soil water storage. Over a time period of 5–10 years, it is reasonable to assume that changes in soil water storage will be sufficiently small (Zhang et al., 2001). Here, I divided the evaluation period into a number of 7-year subperiods with the exception for the final subperiod, which varied from 7 to 13 years in length depending on the length of the evaluation period.

Determining

For a short period, the integral path

Summary of the long-term hydrometeorological characteristics of the
selected catchments

To evaluate the LI method, I compared it with the existing methods,
including the decomposition method, the total-differential method, and the
complementary method. The total-differential method approximated

The decomposition method (Wang and Hejazi, 2011) calculated

The complementary method (Zhou et al., 2016) uses a linear combination of the
complementary relationship for runoff to determine

I collected runoff and climate data from 19 selected catchments evaluated in previous studies (Table 1). The change-point years given in these studies were directly used to determine the reference and evaluation periods for the LI method. As mentioned above, the LI method further divides the evaluation period into a number of subperiods. For the sake of comparison, the final subperiod of the evaluation period was used as the evaluation period for the decomposition, the total-differential, and the complementary methods (it can be equally considered that all of the four methods used the final subperiod as the evaluation period, but the LI method cares about the intermediate period between the reference and the evaluation periods, and the other methods do not). Eight of the 19 catchments had a reference period comprising only one subperiod (Table 1), and the others had two to seven subperiods.

Comparisons between the LI method and the decomposition method.

Comparisons of the estimated contribution to runoff from the changes
in

The 19 selected catchments have diverse climates and landscapes, with 12 from
Australia and seven from China (Table 1). The catchments span from tropical
to subtropical and temperate areas and from humid to semi-humid and semiarid
regions, with the mean annual rainfall varying from 506 to

Comparisons of

Effects of precipitation (

Table 3 lists the resultant values of

Comparisons of

Comparisons of

Figure 4a compares the resultant

The total-differential method is predicated on an approximate equation,
i.e., Eq. (7). The LI method reveals that the precise form of the equation is

Performances of Eq. (2) to be used to predict

As with the LI method, the complementary method produced

The LI method highlights the role of the evolutionary path in determining the resultant partition. Yet it seems that no studies have accounted for the path issue while evaluating the relative influences of drivers. The limit of the LI method is high data requirements for obtaining the evolutionary path. When the path data are unavailable, the complementary method can be considered as an alternative. The complementary method is free of residuals; moreover, it employs data from both the reference and the evaluation periods, thereby generally yielding sensitivities closer to the path-averaged results than the total-differential method.

While using the Budyko models, a reasonable timescale is relevant to meet
the assumption that changes in catchment water storage are small relative to
the magnitude of fluxes of

The mutual independence of the drivers is crucial for a valid
partition. In the present study,
although annual

The LI method revises the concept of sensitivity at a point as the
path-averaged sensitivity. Mathematically, the LI method is unrelated to a
functional form and hence applies to communities other than just hydrology.
For example, identifying the carbon emission budgets (an allowable amount of
anthropogenic

This study presented the LI method using time-series data, but it applies equally to the case of spatial series of data. Given a model that relates fluvial or eolian sediment load to the influencing factors (e.g., rainfall and topography), for example, the LI method can be used to separate their contributions to the sediment-load change along a river or in the along-wind direction.

Based on the line integral, I created a mathematically precise method to partition the synergistic effects of several factors that cumulatively drive a system to change from one state to the other. The method is relevant for quantitative assessments of the relative roles of the factors behind the change in the system state. I applied the LI method to partition the effects of climatic and catchment conditions on runoff within the Budyko framework. The method reveals that, in addition to the change magnitude, the change pathways of climatic and catchment conditions also play a role. Instead of using the runoff sensitivity at a point, the LI method uses the path-averaged sensitivity, thereby ensuring a mathematically precise partition. As a mathematically accurate scheme, the LI method has the potential to be a generic attribution approach in the environmental sciences.

Let

Thus,

The right-hand side of the equation

Preliminary theorem: Given an open, simply connected region

Runoff (

The result differs depending on

As expected, the sum of

If the interval [

The result can readily be extended to a function of three variables.
Applying the mathematic derivation determined above to the MCY equation
results in a precise form of Eq. (7):

Given a one-dimensional function

The data used in this study are freely available by contacting the author.

The supplement related to this article is available online at:

MZ designed the study, analyzed the data, and wrote the manuscript.

The author declares that there is no conflict of interest.

I thank Yuanqiong Zheng for his assistance with the mathematical proof in Appendix A.

This research has been supported by the National Natural Science Foundation of China (grant no. 41671278) and the Guangdong Academy of Sciences' Project of Science and Technology Development (grant nos. 2019GDASYL-0103043, 2019GDASYL-0502004, 2019GDASYL-0401003, 2019GDASYL-0301002, 2019GDASYL-0503003, and 2019GDASYL-0102002).

This paper was edited by Erwin Zehe and reviewed by two anonymous referees.