We analyze the inter event time distribution of fluid-injection-induced earthquakes for six catalogs collected at geothermal injection sites at Soultz-sous-Forêts and Basel. We find that the distribution of waiting times during phases of constant seismicity rate coincides with the exponential distribution of the homogeneous Poisson process (HPP). We analyze the waiting times for the complete event catalogs and find that, as for naturally occurring earthquakes, injection induced earthquakes are distributed according to a non homogeneous Poisson process in time. Moreover, the process of event occurrence in the injection volume domain is a HPP. These results indicate that fluid-injection-induced earthquakes are directly triggered by the loading induced by the fluid injection. We also consider the spatial distance between events and perform a nearest neighbor analysis in the time-space-magnitude domain. Our analysis including a comparison to a synthetic catalog created according to the ETAS model reveals no signs of causal relationships between events. Therefore, coupling effects between events are very weak. The Poisson model seems to be a very good approximation of fluid induced seismicity.
 The analysis of waiting times between earthquakes [see, e.g., Bak et al., 2002; Corral, 2006] has contributed to the development of statistical models of seismicity. These models are essential to effectively assess seismic hazards. However, the waiting time distribution of earthquakes induced by the injection of pressurized fluids into geothermal and hydrocarbon reservoirs has not yet been studied. It has been established to model the occurrence of naturally triggered earthquakes according to a non homogeneous Poisson process in time [see Ogata, 1998; Shcherbakov et al., 2005]. The inhomogeneous nature of the process reflects changes of the seismicity rate [see Toda et al., 1998] which are attributed to the occurrence of aftershock sequences. We analyze seismic sequences induced by borehole fluid injections into geothermal reservoirs at Soultz-sous-Forêts [see Baria et al., 1999] and Basel [see Häring et al., 2008] to test whether their waiting time distributions suggest the Poisson nature of fluid induced earthquakes. Further, we replace the time by the injected fluid volume and analyze the distribution of fluid volume injected between the occurrences of successive events. To deepen our analysis we include information about the spatial distance between events and perform a nearest neighbor analysis in the time-space-magnitude domain according to Zaliapin et al. . We start the analysis for simple cases of constant seismicity rate.
2. Stationary Induced Seismicity: A Homogeneous Poisson Process
 A sequence of independently occurring events is described by a Poisson process. In the most simple case of a Poisson process with constant intensity λ(expected number of events per unit time), the process is called a homogeneous Poisson process (HPP). The probability P(n, λ, t) to have n events in the time interval [0, t] is then given by:
with corresponding probability density function (PDF) of inter event times (IET) between successive events:
Because this relation depends on the intensity λ, it is reasonable to analyze the distribution of normalized IET given by: Δτ = Δtλ [see also Corral, 2006]. This normalization results in an expected value 〈Δτ〉 = 1 and a pdf(Δτ) = e−Δτ.
Figure 1 shows the temporal distribution of induced seismicity and applied fluid flow rates during six injection experiments. From the earthquake catalogs we select phases of approximately constant event rate (see Figure 1) and calculate normalized inter event times between successive events according to Δτi = (ti − ti−1) . Here is the mean seismicity rate of a stationary phase. Figure 2 (left) presents the number of normalized IET Δτ within logarithmically binned time intervals for the identified stationary phases, 400 events simulated according to a HPP and the distribution e−Δτ of the HPP. In Figure 2 (right) the PDF of IET are shown, which are obtained after dividing the number of counts in a time interval by its length and normalizing the overall probability to one. Apart from negligible deviations that also occur for the simulated events, the distributions of Δτ coincide with the pdf(Δτ) = e−Δτ of the HPP. This implies that successively occurring events are not causally related to each other. Before we discuss possible sources and implications of this result in detail, we analyze the complete catalogs shown in Figure 1.
3. Complete Induced Seismic Sequences: A Non Homogeneous Poisson Process
 Usually, the occurrence of naturally triggered earthquakes is modeled according to a non homogeneous Poisson process (NHPP) in time. If the occurrence of fluid-injection-induced earthquakes can also be described in this way, the probability to induce n events by a fluid injection in the time interval [0, t] can be calculated according to:
where λ(t) corresponds to the time dependent intensity of the process given by the seismicity rate. In Figure 3 we compare the IET distributions of the complete seismicity catalogs to IET distributions calculated for events simulated according to a NHPP with time dependent intensities corresponding to the seismicity rates shown in Figure 1. The probability density functions calculated from the simulated NHPP and the induced events coincide over the whole value range of Δt. Thus, the NHPP model can explain the distribution of fluid injection induced events in time.
4. Inter Event Volume
 The temporal distribution of fluid-injection-induced seismicity is given by a NHPP in all six case studies. We now analyze why the Poisson process is inhomogeneous, that is, why the intensity of the process is changing with time. We start with a comparison to naturally triggered seismicity.
