## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematics of a Threshold-Based Footprint

_{0}, before the stock is used. Then, S

_{0}is drawn down by a gross Withdrawal, W, due to the aggregated direct actions of all processes in the system. Stock-reducing withdrawals are positive by sign convention. The term “stock” is used generally and not strictly, and the “stock” could be one of a variety of environmental quantities: stock, flow, resource, event magnitude, population, or incidence. For instance, in the water footprint example in Section 2.2, the metric of interest is surface water flow.

^{T}is the nonnegative difference between F and the Threshold T (Equation (1), Figure 1). The stock’s Threshold is often a limit on the sustainable consumption or degradation of that stock. For example, an environmental flow requirement R would be the difference between the flow Q and the surface water stock’s threshold, such that R = Q − T. The Free Footprint F

^{f}is the portion of F falling below the threshold T. This free portion of the footprint is discounted and characterized as having no impact because it has negligible environmental or economic cost (Equation (2), Figure 1). The relative Threshold T` = T/S is the Threshold expressed as a fraction of the Stock. Units of W, F, and T are those of S; c is unitless. If T = 0 then all resource withdrawals are adverse and F

^{T}= F; if T = S then all impacts are discounted and F

^{T}= 0. F = F

^{f}until F > T. Observe in Equations (1) and (2) that the inventory of the stock S does not appear in the footprint calculation, but T does appear and is presumably related somehow to S. If F

^{T}= 0 the resource is “abundant” (negligible marginal value and cost), at least from the point of view of this decision making process. If F

^{T}> 0 the resource is scarce from the point of view of this decision making process and has non-negligible marginal value and cost.

^{T}+ F

^{f}. By definition, if c ≤ T`, then F

^{T}= 0 and F = F

^{f}. The Free Fraction R of the footprint, R = F

^{f}/F, is a sort of efficiency metric for the footprint. For example, if a river has a flow S of 1 Million m

^{3}/year and an adverse threshold T of 100,000 m

^{3}/year, and the total footprint F is 150,000 m

^{3}/year, then F

^{f}= 100,000 m

^{3}/year and F

^{T}= 50,000 m

^{3}/year; R = 100,000/150,000 and thus two thirds of this footprint is “free”.

_{0}

^{T}, where S

_{0}

^{T}= S

_{0}− T

_{0}, and the initial free capacity below the threshold is S

_{0}

^{f}, where S

_{0}

^{f}= T

_{0}(Figure 1). After an initial footprint F

_{0}is applied, the remaining adverse capacity is S

_{1}

^{T}(Equation (3), Figure 1), and the remaining free capacity is S

_{1}

^{f}(Equation (4), Figure 1).

_{x}is the footprint of an individual process x. F

_{x}is not bounded by F when there are multiple processes, because the net impacts of different processes may be positive or negative and may offset, such that F = ∑

_{x}F

_{x}. As with the aggregated footprint, the process’s footprint is the sum of the free and the adverse components, so F

_{x}= F

_{x}

^{T}+ F

_{x}

^{f}. Processes may possess their own thresholds, T

_{x}; a U.S. Environmental Protection Agency (EPA) regulation placing a Total Maximum Daily Load (TMDL) limit on a factory’s emission of water pollution is an example of a process having a threshold.

_{1}= T. However, each subsequent and junior processes (x > 1) would have its threshold set at current free capacity of the stock, S

_{1}

^{f}(Equation (4)), after the sum total of all prior footprints F

_{0}were deducted, such that F

_{0}= ∑

_{x_prior}F

_{x}. Without any seniority one might choose to weight the threshold, free footprint, and adverse footprint of a process by its contribution to the footprint by using the weighting factor b = F

_{x}/F, so F

_{x}

^{T}= b F

^{T}, F

_{x}

^{f}= b F

^{f}, and T

_{x}= b T. Or, one could use a different weighting factor for a more progressive attribution system.

#### 2.2. Application of the Threshold Concept to Water Footprints and Impact Metrics

^{T}. The Free Water Footprint (FWF) is equal to F

_{f}. If BWF is used in LCA, it is an inventory, volumetric, or pressure type LCA metric, whereas TWF is a mid-point LCA metric. However, both use identical volumetric units, for instance cubic meters or gallons. Crucially, for example, for a company that seeks to measure the impact of its water footprint, for T > 0 it is not possible to calculate TWF or FWF without full knowledge of the net aggregated BWF of all other processes impacting that water stock, as well as the stock’s threshold. TWF and FWF therefore have a fundamentally higher burden of information than BWF.

^{*}

_{SRF}where the square root of the VF is utilized, reflecting lower flow variability and less low-flow seasonal water stress in an SRF basin, such that WSI* = WSI × VF

^{1/2}. Note that this definition of WSI assumes a constant relationship between the WTA ratio and the water stress in the basin.

