## 1. Introduction

[2] Over recent years, a number of hydrological studies have investigated multiple steady states [*Dent et al.*, 2002; *Anderies*, 2005; *D'Odorico et al.*, 2011; *D'Odorico and Porporato*, 2004; *Heffernan*, 2008; *Hilt et al.*, 2011; *Peterson et al.*, 2009a; *Rennermalm et al.*, 2010; *Ridolfi et al.*, 2006; *Runyan and D'Odorico*, 2010]. For those using physical based models, different quasi-equilibrium states and fluxes can emerge for one set of parameters and forcing, depending on the initial conditions. Between two steady states is a threshold, and the state variable distance to it from a steady state is a measure of the cumulative disturbance the system state can withstand before switching steady states. Often, this *distance* is adopted as the measure of a state's resilience (see section 2.2 of *Peterson* [2009], for a detailed review). For the multiple steady states to have the potential to emerge the system must have a positive biophysical feedback, but the emergence of the multiple steady states depends upon the strength of this feedback relative to the forcing. These concepts, and associated methods, draw heavily from the field of ecosystem resilience; where *Holling* [1973] defined resilience as the magnitude of the disturbance that a system can absorb without undergoing a regime shift, while *Scheffer et al.* [2001] states that a sufficiently severe perturbation of an ecosystem state may bring the system into the basin of attraction of another state.

[3] The potential importance of resilience concepts to hydrology has been highlighted in a number of reviews [*Rodriguez-Iturbe et al.*, 2007; *Dent et al.*, 2002; *Asbjornsen et al.*, 2011]; and hydrological modeling has been criticized for often implicitly assuming no positive feedbacks, and the existence of only one steady state, and thus assuming the system to be infinitely resilient to, say, climatic disturbances [*Peterson et al.*, 2009a]. While the field of ecosystem resilience has been modeling multiple steady state systems since *Holling* [1973] and *May* [1977], only recently have hydrologists begun to numerically explore these concepts. Quantitative investigations of multiple hydrological attractors have been undertaken into vegetation-soil moisture [*D'Odorico et al.*, 2005; *Guttal and Jayaprakash*, 2007; *D'Odorico et al.*, 2008]; unconfined aquifer-vegetation interactions [*Peterson*, 2009; *Peterson et al.*, 2009a; *Anderies*, 2005; *Ridolfi et al.*, 2006; *Runyan and D'Odorico*, 2010]; wetlands [*D'Odorico et al.*, 2011; *Heffernan*, 2008]; and peatlands [*Rennermalm et al.*, 2010]. Like quantitative ecosystem resilience studies, many of these have ignored the spatial dimension. Recently, a spatial dimension has been incorporated into studies of soil moisture vegetation [*Guttal and Jayaprakash*, 2009; *van Nes and Scheffer*, 2005; *Dakos et al.*, 2010; *van de Koppel and Rietkerk*, 2004; *von Hardenberg et al.*, 2001]; groundwater flow [*Peterson et al.*, 2009a, 2009b]; interconnected lakes [*Hilt et al.*, 2011]; and lake nutrients [*Serizawa et al.*, 2009] to reveal complex dynamics not apparent from the 1-D modeling. Despite the appeal of these concepts and the emergence of complex behavior from simple models, very few studies have undertaken field investigations of multiple hydrological attractors [*D'Odorico and Porporato*, 2004; *Heffernan*, 2008].

[4] To quantify the steady states (henceforth referred to as *attractors*) within a model, *Ludwig et al.* [1997] detailed a number of fundamental methods. A simple method is to conducted time-integration simulations from very different initial state variable values using nonstochastic forcing [e.g., *van Nes and Scheffer*, 2005; *Peterson et al.*, 2009b]. If the simulations converge to different steady states then multiple attractors exist. However, while simple to implement, this technique cannot easily identify the threshold between the attractors (henceforth referred to as *repellor*) and is cumbersome for exploring how the number of attractors change with a model parameter. A more advanced technique, called equilibrium or limit cycle continuation analysis or codim 0 bifurcation [*Kuznetsov*, 2004], has been adopted by some to quantify how the number of attractors change with a single model parameter [*Peterson*, 2009; *Ridolfi et al.*, 2006; *Runyan and D'Odorico*, 2010; *Ludwig et al.*, 1997]. It is a powerful technique for understanding complex nonlinear model behavior but very often it is undertaken using complex numerical methods, such as that of *Dhooge et al.* [2003], that render the concept opaque to most readers [e.g., *Peterson et al.*, 2009a]. An exception is that by *Ludwig et al.* [1997], in which two ecosystems are modeled by single ordinary differential equations (ODE) and the ODEs are analytically rearranged to quantify the attractor and repellors with a change in a model parameter. While being a worthy and transparent example of one-parameter equilibrium continuation analysis, there remains a considerable gap between the broader mathematics of continuation analysis [*Kuznetsov*, 2004] and those applied to resilience science. Recently, more advanced studies have quantified the number of attractors in two parameter space; however, the methods are often insufficiently detailed or require repeated time-integration simulations from differing initial conditions and parameter combinations [*van de Koppel and Rietkerk*, 2004; *van Nes and Scheffer*, 2005; *Møller et al.*, 2009; *Anderies et al.*, 2006], rather than formal two-parameter continuation methods. To encourage wider application within hydrology, this paper presents an analytical demonstration of one and two-parameter continuation analysis of a 1-D ODE model for an unconfined aquifer. The shape of each attractor basin is then quantified using a method based on Lyapunov stability curves and the results compared against those from simulations under stochastic forcing. New insights into the switching between attractors are identified and then drawn upon to propose new measures of resilience. While the 1-D groundwater model is very simple, relative to both conventional groundwater models and other ecosystem resilience models, this simplicity allowed the clear demonstration and development of a wide range of resilience analysis methods. Some of the methods may be problematic to apply to more complex models, but the authors hope that the clear presentation of resilience techniques will encourage adoption of resilience concepts and techniques within hydrology and expand the tools available to resilience scientists.