 A frequently used statistical model for the process of earthquake occurrence is the Epidemic Type Aftershock Sequence (ETAS) model [see, e.g., Ogata, 1998]. According to the ETAS model the seismicity consists of independent background events and aftershocks. Background events are directly triggered by tectonic loading. They are distributed according to a HPP in time, because the tectonic loading rate is approximately constant. In contrast, aftershocks are caused by the occurrence of a background event. Each aftershock again has a certain probability to trigger its own aftershocks and so on. This leads to nested aftershock sequences that result in seismicity rate (intensity) changes and thus are the cause of inhomogeneities. Based on this idea we build a similar model for the case of fluid-injection-induced seismicity. To avoid misunderstandings we note that at all analyzed injection locations the background seismicity caused by tectonic loading is vanishing small. For example in Basel the background rate is λb = 3.38 * 10−4ev/day [Bachmann et al., 2011]. It means that all events included in the catalogs are a consequence of the injection of fluids and not of tectonic loading. The analogue to the tectonic loading rate in the fluid-injection-induced case is given by the applied injection flow rate. The intensity of independent background events should hence be proportional to the flow rate. Indeed it has been shown by Shapiro and Dinske  that the cumulative number of events induced by a fluid injection will be proportional to the cumulative fluid volume injected into the borehole, if the flow rate is a non decreasing function. We therefore consider the process of event occurrence in the injection-volume domain and analyze the inter event volume (IEV) ΔV, that is, the fluid volume injected between the occurrence of successive events. The cumulative number of events in the time domain is given by Nev(t) = C1Qc(t), with Qc(t) the cumulative fluid volume injected until the time t and C1 a constant characterized by the seismo-tectonic state at the injection region [see Shapiro et al., 2010]. By transferring this expression to the volume domain we obtain Nev(Qc) = λVQc. If the process of event occurrence is a HPP in the volume domain, λV corresponds to the expected number of events per injected unit volume. The probability to induce n events by injection of a fluid volume Qc is then given by:
with corresponding PDF of normalized IEV ΔVn = ΔVλV:
Figure 4 shows the distribution of ΔVn for the six case studies. All events induced during injection of fluid have been analyzed. Apart from negligible deviations all PDF coincide with the exponential PDF of the HPP (equation (5)). Successively occurring events are hence not causally related to each other or this relation is very weak. Thus, intensity (seismicity rate) changes are dominantly caused by flow rate (loading rate) changes and not by the occurrence of aftershocks. Figure S1 in the auxiliary material supports this finding. It shows that even the strongest events in the Basel catalog leave no signatures in the seismicity rate.
 Note, that we do not consider events occurring after termination of fluid injections in our analysis of IEV to make the results as clear as possible. Because during an injection with non decreasing flow rates the complete fluid flow inside the reservoir contributes to the triggering process of seismic events, the number of induced events is proportional to the applied flow rate. If the injection of fluid is terminated, the fluid flow inside the reservoir that contributes to the triggering process is limited to the reservoir volume characterized by a non decreasing flow [see Parotidis and Shapiro, 2004]. This limitation results not in an immediate drop down of directly triggered events to zero but in a decrease of the intensity analogously to Omori's law [see Langenbruch and Shapiro, 2010]. The a priori assumption of a constant background rate in the time domain is hence not reasonable if flow rates are not constant or if events after injection termination are analyzed.
5. Nearest Neighbor Analysis
 The inter event volume analysis resulted in an exponential distribution implying that successively occurring events are not causally related to each other. However, there are two possible explanations for our finding. First, all events could be triggered by the loading caused by the injection of fluid. In this case, fluid-injection-induced seismicity consists only of independent background seismicity distributed according to a HPP in the injection-volume domain. The stress perturbation caused by the occurrence of a background event would then not be sufficient to trigger aftershocks. Alternatively, Touati et al.  argue based on the ETAS model that, in case of a high background rate, aftershock sequences triggered by different background events may overlap in time. Because the background events are independent (HPP),events from different aftershock sequences, which may be successive in the complete catalog, are also independent.
 To investigate whether aftershock sequences are hidden in the catalogs we consider also the spatial distance between events and perform a nearest neighbor analysis in the time-space-magnitude domain. The time-space-magnitude distance between two events i and j is defined as [see Zaliapin et al., 2008]:
Here tij = ti − tj is the inter event time, rij the inter event distance, d the fractal dimension of earthquake hypo-centers, mi the magnitude of the event with index i and b the b-value of the Gutenberg-Richter relation. We replace tij in equation (6) by the fluid volume ΔVij injected between events i and j to achieve a homogeneous flow of events. Two events i and j are defined as nearest neighbors if their distance is given by: ηj = mini nij.
