HSHMs represent intervals in space and/or time characterized by hydrobiogeochemical activity rates or fluxes that differ significantly from the background conditions, thus exerting disproportionate influences over an ecosystem's dynamics. We define Ω* as the volume (subdomain) within which the hot spot may be located and t* the time at which the hot moment occurs. An indicator random variable, IHSHM(Ω*,t*), is used to identify whether an HSHM occurs at (Ω*,t*) or not. If user-defined critical conditions needed to trigger an HSHM are met at (Ω*,t*), then IHSHM(Ω*,t*)=1, and it is equal to zero otherwise. These user-defined critical conditions may require concentration or reaction rates above certain thresholds or require both concentration and rate limits depending on the type of HSHM. What makes the indicator a random variable is the uncertainty in the spatial and temporal distribution of the quantities triggering the HSHM event in real-life applications. Following the original definition by McClain et al. (2003), in our method, IHSHM(Ω*,t*) can take a value of 0 or 1, depending on whether suitable thresholds are exceeded or not as follows: 1IHSHMΩ*,t*=1,ifCx,t*>Cth;x⊆Ω*0,otherwiseorIHSHMΩ*,t*=1,ifRx,t*>Rth;x⊆Ω*0,otherwise, where C(x,t*) and R(x,t*) are the concentration and reaction rate at position x and time t*, respectively. Cth and Rth represent the concentration and reaction rate thresholds, respectively, which identify whether an HSHM is triggered or not. Defining indicators with concentration or reaction rate depends on the target of an HSHM. For example, indicators defined by concentration are preferred with transport-driven HSHMs, whereas biogeochemically driven HSHMs may require indicators defined by both concentration and reaction rates. The threshold values can also be based on regulatory limits or defined by the user.

The critical values of Cth and/or Rth are keys to an effective application of the above framework and should be determined based on the specific scenario under investigation. For example, in the case of contaminants that are associated with significant environmental or health risks (e.g., nuclear waste or a cancerous substance), Cth=0 or Rth=0 can be used so that the HSHM will be triggered as soon as there is the presence of such contaminants and relevant chemical reactions. As an alternative, a limit in the total accumulated mass or fluxes within hot spots may also be set, such as suggested by the EPA (USEPA, 2001), but in this case the definition (1) of the indicators should be modified. For water quality parameters, Cth=MCL or Rth=R* can be assigned, where MCL represents the maximum concentration limit for a specific solute, whereas R* could represent a critical reaction rate. The critical thresholds can be determined based on statistics, such as percentiles and extremes as defined by regulations or analytical studies. Alternatively, Cth and Rth could also be chosen based on the experts' domain knowledge or from well-documented studies in similar environments. Through the flexibility to adopt different choices for activation thresholds, our approach could allow users to compare relevant indicator models and assess their applicability by testing how different thresholds would influence the probability of the HSHM occurring and assessing said probabilities against risk tolerance and/or regulations.

Following the definition of IHSHM(Ω*,t*) as a binary random variable (Eq. 1), we propose to model it with a Bernoulli distribution, such as IHSHM(Ω*,t*)∼Bernoulli(<IHSHM(Ω*,t*)>), where <.> is the ensemble-averaging operator. An important characteristic of the Bernoulli distribution is that all the statistical moments of IHSHM(Ω*,t*) can be expressed as a function of the ensemble mean <IHSHM(Ω*,t*)>. For example, the variance is given by var(IHSHM(Ω*,t*))=<IHSHM(Ω*,t*)>⋅(1-<IHSHM(Ω*,t*)>). Thus, being able to fold the HSHM physics into an indicator formulation, a simplified approach is presented through Eq. (1).

Case-based formulation of the Bernoulli distribution of IHSHM(Ω*,t*) requires the incorporation of the mechanisms that govern the development and occurrence of HSHMs into the indicator model. To facilitate this undertaking, we propose to decompose IHSHM(Ω*,t*) into a type-A (static) indicator random variable, Is(Ω*), and a type-B (dynamic) indicator random variable, Id(Ω*,t*). Definitions of the type-A and type-B indicators are provided herein.

