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Edited by: Mattia Pistone, Université de Lausanne, Switzerland

Reviewed by: Tom Sheldrake, Université de Genéve, Switzerland; Catherine Annen, Université Savoie Mont Blanc, France

This article was submitted to Volcanology, a section of the journal Frontiers in Earth Science

†Present Address: Kendra E. Murray, Department of Geosciences, Idaho State University, Pocatello, ID, United States

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Flood basalt volcanism involves large volumes of magma emplaced into the crust and surface environment on geologically short timescales. The mechanics of flood basalt emplacement, including dynamics of the crustal magma transport system and the tempo of individual eruptions, are not well-constrained. Here we study two exhumed dikes from the Columbia River Flood Basalt province in northeast Oregon, USA, using apatite and zircon (U-Th)/He thermochronology to constrain dike emplacement histories. Sample transects perpendicular to the dike margins document transient heating of granitic host rocks. We model heating as due to dike emplacement, considering a thermal model with distinct melt-fraction temperature relationships for basaltic magma and granitic wallrock, and a parameterization of unsteady flow within the dike. We model partial resetting of thermochronometers by considering He diffusion in spherical grains as a response to dike heating. A Bayesian Markov-Chain Monte Carlo framework is used to jointly invert for six parameters related to dike emplacement and grain-scale He diffusion. We find that the two dikes, despite similar dimensions on an outcrop scale, exhibit different spatial patterns of thermochronometer partial resetting away from the dike. These patterns predict distinct emplacement histories. We extend previous modeling of a presumed feeder dike at Maxwell Lake in the Wallowa Mountains of northeastern Oregon, finding posterior probability distribution functions (PDFs) that predict steady heating from sustained magma flow over 1–6 years and elevated farfield host rock temperatures. This suggests regional-scale heating in the vicinity of Maxwell Lake, which might arise from nearby intrusions. The other dike, within the Cornucopia subswarm, is predicted to have a 1–4 year thermally active lifespan with an unsteady heating rate suggestive of low magma flow rate compared to Maxwell Lake, in a cool near-surface thermal environment. In both cases, misfit of near-dike partial resetting of thermochronometers by models suggests either heat transfer via fluid advection in host rocks or pulsed magma flow in the dikes. Our results highlight the diversity of dike emplacement histories within the Columbia River Flood Basalt province and the power of Bayesian inversion methods for quantifying parameter trade-offs and uncertainty in thermal models.

The emplacement of flood basalt provinces involves a massive flux of magma into the crust from the mantle over 1–10 Myr timescales. Magmas that make it to the surface erupt to produce deposits with volumes of up to 10^{3}−10^{4} km^{3} that are the largest known effusive eruptive events. Large volumes of intrusive magmatism accompany surface eruptions, as constrained by exhumed networks of large dikes and sills (e.g., Burgess et al.,

Eruptive rates estimated based on radioisotopic dating of flow sequences suggests average eruption rates between ~ 10^{−2} and ~ 10^{1} km^{3}/yr (e.g., Burgess and Bowring, ^{2} − 10^{4} km^{3}/yr. Differences in these two estimates are not surprising, given that the geochronology considers an average over many flows and may include eruptive hiatuses. Such differences are an indication that eruptive tempo is highly unsteady over the duration of a typical flood basalt province.

Here we examine the problem of flood basalt eruption longevity and tempo through the lens of an exposed crustal transport network. We focus on two 8–10 m wide dikes exposed within the Chief Joseph Dike Swarm of the Columbia River Flood Basalts (CRFB) in northeastern Oregon, USA (

Map showing locations of the Maxwell Lake and Lee dikes, along with distribution of Columbia River Flood Basalt Chief Joseph Dike Swarm segments (green square on inset map).

To invert thermochronometric data for dike emplacement involves forward modeling of heat transport away from the dike and resetting of thermochronometers, both of which involve substantial uncertainties. We implement a Bayesian Markov-Chain Monte Carlo (MCMC) inversion as an objective information-gathering approach, in which many forward models with parameters stochastically sampled from prior constraints generate a posterior probability distribution function (PDF) representing the extent to which the data constrain unknowns of interest. In our case, these unknowns include dike longevity and time variation in magma flow, background temperature of the crust, thermal conductivity of wall rock, and parameters associated with diffusion of He in apatite and zircon grains.

MCMC inversion predicts that the two dikes have well-resolved but distinct thermal histories. The Maxwell Lake dike shows evidence for steady heating by magma over 1–6 years, within a warmed background environment at temperatures higher than a normal geotherm at paleodepth of ~ 2 km. The Lee dike, slightly thicker than Maxwell Lake, heated host rocks for a similar total duration of 1–4 years but at a highly unsteady rate that implies lower overall magma flux, in a normal geotherm at exposed paleodepth of < 1 km.

In what follows, we first introduce the regional geological context and prior work on the Maxwell Lake and Lee dikes. Sections 3–5 develop the methods for data analysis, forward modeling, and inversion. We then present the MCMC results in section 6, ending in section 7 with a discussion of implications for CRFB eruptive phenomenology and the general merits of the MCMC inversion approach for volcanologic data.

