The application of a technique from quantum dynamics to the governing equation for hydraulic head leads to a trajectory-based solution that is valid for a general porous medium. The semi-analytic expressions for propagation path and velocity of a change in hydraulic head form the basis of a travel-time tomographic imaging algorithm. An application of the imaging algorithm to synthetic arrival times reveals that a cross-well inversion based upon the extended trajectories correctly reproduces the magnitude of a reference model, improving upon an existing asymptotic approach. An inversion of hydraulic head arrival times from cross-well slug tests at the Widen field site in northern Switzerland captures a general decrease in permeability with depth, which is in agreement with previous studies, but also indicates the presence of a high-permeability feature in the upper portion of the cross-well plane.

Understanding the spatial variation in subsurface flow properties is important for many applications, such as groundwater extraction and storage, hydrocarbon production, geothermal energy generation, and waste water disposal. Advanced production processes like hydraulic fracturing require the development of high-resolution reservoir models necessary to capture the influence of the fractures (Zhang et al., 2014; Fujita et al., 2015). Often there are very few observations with which to infer such properties: typically measurements from a few wells intersecting a formation of interest. However, developments such as cross-well transient pressure testing (Hsieh et al., 1985; Paillet, 1993; Karasaki et al., 2000); and hydraulic tomography (Tosaka et al., 1993; Gottlieb and Dietrich, 1995; Butler et al., 1999; Yeh and Liu, 2000; Vasco and Karasaki, 2001; Bohling et al., 2002, 2007; Brauchler et al., 2003, 2010, 2011, 2013; Zhu and Yeh, 2006; Illman et al., 2007, 2008; Fienen et al., 2008; Bohling, 2009; Cardiff et al., 2009, 2013a, b; Huang et al., 2011; Sun et al., 2013; Paradis et al., 2015, 2016), have improved the ability to resolve two- and three-dimensional variations in hydraulic properties. New techniques, including fiber-optic temperature and pressure observations, and geophysical observations sensitive to pressure changes (Yeh et al., 2008; Rucci et al., 2010; Marchesini et al., 2017), will further improve spatial and temporal coverage and generate large data sets. Finally, the joint interpretation and inversion of geophysical and hydrological data leads to better constrained imaging of flow properties (Rubin et al., 1992; Hyndman et al., 1994, 2000; Vasco et al., 2001; Vasco, 2004; Kowalsky et al., 2004; Day-Lewis et al., 2006; Brauchler et al., 2012; Lochbühler et al., 2013; Soueid Ahmed et al., 2014: Ruggeri et al., 2014; Jimenez et al., 2015; Binley et al., 2015; Linde and Doetsch, 2016).

The characterization of complicated aquifer and reservoir models using sizable data sets points to the need for robust and efficient approaches for modeling pressure propagation. To this end, there are a number of approaches that aim to reduce the computational burden and data handling requirements associated with hydraulic tomography. For example, there are methods that reduce the governing equation to a simpler form for the moments of the transient head or pressure variation (Li et al., 2005; Yin and Illman, 2009; Zhu and Yeh, 2006). There are also approaches for the analysis of sinusoidal and oscillatory pumping tests that are based upon the phase shifts and amplitude differences between observed and calculated pressure variations, using these phase shifts to infer properties between two wells (Bernabe et al., 2005; Black and Kipp, 1981; Cardiff et al., 2013b; Kuo, 1972; Rasmussen et al., 2003; Renner and Messar, 2006). Another technique relies upon a measure of the arrival time of a pressure pulse or disturbance as a basis for transient travel-time imaging or tomography (Vasco et al., 2000; Kulkarni et al., 2001; Brauchler et al., 2003, 2007, 2010, 2011, 2013; He et al., 2006; Hu et al., 2011; Vasco and Datta-Gupta, 2016). Finally, there are methods that attempt to find lower-dimensional representations of the model or of the matrices describing the forward and inverse problems. These methods include principal component analysis (Lee and Kitanidis, 2014), Karhunen–Loèève expansions (Zha et al., 2018), and reduced-order models (Liu et al., 2013).