Type-A (static) contributors. This category covers discrete spatial elements (and their associated critical states) that could trigger an HSHM once they come into contact with type-B contributors (see discussion below). Critical states are the range of values needed to trigger an HSHM (either in stand-alone mode or when coupled with type-B contributors).

Both indicators of the type-A and type-B contributors assume a value of either 0 or 1. If one of these indicators takes a value of 1, it can be viewed as an HSHM contributor. However, for an HSHM to occur, both indicators must have a value of 1 at the same location and time. This idea can be expressed as follows: 2PIHSHMΩ*,t*=1=PIsΩ*=1,IdΩ*,t*=1=PIsΩ*=1⋅PIdΩ*,t*=1|IsΩ*=1=PIdΩ*,t*=1⋅PIsΩ*=1|IdΩ*,t*=1. Based on the mechanisms of HSHMs, we can distinguish three different HSHM categories as discussed below. These categories can be used to guide the application of the above statistical framework in a variety of complex HSHM scenarios, and they can also be used to develop analytical or numerical solutions for both type-A and type-B contributors.

In this section, we consider HSHMs that are defined by static indicators only (Fig. 1a). This list can include zones of high, persistent concentration and reactivity that are due to the subsurface or the ecosystem's unique hydrological and biogeochemical properties. For example, the accumulation of contaminants in the subsurface (e.g., the high nuclide concentration in the subsurface at the Hanford site) could lead to the evolution of persistent, high-reactivity zones. An aquifer's reactivity is another example that could distinguish certain regions with high reactivity compared to surrounding areas (Loschko et al., 2016). Such high-reactivity spots (hereafter denoted as Ω*) can be characterized by static indicator RVs due to the persistence of high concentration or reactivity. The static indicators are defined as follows: 3IHSHMΩ*=IsΩ*=1,ifZΩ*⊆Zs*0,otherwise, where Zs* represents the conditions needed to trigger a hot spot at Ω*, and Z(Ω*) represents the corresponding local conditions at Ω*. Notice that Ω* is a volume centered at a selected position of the domain where the probability of developing an HSHM is evaluated.

HSHMs can also result from dynamic processes encountering specific local conditions at Ω* (Fig. 1b). This is the situation described by Eq. (2), where the type-A indicators are determined first and then used jointly with the type-B indicators for complete HSHM characterization. For example, Bundt et al. (2001) concluded that locations (Ω*) intersected by preferential flow paths are possible biological hot spots for soil microbial activities. Meanwhile, dynamic factors, such as snowmelt or rainfall infiltration control contaminant transport via the preferential flow paths, thus constitute the hot moment component. Additional case studies are presented in Sect. 3.

For HSHMs induced by both type-A and type-B indicators, the static locations are selected first, based on their HSHM-related properties. After this, we can focus on characterizing the HSHM dynamics as they relate to the relevant locations. A selected location, Ω*, could become an HSHM site based on characteristics defined through the following type-A and type-B indicators, respectively: 4IsΩ*=1,ifZsΩ*⊆Zs*0,otherwise,5IdΩ*,t*=1,ifZdΩ*,t*⊆Zd*0,otherwise,6IHSHMΩ*,t*=1,ifZsΩ*⊆Zs*,andZdΩ*,t*⊆Zd*0,otherwise, where Zd* represents the critical conditions needed to trigger the hot moment, whereas Zd(Ω*,t*) represents the corresponding, critical-state local conditions at t* and Ω*. The statistical model of IHSHM(Ω*,t*) can be expressed using the statistical models of Is and Id as shown in Eq. (2).

A confluence of dynamic processes could result in the formation of an HSHM (Fig. 1c). Unlike the previous scenarios where static locations can be determined through known characteristics provided by geophysical or other types of data, HSHMs can also emerge due to a confluence of dynamic processes. For example, Gu et al. (2012) analyzed how streamflow fluctuations could trigger a nitrogen HSHM. In their example, the dynamics of the streamflow and groundwater controlled the transport and mixing of the chemical reactants, thus triggering the occurrences of HSHMs. For this case, the static locations of Ω* are determined by the confluence of multiple dynamic processes, not being restricted by a set of local conditions. In this case, only type-B indicators need to be modeled.