 In Figure 5 we examine the magnitude normalized volume and space components of the nearest neighbor distance η, namely V = ΔVij and R = rijd Examples are shown for (a) the Basel case study (b) randomly selected events of the Basel case study used as background events for (c) a ETAS simulation (see Appendix). Zaliapin et al.  demonstrate that two distinct clusters appear if the event catalog contains two different classes of events. The first cluster around the line of log10(V) + log10(R) = const corresponds to stationary but possibly space-inhomogeneous Poisson seismicity, whereas a second cluster around log10(R) ≈ const corresponds mainly to aftershock clustering. We observe that the components in Figure 5a form a single cluster around the line of log10(V) + log10(R) = const, indicating Poisson seismicity. In case Figure 5b we observe no changes in the relative distribution of the volume and space components. The data set shown in Figure 5c the ETAS simulation contains a significant number of aftershocks. While the synthetically added aftershocks leave no signatures in the volume component a clustering at low values of the space component is visible. This shows that if a significant number of aftershocks would be present in the Basel data signatures should be identifiable in the space component of the nearest neighbor distance. However, Figure 5c also illustrates that due to the high occurrence rate of events in time/volume the cluster build by causally related event pairs (background-aftershock (blue) and aftershock-aftershock (black)) tend to some extend into the Poisson cluster (background-background (red) and aftershock-background (green)) making a clear classification of events as background events and aftershocks impossible. Nevertheless, as shown in Figure 5a no signs of aftershock triggering are identifiable from the nearest neighbor analysis for the induced seismicity in Basel. This analysis confirms that aftershock triggering is not significant for the Basel case study.
 We observe that the inter event time distribution of fluid-injection-induced earthquakes for six catalogs collected at geothermal injection sites at Soultz-sous-Forêts and Basel during phases of constant seismicity rate coincides with the exponential waiting time distribution of the HPP. This implies that successively occurring events are not causally related to each other. The waiting times for the complete event catalogs are distributed according to a NHPP in time. Our finding that the occurrence of events is given by a HPP in the volume domain strongly supports the idea that, in contrast to naturally triggered earthquakes, seismicity rate changes are primarily related to changes of the injection flow rate and not to the occurrence of aftershocks.
 The nearest neighbor analysis has revealed no signs of causal relationships between fluid-injection-induced events. The absence of aftershock signatures can be to some extend related to the high occurrence rate of earthquakes in space and time. However, the comparison to the ETAS model shows that, if a significant number of causally related events would be present, signatures should be identifiable by the nearest neighbor analysis. Therefore, coupling effects between events are weak. Our results demonstrate that the Poisson model can be applied to calculate occurrence probabilities of fluid-injection-induced earthquakes for seismic hazard assessment.
 We briefly explain the ETAS simulation method used to build the synthetic catalog analyzed in Figure 5c. First, we randomly select 50% of the Basel events occurring during injection of fluid. By randomly selecting the events we break a part of possibly existing causal relations. The selected events are used as background events in the ETAS simulation. We calculate the occurrence of aftershocks corresponding to the conditional intensity function λ(t) of the ETAS model [see Ogata, 1998] given by:
where λb(t) is the time dependent temporal intensity of background events randomly chosen from the Basel catalog, p, c and A = K/cp are the parameters of the Omori law, α is the productivity parameter and m0 is the magnitude of completeness. The second part of equation (A1) describes the probability of aftershock occurrence based on the history of event occurrence. The magnitudes of aftershocks are independently chosen from the Gutenberg-Richter distribution derived from the Basel catalog. To determine the location of an aftershock we first calculate the source radius rs of each event according to Brune  and Kanamori , as function of moment magnitude Mw and stress drop Δσ:
A probable aftershock is located on a spherical shell with radius rs around the mother event. Each position on the shell has the same probability to be the location of an aftershock. For the simulation we apply the following parameters derived from the Basel injection: Δσ = 2.3 MPa [Goertz-Allmann et al., 2011], m0 = 0.45, b = 1.5 and classical parameters of the Omori law [see, e.g., Touati et al., 2011]: A = 10, p = 1.2, c = 0.01 days. We use a productivity parameter α = 1.3 for simulation, because this choice results in an consistent number of events in the simulated and the real earthquake catalog.
 We thank the Federal Ministry for the Environment, Nature Conservation and Nuclear Safety as a sponsor of the project MAGS and the sponsors of the PHASE consortium project for supporting the research presented in this paper. Furthermore, we thank A. Jupe for provision of the Soultz data sets. The Basel data are kindly provided by M. Häring and are courtesy of Geothermal Explorers.
 The Editor thanks Mark Naylor and an anonymous reviewer for their assistance in evaluating this paper.