There are at least three advantages associated with the use of travel times, an alternative to the direct treatment of the entire transient head or pressure waveforms. First, the arrival of the early onset of the transient pressure pulse can be much sooner than the time at which steady-state conditions are achieved. Thus, cross-well slug tests can be conducted rapidly, facilitating improved spatial coverage. Second, the relationship between such travel times and hydraulic diffusivity is quasi-linear and convergence to a solution is not as sensitive to the initial model as it is for the direct inversion of transient pressure waveforms (Cheng et al., 2005). Third, the interpretation and reduction of transient head or pressure waveform data can be more complicated due to the sensitivity of amplitudes to various factors such as the packer coupling, the calibration of the receiver transducers, and the conditions surrounding the borehole.

Previous trajectory-based formulations of pressure arrival-time tomography relied upon an asymptotic approach that assumes smoothly varying properties (Vasco et al., 2000; Brauchler et al., 2003, 2007; He et al., 2006; Vasco, 2008; Vasco and Datta-Gupta, 2016). This assumption is certainly violated in many commonly encountered situations, such as a layered sedimentary environment and in the presence of faults or fractures. Here we apply a newly developed trajectory-based technique for travel-time tomography that dispenses with the assumption of smoothly varying properties, enlarging its range of validity to any model that may be treated using a numerical simulator (Vasco, 2018; Vasco and Nihei, 2019). The semi-analytic approach provides insight into factors controlling the propagation of a pressure transient in a complex porous medium. As shown here, the expression for the trajectories may form the basis for efficient sensitivity computations. These sensitivities are particularly useful in inverting transient pressure propagation times and in hydraulic travel-time tomography. All of the sensitivities required for the interpretation of a pressure test can be obtained in a single numerical simulation of the test. We apply the method to cross-well hydraulic tomographic imaging, considering both synthetic and field pressure arrival times.

In this section we describe our iterative algorithm for updating an aquifer model in order to improve the fit to a set of observed arrival times. We shall only discussion the elements of the derivation of Vasco (2018), as well as a perturbation technique, which are essential for understanding the inversion procedure. The approach involves a number of steps, beginning with the equation governing the transient variation in hydraulic head, and ending with a linear system of equations to be solved for the aquifer parameters. As an overview, the major steps of the methodology are shown schematically in Fig. 1. The approach is an offshoot of trajectory-based techniques developed in quantum dynamics for the study of large chemical systems (Wyatt, 2005; Liu and Makri, 2005; Goldfarb et al., 2006; Garashchuk, 2010; Garashchuk and Vazhappilly, 2010; Garashchuk et al., 2011; Gu and Garashchuk, 2016). As shown in Vasco (2018), the trajectory-mechanics treatment leads to a set of coupled ordinary differential equations that may be solved numerically, as is done in quantum mechanics. However, one can take advantage of existing numerical simulators to compute one of the unknown vector fields, reducing the system to a single set of equations for the trajectory (Vasco, 2018). The result of this analysis is a semi-analytic expression for the path of a transient pulse. This expression, along with a perturbation technique, provides a basis for an efficient method for imaging spatial variations in hydraulic diffusivity in the subsurface – a form of travel-time tomography. We illustrate the procedure with applications to both synthetic and observed arrival times in the section that follows this description.

Schematic illustration of the approach used to obtain the sensitivities that form the basis for the linearized, iterative, cross-well imaging algorithm.

We begin with the equation governing the evolution of a
transient variation in hydraulic head

From this point on, we shall assume that the hydraulic head has been normalized by dividing
both sides of Eq. (

A primary application of the trajectories described above will be to estimate flow properties between boreholes via hydraulic tomographic imaging. In this procedure a series of pumping tests are conducted in isolated segments of one borehole. During each test a rapid injection is used to generate a transient fluid pressure pulse that propagates to pressure sensors in an adjacent well. For an impulsive source, the time at which the peak pressure is observed in the adjacent borehole is defined as the arrival time. For the inverse problem we determine the flow properties from the arrival times observed in isolated sections of the monitoring well. In order to solve the inverse problem we must relate the travel time of the pressure pulse to the hydraulic properties of the medium.

Our approach to the solution of the nonlinear inverse problem will be iterative
in nature.
That is, in order to estimate flow properties we begin with an
initial model and progressively update it, solving the forward problem
of reservoir simulation at each step.
We shall need model parameter sensitivities, which are
the partial derivatives of each observation with respect to changes in
each of the model parameters (Jacquard and Jain, 1965), for every iterative update.
We will be interested in transient pressure arrival times that are
defined as the time at which the peak of a pressure pulse is
observed at a measurement point.
Expression (Eq.