We can consider the case where an HSHM is predicated on m dynamic processes, dj, where Id,j(Ω*,t*) is the dynamic for each dynamic dj at Ω* and time t*. The hotspot location Ω* is determined by the confluence of all dynamic processes at time t*. These dynamic processes are not necessarily independent. Therefore, generally, the statistical model for the comprehensive dynamic indicator (which covers all dynamic contributors) assumes the following form: 7PIdΩ*,t*=1=PId,1Ω*,t*=1,…,Id,mΩ*,t*=1. In situations where the various dynamic contributors can be viewed as independent (e.g., Destouni and Cvetkovic, 1991) – i.e., where the reactants travel via different paths – then, assuming independence, we can state that 8PIdΩ*,t*=1=∏j=1mPId,jΩ*,t*=1. Here, the mean of the dynamic indicator becomes 9<IdΩ*,t*>=∏j=1m<Id,jΩ*,t*>. If Ω* is a hot spot, then Eq. (9) also defines <IHSHM(Ω*,t*)>. However, if Ω* is not a hot spot, then we need to resort to coupled statistical modeling, as suggested by Eq. (2).

The statistical framework provides several benefits. Unified formulations of HSHMs through indicators provide us with a platform to evaluate alternative HSHM models thoroughly and objectively. For example, the Akaike information criterion (AIC, Akaike, 1974) and Bayesian information criterion (Schwarz, 1978) can be used to rank between alternative indicator formulations and evaluate their ability to explain HSHM observations. Smaller AIC and BIC values indicate more information preserved in a given indicator HSHM model and imply better model quality than other indicator models. On the other hand, if larger AIC and BIC values are observed, important processes for HSHMs are likely missing, indicating the necessity of increasing site characterization and refinement of conceptual models.

In addition, informative priors constructed from similar HSHM sites (Cucchi et al., 2019; Li et al., 2018) could advance early stage planning for HSHM site investigation. Knowledge from studies at similar HSHM sites can be summarized into prior distributions, which can account for variabilities within and between sites. For example, Cucchi et al. (2019) demonstrated how the distribution of hydraulic parameters at unknown target sites can be predicted using information from hydrologically similar sites with existing tool packages such as exPrior. Goal-oriented site characterization also becomes feasible with informative priors; for example, Li et al. (2018) demonstrated the usefulness of informative priors in reducing model uncertainty and potential risks for estimating groundwater drawdown at Mintang tunnel in China. Therefore, through integration with statistical concepts, unified formulations of HSHMs enable us to integrate Bayesian concepts to obtain combined and less risky estimations of HSHMs at new sites, which can help us gain better understanding of the underlying mechanism.

In this section, we use Wainwright et al. (2015) as an example to illustrate our process to construct IHSHM(Ω*,t*) following Eq. (3), where an HSHM is triggered by static contributors only (Sect. 2.1). NRZs within floodplain environments at the Rifle site are considered biogeochemical hot spots because they represent elevated concentrations of uranium, organic matter, and geochemically reduced minerals, and they have been found to contribute significant carbon fluxes to the atmosphere and to local rivers (Arora et al., 2016a). Due to its characteristics, we considered the spatial distribution of an NRZ to be a static-mechanism-based hot spot. Wainwright et al. (2015) used geophysical data (e.g., induced polarization) to map the distribution of an NRZ at the subsurface level. They found that the phase shift (ϕ) from the induced polarization data of the NRZ was within [4.5, 5] mrad compared to non-NRZ locations at ϕ⊆[1, 3.5] mrad. Thus, ϕ can be used to construct the static indicator with a critical condition of [4.5, 5] mrad. Therefore, 10IsΩ*=1,ifZϕΩ*⊆[4.5,5]mrad0,otherwise. Other static attributes, including but not limited to elevation, hydraulic conductivity, and resistivity, can also be used to define the critical conditions to construct the static indicator for hot spots through Bayesian conditioning.

The second case we present here utilizes Eqs. (4)–(6), where HSHMs are triggered when dynamic contributors coincide at hot spots determined by static contributors. Here, we present the case investigated by Duncan et al. (2013), where riparian hollows representing less than 1 % of the total catchment area contributed more than 99 % of the total denitrification within the watershed. In their study, the denitrification rates peaked during the base flow (midsummer) period, when the riparian hollows were partially oxygenated and the hydrologic fluxes were at a minimum. The site was considered to have low inorganic N availability and, thus, nitrate was supplied via nitrification. The highest rates of denitrification were therefore tied to nitrification and the partially aerated conditions.