## 3. Results

_{0}= 1; hereafter these units will be omitted for simplicity so all results are unitless fractions of a river’s total flow. The experiment explores combinations of thresholds T`, consumption coefficients c, and Withdrawal-to-Availability (WTA) ratios, so that these five metrics can be compared side by side on a unitless basis. This comparison will make it clear that TWF can give quantitatively similar results to RED depending on the choice of threshold, and that the logistic form chosen by WSI implies an approximate “effective” threshold assumed by RED, a threshold that varies based on the combination of c and WTA.

## 4. Discussion

_{x}can be calculated with knowledge of withdrawal and the consumption coefficient. F

^{T}, F

^{f}, F

_{x}

^{T}, and F

_{x}

^{f}require knowledge of the footprint and the threshold. The threshold T for the stock is obtained using external information which presumably includes reference to the initially available inventory of the stock, S

_{0}. However, the threshold T

_{x}for an individual process, and by extension its adverse and free footprints, can only be calculated with external knowledge of seniority or weighting between processes, as well as the footprints of all processes. If F > T at the level of a stock, it is impossible to calculate an individual process’s Threshold-based Footprint F

_{x}

^{T}(a mid-point metric) without knowing the details of all the other processes’ footprints and thresholds, along with their weightings or priorities. If F < T (an abundant stock) it is known that F

_{x}

^{T}= 0 and F

_{x}

^{f}= F

_{x}, so the Blue Water Footprint of this individual process is sufficient information to calculate its mid-point impact. In other words, a factory or city or power plant needs to know the sum total of all other human and natural agents’ water uses and contributions (any significant uses that are upstream or sharing the water stock), along with the water stock’s adverse resource impact threshold, in order to calculate the impact of its own water use. In a system with seniority or priority the situation is even worse: each water user needs to know each other user’s water use and also each other user’s individual threshold and priority.

_{x}

^{T}, in the absence of systemic and publicly transparent water use and availability data.

^{T}is the most urgent objective. This is accomplished by minimizing F, W, and c, but these metrics are equally dependent on S

^{f}and T which can be relaxed through investment in infrastructure or manipulated by changing environmental laws and standards. Resource managers may also seek to maximize the Free Fraction R, or alternatively the ratio of “free” to “adverse” footprints. From the process manager’s differing point of view (e.g., as a company with a commitment to sustainability), the objective is to minimize this individual process’s adverse footprint F

_{x}

^{T}, which can be done either by reducing the process’s own footprint, or by offsetting that footprint by achieving reductions in another process’s footprint against the same resource stock.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Various Threshold-Based Water Scarcity and Stress Indices

^{C}(Equation (A2))) of the aggregated footprint F to the Initial Capacity S is identical to the commonly employed Consumption-to-Availability scarcity index (CTA) [47]. Two alternative indices of scarcity are the Threshold-based Adverse Impact Index I

^{T}and the Threshold-based Free Impact Index I

^{f}(EquationS (A3) and (A4)). The Threshold-based Scarcity Index I, is the ratio of the total footprint to the threshold (Equation (A5)), such that the stock is in a “scarce” condition when F > T and the critical value of the dimensionless number is 1. The proportion of the aggregated net footprint that is adverse is P

^{T}, and the free proportion is P

^{f}(Equations (A5) and (A6)). The Free Impact Ratio RI is the ratio of free to adverse impacts, R = F

^{f}/F

^{T}(Equation (A7)). Many other straightforward combinations of these metrics are possible.

^{T}is the Process-level Scarcity Footprint Fraction, P

_{x}

^{T}(Equation (A8)). Similarly, the process’s fraction of F

^{f}is the Process-level Free Footprint Fraction, P

_{x}

^{f}, and the process’s fraction of F is the Process-level Footprint Fraction, P

_{x}

## Appendix B. The Economic Interpretation of F^{T}

^{T}as an index for the total cost of net aggregated impacts. Water Scarcity Footprints attempt to address this limitation by indexing for the economic condition of scarcity, without attempting to directly address the value of the resource [35,65]. A threshold-based footprint fits this general category.

_{r}on the stock equals the initial capacity S of the stock. The Adverse Impact Threshold for this specific stock T is chosen at the highest value of F where the Opportunity Cost is close enough to zero to be “acceptable” in some socio-environmental-political value judgment. The stock is considered to be “scarce” with adverse marginal impacts and unacceptably high marginal costs when F > T, and the stock is “abundant” when F < T. Multiple cost curves and thresholds may exist for different types of impacts; in this case, the lowest T should typically be chosen. In the case where water LCA assesses impacts on a sensitive wetland, the acceptable impact threshold might be close to zero [69]. For a nonrenewable resource that is subject to market pricing, this threshold will usually be zero.