Model parameter sensitivities, in this case relating small changes in
the slowness along the trajectory,

In order to update the model and the head or pressure field
using a numerical simulator, we shall need to map the
updated slowness estimates into the reservoir model parameters

Several trajectory-based methods for pressure arrival-time tomography
(Vasco et al., 2000; Brauchler et al., 2003; He et al., 2006;
Hu et al., 2011; Vasco and Datta-Gupta, p. 131, 2016)
utilize a high-frequency asymptotic solution to the diffusion equation.
A major assumption of such solutions is that the pressure variation is rapid in time
(Virieux et al., 1994)
or that the dominant frequencies in a Fourier transform of the trace are high.
Equivalent results can be obtained if we assume that the medium
properties are smoothly varying in comparison with the length scale
associated with the propagating pressure transient or that parameters take on
values in a particular range (Cohen and Lewis, 1967).
In that case we can neglect the divergence term on the right-hand side of
Eq. (

From the high-frequency asymptotic solution and the ray equations,
Vasco et al. (2000) derived a semi-analytic expression,
in which the square root of the peak arrival time is given by the line integral
along the trajectory

In Vasco (2018) the limitations of the high-frequency asymptotic approach are discussed and illustrated. In particular, it is shown that for abrupt boundaries and sharp layers, the trajectories calculated using the eikonal equation bend too strongly into high-permeability regions of a half-space or layer. This leads to deviations in the trajectories from regions with high model parameter sensitivity, and the potential for errors when updating a simulation model. In the next section we will explore these limitations in the context of hydraulic tomography, using both synthetic and experimental data.

A reservoir model is typically defined over a two- or three-dimensional grid
that is used by a numerical reservoir simulator.
For such a discrete model with properties defined on a grid of cells,
and where one assumes constant values within each cell of the model,
we can break up the path integrals Eqs. (

In the iterative, linearized inversion scheme that we shall
adopt here, we start with an initial model, perhaps derived from well logs,
denoted by

Due to errors in the data and modeling approximations, we do not
expect that the system of Eq. (

The linearized expression Eq. (

Cross-hole hydraulic travel-time tomography and cross-well slug tests are valuable approaches for imaging spatial variations in flow properties (Paillet, 1993; Yeh and Liu, 2000; Vasco and Karasaki, 2001; Bohling et al., 2002; Butler et al., 2003; Brauchler et al., 2007, 2010, 2011). Such tests can resolve features between boreholes, similar to cross-well geophysical imaging, and are directly sensitive to flow properties. In this section we set up a synthetic hydraulic tomographic test, roughly based upon a field experiment at the Widen site in Switzerland. Following that, we analyze data from the actual field experiment, using them to image the spatial variations of permeability between two shallow boreholes.

The overall setup of the test example is shown in Fig. 2, along with the reference model.
A set of sources in each well, denoted by filled squares and open circles,
transmit transient pressure signals to various receivers located in the adjacent borehole.
The reference distribution, a three-dimensional permeability model with a
dominantly vertical variation in properties, was generated stochastically.
That is, a uniform number generator was used to derive permeability multipliers
between 1 and 12 for each layer in the model.
A uniform random variation of 50 % was introduced within each layer
and this variation was smoothed using a three-point moving window.
The model extends an additional 5 m in the

Reference model for the cross-well test example. A cross-section through the permeability model representing the cross-well plane. The cross-well configuration, for imaging flow properties between two boreholes, consists of pressure sources in the two wells (filled squares and open circles) transmitting transient pulses to receivers (open circles) in an adjacent well. The source–receiver geometry mimics that of the field experiment conducted in Widen, Switzerland. The color scale varies linearly between permeability multipliers from 1 to 12.