The static indicator needs to be constructed based on the micro-topographical features within the riparian zone. Specifically, the topographic wetness index (TWI) (Beven and Kirkby, 1979; Sørensen et al., 2006) was used in Duncan et al. (2013) to delineate the riparian hollows from other riparian locations. Terrain analysis indicated TWI threshold values of 6.0 and 8.0 for riparian hollows under wet and dry conditions, respectively, whereas 4.8 and smaller TWI values corresponded to other riparian locations (e.g., hummocks). Thus, the static indicator can be constructed using the TWI values within the riparian zone to determine the hotspot locations – the hollows. Hence, 11IsΩ*=1,ifZTWIΩ*>6(wetcondition)or8(drycondition)0,otherwise. Multiple dynamic processes control the denitrification rate at the riparian hollows. As examined by Duncan et al. (2013), a partially aerated condition (CO2>5 %) is needed to support nitrification, which supplies the nitrate for denitrification. As quiescent, non-storm periods during base flow favor the coupled nitrification–denitrification mechanism, this is another key process that needs to be represented by a dynamic indicator. Although Duncan et al. (2013) did not mention specific concentration ranges for nitrogen species, the major components, such as organic N, should be available. Therefore, we can construct the dynamic indicators as follows: 12PIdΩ*,t*=1=PId,O2Ω*,t*=1,Id,HydroΩ*,t*=1,Id,NΩ*,t*=1, where Id,Hydro(Ω*,t*) is the dynamic indicator representing the streamflow stages; this will be 1 if the base flow conditions are met. Additionally, here, Id,N(Ω*,t*) is the dynamic indicator for the transport of the nitrogen species in the subsurface that support the coupled nitrification–denitrification mechanism. 13Id,O2Ω*,t*=1,ifCO2Ω*,t*>5%0,otherwise,14Id,HydroΩ*,t*=1,ift*⊆baseflowperiods0,otherwise,15Id,NΩ*,t*=1,ifCNΩ*,t*>00,otherwise It is noted that these dynamic processes are not statistically independent. Usually, when one condition is met (e.g., base flow conditions), other conditions may consistently be satisfied (e.g., the transport of nitrogen in riparian hollows). Alternatively, numerical modeling approaches can be used to construct the dynamic indicators based on the critical conditions at riparian hollows (Ω*), where we could directly target N2 fluxes using a Monte Carlo approach. The statistical formulation used here is constructed specifically for the mechanisms described by Duncan et al. (2013). Thus, the detailed threshold limits could change under other denitrification HSHM cases, such as the case presented in Hill et al. (2000), who focus on desert landscapes, or the one by Harms and Grimm (2008), where the monsoon season is influential for the nitrogen transport. Nonetheless, the general formulation of HSHMs using indicators is still applicable.

HSHMs can also be triggered by the confluence of multiple dynamic processes that lead to the convergence of complementary reactants at Ω*. Complementary reactants can be mobilized and transported via different hydrologic flow paths. They can converge at hotspot locations and trigger hot moments during the mixing. Following the statistical framework developed in this study, Eqs. (7) to (9) are suitable for this condition. In order to illustrate how the dynamic indicators are constructed, we consider here the case reported by Gu et al. (2012), where high biogeochemical activity was observed at the interface of groundwater and surface water during the stream stage fluctuations, which resulted in significant in-stream denitrification and NO3- removal.