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**Figure 1.**Schematic of the Threshold-based Footprint concept. The vertical axis gives the marginal value, impact or cost of water use, and the horizontal axis gives the Footprint. The cost of the footprint increases slowly at first as the footprint rises, but beyond a threshold it increases sharply and in unbounded fashion as the resource becomes “scarce” and the marginal value, impact, and cost begin to rise. Illustrations of footprint components during “scarcity” (F > T) and “abundant” (F ≤ T) conditions are given. Section 2.1 defines mathematical symbols and equations. Commonly employed scarcity and stress indices [35] may be expressed using these mathematics (Appendix A). A discussion of the interpretation of impact and cost is provided in Appendix B.

**Figure 2.**Ratio of the surface water flow (Blue) Water Footprint (BWF = D = F

^{T}) to the streamflow depletion threshold, T, for each stream segment of the Kalamazoo River in Michigan, USA. Dark grey colors where D/T > 1 indicate the presence of scarcity where F

^{T}> 0. This is Figure 5 [59] reproduced with permission of ASCE Press and the Authors.

**Figure 3.**Comparisons between the Relevant for Environmental Deficiency (RED) midpoint impact indicator (in red, with RED Strongly Regulated Flow (SRF) version in blue) and Threshold-based Water Footprint (TWF) (in green), with different choices of threshold T` and mean consumptive use coefficient c, plotted against the Withdrawal-to-Availability index (WTA). BWF is coincident with the c:1 line and bounds RED and TWF. (

**a**) T` = 0, c = 1: “Simple Withdrawal” case where all withdrawn water is consumed and counts as adverse impact, resulting in large differences between RED and TWF and additionally Free Water Footprint (FWF) = 0. (

**b**) T` = 0, c = 0.5: “Symmetry” case resulting in a symmetry of RED and Free from Environmental Deficiency (FED) metrics about TWF and FWF, bounding of TWF and FWF by RED and FED, and TWF = FWF. (

**c**) T` = 1/3, c = 2/3; “Reversal” case where RED → FWF and TWF → FED as WTA → 1, and TWF = 0 when WTA < 0.5. (

**d**) T` = 0.25, c = 0.75; “Convergence” case where TWF converges to FWF at WTA = 1, and RED and FED metrics are bounded by TWF and FWF below WTA = 0.58.

**Figure 4.**Water impact indices for the SRF irrigation-dominated best-fit special case where T` = 0.125 and c = 0.7. Note the similarity between RED SRF, in red, and TWF, in green. RED SRF and TWF are approximately equal below WTA = 0.20 and at WTA = 0.63. BWF is coincident with the c:1 line, and is somewhat higher than RED and TWF.

**Table 1.**The effective flow alteration thresholds T` that are approximately implied by RED range from 1% to 18% depending on the details. Table gives values of T` that are the best-fit between RED and TWF, for each consumptive use coefficient (increments of 0.1), achieved by varying T` and WTA. Also shown is the Root Mean Squared Error (RMSE) of fit and the local-maxima or peak value of FED.

RED | RED SRF | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

c | T` | RMSE | Peak FED | @ WTA | Notes | T` | RMSE | Peak FED | @ WTA | Notes |

0 | - | 0 | 1 | 1 | - | 0 | 1 | 1 | ||

0.1 | 0.010 | 0.008 | 0.900 | 1 | 0.018 | 0.009 | 0.902 | 1 | ||

0.2 | 0.019 | 0.016 | 0.800 | 1 | 0.036 | 0.018 | 0.804 | 1 | ||

0.3 | 0.029 | 0.024 | 0.700 | 1 | 0.054 | 0.027 | 0.705 | 1 | ||

0.4 | 0.038 | 0.032 | 0.600 | 1 | 0.071 | 0.036 | 0.607 | 1 | ||

0.5 | 0.048 | 0.040 | 0.500 | 1 | 0.089 | 0.045 | 0.509 | 1 | ||

0.6 | 0.058 | 0.048 | 0.401 | 0.4 | FED local maxima | 0.107 | 0.054 | 0.411 | 0.54 | FED local maxima |

0.7 | 0.067 | 0.056 | 0.301 | 0.37 | FED local maxima | 0.125 | 0.064 | 0.351 | 0.49 | FED local maxima |

0.8 | 0.077 | 0.064 | 0.248 | 0.34 | FED local maxima | 0.143 | 0.073 | 0.333 | 0.46 | FED local maxima |

0.9 | 0.086 | 0.072 | 0.238 | 0.33 | FED local maxima | 0.161 | 0.082 | 0.319 | 0.44 | |

1 | 0.096 | 0.080 | 0.228 | 0.31 | 0.179 | 0.091 | 0.306 | 0.42 | ||

1 | 0.000 | 0.122 | 0.228 | 0.31 | 0.000 | 0.190 | 0.306 | 0.42 | ||

RMSE/c = 0.08 | RMSE/c = 0.09 |

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