The reservoir simulator TOUGH2 (Pruess et al., 1999) was
used to model the complete set of cross-well slug tests that comprised the full synthetic experiment.
The computations were conducted using a three-dimensional mesh with
constant pressure boundary conditions, simulating a 300 s transient pressure
test for each source.
This interval provided enough time for any head variation to propagate from a source
to the receivers due to the high background permeability of

In order to image the permeability variations between the boreholes we conducted a series of linearized inversion steps,
where we solve the system of Eq. (

The sum of the squares of the residuals for the eikonal-based

The regularization weightings for each approach,

The spatial variation in the permeability multiplier
resulting from inversions based upon the eikonal equation

At each iteration we solve for a permeability multiplier,
a factor that is multiplied by the background permeability of
the uniform starting model to get the estimated permeability.
A total of 10 iterations for the eikonal-based algorithm took 6 s, whereas 10 iterations
for the extended trajectory approach took 129 min, illustrating the computational
advantage provided by an inversion approach based upon the eikonal equation.
In Fig. 3 we plot the misfit reduction as a function of the
number of iterations for both the high-frequency inversion algorithm (eikonal) and an inversion based
upon the extended trajectories computed using Eq. (

Observed versus calculated arrival times for both the eikonal-based

The final updated high-frequency solution, plotted in Fig. 4, contains higher permeabilities between about 5.5 and 7.0 m. However, the amplitude of the permeability multiplier is less than that of the reference model (Fig. 2). Furthermore, the amplitude of the high-permeability feature at around 9.0 m is underestimated, perhaps due to its narrow width of less than 1 m. The iterative inversion based upon the extended trajectories does image the two higher-permeability zones seen in the reference model (Fig. 2). The estimated amplitude of the features appears to be closer to those of the reference model but it does overestimate the permeability of the lower feature and underestimates the permeability of the upper zone.

A better idea of the differences in the magnitude of the two solutions is conveyed in Fig. 5,
where we plot the depth variation of the reference, eikonal-based, and extended trajectory-based
models.
That is, we display the depth variation of the average of the two models,
along with the upper and lower permeability multiplier values obtained in each depth interval.
It is evident that the solution provided by a conventional imaging algorithm which uses the
eikonal equation displays permeability changes with depth that are much smoother than the
reference model.
The extended approach does contain

Figure 6 provides more information regarding the misfit reductions for the inversions based upon the eikonal equation paths and the extended paths. It displays the calculated travel times plotted against travel times calculated using the reference model shown in Fig. 2. Both the initial travel times, calculated using the homogeneous background model used to start the inversions, and the final travel times based upon the models obtained at the conclusion of the algorithms, are shown in the plots. The initial travel-time estimates are all larger than the actual values calculated using the reference model. This is to be expected because the largest anomalies are the approximately order-of-magnitude increases associated with the upper and lower high-permeability layers in the model. The high-permeability channels promote rapid pressure propagation between the boreholes. The eikonal equation-based algorithm does reduce the average of the calculated travel times but does not lead to good fits. The inversion based upon the extended trajectories produces relatively good fits to the reference travel times.

An important aspect of the inverse problem is an assessment of the resulting model parameters and
estimates of their reliability.
As noted in Sect. 2, the calculation of two key components of the model assessment –
model parameter resolution and model parameter covariance or uncertainty – follow from the
generalized inverse

We end our treatment of the synthetic test with a discussion of some validation calculations,
in which additional sources were introduced to mimic independent pumping tests.
Two tests were simulated, with one source at the left edge of the model shown in Fig. 2, at
a height

Validation exercise in which arrival times for two tests that were not used in the inversion are calculated based upon the final models estimated using the eikonal and extended approaches. These calculated times are plotted against travel times computed using the reference model.

The Widen field site, adjacent to the Thur River in northern Switzerland (Fig. 9), has been the subject of numerous geophysical and hydrological studies (Lochbühler et al., 2013). The primary goal of the work at the Widen site is to understand the hydrologic, ecologic, and biochemical effects of river restoration. The geophysical and hydrological experiments focused upon a sandy gravel aquifer that is in contact with an unrestored section of the river (Doetsch et al., 2010). The area was penetrated by a number of boreholes and is relatively well characterized. Borehole cores revealed that the roughly 7 m thick sandy gravel aquifer is overlain by a silty sand layer and that it sits atop a thick impermeable clay aquitard. Early work at the site included individual and joint inversions of cross-well seismic, radar, and electrical resistance tomography for a zoned model (Doetsch et al., 2010). The model was consistent with the three-layer structure defined by the existing boreholes. This study was followed by several others, including a cross-hole ground-penetrating radar investigation (Klotzsche et al., 2010), and three-dimensional electrical resistance tomographic (ERT) imaging of river infiltration into the site (Coscia et al., 2011, 2012). The three-dimensional ERT imaging indicates that the highest flow velocities occur in the middle of the aquifer, whereas the lowest speeds are at the base of the sequence in clay and silt-rich gravels. A joint inversion of geophysical and hydrological data (Lochbühler et al., 2013) between several well pairs was used to constrain spatial variations in reservoir storage and hydraulic conductivity. That study imaged the large-scale decrease in hydraulic conductivity with depth.