In their study, hot spots form around the near-stream riparian subsurface during river stage fluctuations, where active biogeochemical reaction (e.g., denitrification) requires both O2 depletion and the simultaneous presence of NO3- and the dissolved organic carbon (DOC). Specifically, the spatiotemporal distribution of denitrification hot spots coincides with an O2 depletion zone along the DOC infiltration flow paths. In order to determine the mixing of groundwater and surface water during stage fluctuations, Gu et al. (2012) defined bank storage volume V(t) and maximum bank storage volume Vmax. The flood hydrograph was subdivided into the rising limbs, recession limbs and return flow, the latter representing the slow restitution of part of the water that infiltrated during the previous stages. Considering the different dynamics of these components, they observed that the largest infiltration rate occurred prior to the maximum stage rise, while Vmax=5 m3 m-1 (critical condition) occurred in the recession limb of the flood event. Instead, maximum return flow occurred toward the end of the recession curve before the stream hydrograph stabilizes. Maximum NO3- rate removal occurred when the return flow phase was almost complete and then decreased until the depletion of NO3-. Through statistical analysis, they found that Vmax, viewed as an integrated index for hydrological exchange, could explain 64 % of the variation in the NO3- removal. Thus, Vmax can be used as the critical state to determine whether or not the hyporheic dynamics is significant to enhance relevant biogeochemical processes. In order for the hot moments to be significant, the stream–riparian zone should also be microbially active. Based on these conditions, the dynamic indicators can be constructed as follows: 16PIdΩ*,t*=1=PId,HydroΩ*,t*=1,Id,ChemΩ*,t*=1, where Id,Hydro(Ω*,t*) represents the dynamic process induced by the hydrologic conditions (e.g., stage fluctuation), and Id,Chem(Ω*,t*) represents the dynamic process controlled by the transport and accumulation of chemical reactants. Based on the critical values or ranges, we formulate the indicators as follows: 17Id,HydroΩ*,t*=1,ZVmaxΩ*,t*≥5m3m-10,otherwise,Id,ChemΩ*,t*=1,ifCO2Ω*,t*issmallandCNO3-Ω*,t*>0andCDOCΩ*,t*>00,otherwise. Typically, because of the complexity of the processes, no analytical solutions are available for formulating the indicators. However, Monte Carlo simulations can be useful in constructing such indicators. For this case, an HSHM at any given location and tim (Ω*,t*) will only be triggered when all of the conditions are met and the ensemble mean of the indicator assumes the following form: 18<IdΩ*,t*>=1N∑i=1NId,iΩ*,t*, where Id,i(Ω*,t*) is the value that the indicator assumes in the ith realization and N is the total number of realizations.

Overall, our choices of the three studies should not limit the generalizability of the indicator statistics approach for deriving statistical formulations for HSHM applications. The critical conditions chosen to construct the indicators are determined solely on the findings from these selected studies, and they will vary under different scenarios.

The heterogeneous structure of hydraulic conductivity leads to significant variability in the transport of solutes in the subsurface, which couples with heterogeneous geochemical properties leading to a spatially varying reactivity (Arora et al., 2019b; Loschko et al., 2016; Sassen et al., 2012; Wainwright et al., 2015). Figure 2 demonstrates the uncertainty associated with HSHMs by looking at the flow fields in two-dimensional log-hydraulic conductivity (Y=ln⁡(K)) fields with streamlines resulting from a uniform mean head gradient, left to right. The three panels differ in terms of the variance, σY2, of the log conductivity. The covariance function used for generating the fields is exponential and isotropic. σY2 is shown to have a profound impact upon the conductivity field. As the variance increases, regions of high and low log conductivity emerge, creating preferential flow paths bypassing the low-conductivity zones as shown by particle trajectories. At smaller variance (i.e., σY2=0.1), particles mainly travel along the mean flow direction with very limited departure from the mean trajectory, which are the straight lines connecting the left and right boundaries. In this situation, the arrival times of solute particles at a critical location (i.e., Ω*, where for example geochemical conditions are favorable for certain types of reactions to occur) are predictable. With large variances (i.e., σY2=2), the streamlines assume a very irregular, hard-to-predict geometry, and we can observe the emergence of flow channels, where particles can move quickly next to stagnant flow regions. Arrival times become more uncertain, because the exact geometry of the streamlines is hard to predict unless the Y field is known deterministically. However, since this is never the case, in another equally likely realization of the Y field, the situation may be different, resulting in significant uncertainties in predicting the particle travel times. Thus, spatial variability of log conductivity is a major uncertainty-inducing factor and, by extension, requires the need for stochastic modeling of HSHMs in situations where the associated processes and attributes are subject to uncertainty. In the following sections, we will present illustrative examples to analyze how subsurface spatial variability influences <IHSHM(Ω*,t*)>, including variance and anisotropy ratio of the log conductivity.