Schematic map of the Widen field site located adjacent to the Thur River in Switzerland, as indicated in the inset. The labeled wells, P2, P3, and P4, were used for several hydraulic tomographic experiments.

Cross-well slug interference tests, as described in Brauchler et al. (2010, 2011), were conducted at the site and are discussed in Lochbühler et al. (2013). In such tests, a near-instantaneous change in hydraulic head in a packed-off section of one well generates a fluid pressure transient in the surrounding region. Pressure transducers in isolated sections of a nearby well are used to measure the pulse that propagates between the wells. Both the travel time of the pulse and its amplitude can be used to infer hydraulic properties between the wells (Vasco et al., 2000; Brauchler et al., 2007, 2011; Vasco, 2008). Cross-well interference slug tests were conducted at two well pairs at the Widen site, as described by Lochbühler et al. (2013). The wells P2, P3, and P4 are roughly in a line that parallels the Thur River at a distance of 15 m from the river bank (Lochbühler et al.; 2013), as shown in Fig. 9. For our work we will focus on the P2–P3 well pair, where P3 is the source well and P2 is the observation well, some 3.5 m to the west. The tomographic system consists of two double-packers in each well, where the extent of the isolated regions was 0.25 m and the spacing of the intervals was 0.5 m. A suite of observed pressure variations for receivers in the observation well are shown in Fig. 10. We are interested in the propagation time of the pulse, as measured by the arrival time of the peak pressure at each observation point, which is referenced to the time at which the peak pressure is obtained in the source interval.

Hydraulic head, from a cross-well slug test, recorded at a set of packed-off intervals in observation well P2 from the Widen field site. Each trace has been normalized in order to have a unit peak amplitude.

The overall inversion methodology was discussed and illustrated above and the details will not be repeated here.
During one step of the iterative linearized algorithm we minimize the weighted sum of the squared misfit,
the model norm, and the model roughness, as given in Eq. (

Mean squared error for an inversion of the hydraulic head arrival times.
The inversion labeled eikonal is based upon the eikonal equation and uses
high-frequency asymptotic trajectories.
The open circles are the mean squared error
calculated using travel times produced by the TOUGH2 numerical simulator.
The filled squares denote the mean squared error as a function of the number
of iterations of an inversion scheme that utilizes the
extended trajectories that follow from Eq. (

Initial (open circles) and final (filled squares) misfits for both
the eikonal-based and trajectory-mechanics-based inversions.

The final models produced by the two inversion algorithms are plotted in Fig. 13. Both models display generally higher permeabilities at shallower depths, with values decreasing as the lower edge of the model is approached. The anomalies are largely horizontal, suggesting a generally layered structure, which is in agreement with previous studies (Klotzsche et al., 2010; Lochbühler et al., 2013; Jimenez et al., 2016; Somogyvari et al., 2017; Kong et al., 2018). The magnitude of the permeability variations is larger in the trajectory-mechanics-based inversion and a high-permeability layer is evident in Fig. 13. These general features are observable in the upper and lower permeability bounds plotted as a function of elevation in Fig. 14. Both models display a decrease in permeability with depth, but the variations in the eikonal-based inversion are somewhat smaller than those of the extended trajectory approach.

We can compare our results to previous work by Lochbühler et al. (2013), where a joint inversion of cross-well ground-penetrating radar travel times and hydraulic tomography (travel times and amplitudes) was discussed. In Fig. 15 the spatial variations of the logarithm of hydraulic conductivity corresponding to our inversion grid are plotted to the same color scale. These results correspond to part of Fig. 4h in Lochbühler et al. (2013). In addition, we extracted the highest and lowest permeability values as a function of depth in the inversion region and the average permeability at each elevation. All results show the same general decrease of permeability with depth in the aquifer, as the clay aquitard is approached. The variations in permeability in the extended approach are of the same order as the joint inversion result. As in the synthetic case, the magnitude of the variations in the eikonal equation inversion is smaller.

Upper (crosses) and lower (open circles) permeability values as a function of elevation within the model. The laterally averaged permeabilities are also plotted as filled squares in each panel.

As a validation effort, we left out data from the fifth source from the bottom in Fig. 13
when conducting the inversion for the permeability multipliers.
This allowed us to use the resulting models of

Validation test in which arrival times from source five, which was not used in the inversion, are calculated based upon the final models estimated using the eikonal and extended approaches. These calculated times are plotted against the observed travel times.

Lastly, we used Eqs. (

The trajectory-mechanics approach described in Vasco (2018) and applied here is very general and can be used to model other hydrological processes such as tracer transport (Vasco et al., 2018a) and multiphase fluid flow. One advantage associated with transient pressure is the rapid propagation of a disturbance in comparison with the velocities associated with fluid transport. Thus, transient cross-well pressure testing can be conducted relatively rapidly in formations with moderate hydraulic conductivity. This is particularly true when transient pressure travel times, such as the arrival time of the peak of a pressure pulse or the peak of the time derivative of the pressure (Vasco et al., 2000) are used. For the Widen field experiment the peaks are observed in the first few seconds of the measured traces in the adjacent borehole. Another advantage of hydraulic travel-time tomography is that the relationship between the arrival times and the hydraulic conductivity or diffusivity is quasi-linear (Cheng et al., 2005). Thus, the final model resulting from an inversion of travel times is less sensitive to the initial or starting aquifer model and less likely to become trapped in a local minimum. Finally, travel-time tomography provides an element of data reduction, from an entire transient pressure waveform, to a single arrival time. This can be advantageous when dealing with many intervals from multiple boreholes, time-lapse pressure changes, or large data sets derived from geophysical observations.

We have presented two examples of hydraulic tomographic imaging, one using synthetic transient pressure arrival times and the other using data from an experiment at the Widen field site on the Thur River in northern Switzerland. We do find that an algorithm based upon the eikonal equation is significantly faster than one utilizing the extended trajectories calculated using a reservoir simulator, taking only about 10 s compared with 129 min. From the synthetic application we find that an imaging technique based upon the eikonal equation, the current method used for trajectory-based modeling, has difficulty accurately imaging large and abrupt changes in permeability. Such rapid spatial changes in flow properties are a common occurrence in geologic media, with the presence of layering and fractures, with correspondingly large variations in hydraulic conductivity. For example, in well logs it is quite common to observe thin layers with permeabilities that are orders of magnitude larger than values in the surrounding formations. Indeed, in our field case at the Widen field site we image an order of magnitude change in permeability in agreement with previous results at the site. While the eikonal equation is much faster and can recover large-scale spatial variations, it is likely to produce smoothed images of sharp features and to underestimate rapid changes in properties. Thus, the approach is useful as a rapid reconnaissance tool, as in real-time imaging, and for regions where the properties are thought to be smoothly varying. This usage is supported by that fact that both the eikonal-based and the extended trajectory-based methods share the quasi-linearity of travel-time inversion approaches (Cheng et al., 2005), and are less sensitive, in comparison with inversions based upon head magnitudes, to the initial or starting model.

For a full analysis and interpretation of field data, however, we recommend the trajectory-mechanics approach; this is due to the fact that it does not invoke assumptions about model smoothness
and is therefore more robust and accurate, yet it retains the semi-analytic
sensitivities that are characteristic of trajectory-based approaches.
The semi-analytic sensitivities are computed after a single simulation, using either
numerical methods to solve the coupled system for

The approach that we have described is useful for imaging permeability
variations between boreholes but it does have some limitations.
The use of slug tests limits the allowable distance between wells that
may be used for imaging variations in

The pressure data from the Widen field test are available on the Zenodo archive (

DWV devised, implemented, and executed the inversion algorithm. JD provided and helped with the data analysis, worked on the comparison with previous studies, and participated in writing the paper. RB conceived and carried out the field experiments at the Widen field site, provided the observations and data reduction, and helped to write the paper.

The authors declare that they have no conflict of interest.

Work performed at Lawrence Berkeley National Laboratory was supported by the US Department of Energy under contract number DE-AC02-05-CH11231, Office of Basic Energy Sciences of the US Department of Energy.

This research has been supported by the US Department of Energy, Office of Basic Energy Sciences (contract no. DE-AC02-05-CH11231).

This paper was edited by Gerrit H. de Rooij and reviewed by two anonymous referees.