In this section we illustrate the proposed indicator approach by means of synthetic case studies developed by using methods of stochastic hydrogeology. The choice of the synthetic case studies does not limit our approaches to broader applications where stochastic modeling with Monte Carlo simulations is applicable. Figure 3 displays the configuration of this case example. Consider the case of an instantaneous point source release of a target compound at location x0 and time t0. HSHMs are triggered at any (Ω*,t*) if the solute is present. Consider the hot spot (Ω*) to be confined within the following volume: w1≤x1≤w1′; w2≤x2≤w2′; w3≤x3≤w3′; the dynamic indicator is therefore defined as follows: 19IdΩ*,t*=1,ifXt*⊆Ω*att*0,otherwise. The injected solute can be modeled in a Lagrangian framework as a particle moving according to the velocity field without changing its volume. The latter is the consequence of neglecting pore-scale dispersion. The expected value of this dynamic indicator at t* is therefore 20<IdΩ*,t*>=∫Ω*fXt*a|x0,t0da, where fX(t*t)(a|x0,t0) is the probability distribution function (pdf) of the particle's trajectory at time t* (Dagan and Nguyen, 1989; Rubin, 2003). If we also assume steady, uniform in the average flow with mild heterogeneity of the log hydraulic conductivity field with the Gaussian displacement pdf, then we can compute <IHSHM(Ω*t*)> analytically using the following equation: 21PIHSHMΩ*,t*=1=<IHSHMΩ*,t*>=<IsΩ*><IdΩ*,t*>=probIdΩ*,t*=1=probXt*⊆Ω*=∏i=1m∫wiwi′fXit*ai|x0,t0dai=∫w1w1′fX1t*a1|x0,t0da1∫w2w2′fX2t*a2|x0,t0da2∫w3w3′fX3t*a3|x0,t0da3=1(2π)32X11t*X22t*X33t*∫w1w1′exp⁡-12a1-Ut*2X11t*da1⋅∫w2w2′exp⁡-12a22X22t*da2∫w3w3′exp⁡-12a32X33t*da3, which can be integrated to yield 22PIHSHMΩ*,t*=1=<IHSHMΩ*,t*>=18erfcw1-Ut*2X11t*-erfcw1′-Ut*2X11t*⋅erfcw22X22t*-erfcw2′2X22t*erfcw32X33t*-erfcw3′2X33t*. For simplicity, but without lack of generality, in Eq. (20) we assumed x0=(0,0,0). The displacement variances Xii, i=1, 2, 3 depend on the spatial distribution of the hydraulic conductivity in the subsurface. Equations (A4) to (A6) present the displacement variances for an axisymmetric exponential covariance function of the log conductivity (Eq. A3) given in Appendix A. In Eqs. (A7) to (A19), we have provided derivations of indicator formulations for other HSHM scenarios, including indicator formulation for complex concentration thresholds and indicator formulation for hot moment durations. Notice that in obtaining Eq. (20) we postulated ergodicity, which in practical terms reflects the actual situation of an instantaneous injection into a source zone with a transverse dimension much larger than the integral scale of the hydraulic conductivity (Dagan, 1990), such that the ensemble mean is representative of the effects of the actual, but unknown, distribution of hydraulic conductivity.

In the following sections, we present the results from the case study described in Sect. 4.2. Specifically, in Sect. 4.3.1 and 4.3.2, we explore how heterogeneity of log-hydraulic conductivity influences the probability of HSHM occurrences. To make results as general as possible, lengths are made dimensionless with respect to the integral scales (IYh in the two horizontal directions and IYv in the vertical one) and time with respect to the following advective timescale: IYh/U, where U is the mean velocity. In the following, we explore the effect of the remaining parameters, i.e., the anisotropy ratio e=IYVIYH and the variance of the log conductivity σY2, on the emergence of an HSHM. We placed Ω* along the mean trajectory at (21IYH, 0, 0) with dimensions as (2IYH, 2IYH, 2IYV). The dimensions of the hot spot are therefore of two integral scales in the three coordinate directions (x1, x2, x3) and are placed at a dimensionless distance of 21 from the point source.