The CRFB are the youngest Large Igneous Province (LIP) on Earth, encompassing ~ 210,000 km^{3} basalt erupted in eastern Oregon, Washington and western Idaho, USA, in the interval between ~ 17 and 5 Ma. Although total volumes erupted are the smallest of known continental LIPs globally, the degree to which primary structures are well-exposed and a rich history of study (Reidel et al.,

CRFB dikes are variably exposed throughout the province, although the fraction of these dikes that fed surface flows is unknown. Dikes are particularly well-exposed in northeastern Oregon, where exhumation of the Wallowa mountains (Zak et al.,

We focus on two CJDS segments here, at Maxwell Lake in the Wallowa Mountains and in the Cornucopia region to the south of the Wallowas (^{4} km^{3}.

Previous work on the Maxwell Lake dike has focused on partial melt observed in wall rocks near the dike margin, which constrains the amount and duration of heat conducted away from the dike during active magma flow and subsequent cooling. Petcovic and Grunder (

The Lee dike segment is ~ 10 m wide and ~ 1 km long, in the Pine Lakes region of the Cornucopia stock in northeastern Oregon (45.0305°N, 117.250°W), dated to 120–123 Ma from biotite Ar-Ar and Zr U-Pb geochronology (unpublished data from Peter Zeitler and George Gehrels, ^{40}Ar/^{39}Ar dates on the most distal sample we collected were 123 and 122 Ma, respectively. The stock is dominantly two-mica trondhjemite (Taubeneck,

_{r} and _{i} in Equation (7).

At the Maxwell Lake dike, we collected ~ 2 kg samples at approximately 2, 5, 11, 20.5, 30, 40, 53.5, 72.5, and 100 m distance from the dike margin. Samples were processed using crushing, sieving, magnetic, and density separation methods at Zirchron, LLC in order to concentrate zircon and apatite crystals. At the University of Michigan HeliUM lab, we selected apatite and zircon grains for (U-Th-Sm)/He and (U-Th)/He analyses, respectively, based on size, clarity, morphology, and the lack of visible inclusions under 120–160x stereo zoom with a Leica M165C microscope. We analyzed three or four apatite and zircon grains per sample. Grain dimensions were digitally measured under the microscope prior to packing each grain in Nb foil. An Alphachron Helium Instrument was used for He extraction. Using a diode laser, apatite grains were heated to ~ 900 °C for 3 min and zircon grains were heated to ~ 1200 °C for 10 min. The extracted He was spiked with 3He, purified using gettering methods, and analyzed on a quadrupole mass spectrometer. A known quantity of ^{4}He was analyzed at regular intervals. We sent degassed grains to the University of Arizona for dissolution and U, Th, and Sm analysis using isotope dilution and solution HR-ICP-MS methods (Guenthner et al.,

Samples from the Lee Dike region were collected in a dike-perpendicular transect at distances ranging from 2 cm to 92 m from the dike margin. Apatite and zircon separates were prepared from 2-kg samples using standard separation procedures similar to those used for the Maxwell Lake samples, and (U-Th)/He measurements were made at Yale University using procedures described in Reiners et al. (

Thermochronometric ages from transects around the Maxwell and Lee dikes are shown in

The low-temperature thermochronologic record of geologically short-duration (days to thousands of years) heating events such as dike emplacement reflects a competition between the rapid rate of the thermal diffusion of heat through rocks and the comparably slow rate of the chemical diffusion of radiogenic daughter products (in the case of (U-Th)/He chronometers, ^{4}He nuclides) in individual apatite and zircon crystals. Both thermal and chemical diffusion depend on large-scale temperature gradients in the domain, although at different scales: thermal diffusion proceeds according to bulk (outcrop-scale) heat capacity and thermal diffusivity, while chemical diffusion is a function of diffusion domain (crystal-scale) size as well as experimentally determined activation energy and diffusivity of the daughter nuclides in the mineral of interest. The patterns of partial resetting around Maxwell Lake and Lee dikes suggest that granitic wallrocks at each location experienced different thermal histories, with a longer or hotter magmatic heat pulse at the Maxwell Lake vs. Lee.

The He content of apatite and zircon crystals is controlled by two processes: the time-dependent radiogenic production of ^{4}He during the alpha-decay of U and Th nuclides and the temperature-dependent diffusive loss of ^{4}He. Assuming a spherical diffusion geometry and uniform distribution of parent nuclides, the He concentration as a function of time ^{*} (scaled by radius

_{a} and follows an Arrhenius relationship

where _{0} is He diffusivity at infinite temperature (also called the frequency factor), and _{r}(

with U(

The general analytical solution for production-diffusion in a sphere (Equation 1), assuming a constant production rate _{r} of He (Wolf et al.,

defining a helium age He/_{r} as a function of time, the parameters that control ^{2} (Equation 2), and an initial He age ^{*}He/_{r}.

Temperature sensitivities of the apatite He and zircon He thermochronometers are commonly described using the closure-temperature concept (Dodson,

We calculate the expected fractional loss of He, _{d}, that an apatite or zircon crystal would experience given a thermal history predicted for a particular location by the thermal forward model described below. We do not use approximate solutions to diffusive He loss (Watson and Cherniak,

where σ_{T} is

We evaluate Equation (5) numerically using 1,000 terms. We calculate the approximate observed fractional resetting from the thermochronologic ages measured along each transect (Reiners,

where _{r} is the time elapsed since the heating event (emplacement of CRFB dikes), and _{i} is the unreset age (crystallization age of the host pluton).

The kinetic parameters, activation energy _{a} and frequency factor _{0}, vary for different minerals and are derived from diffusion experiments on apatite and zircon standards. For a single mineral system, _{a} and _{0} are also known to vary from crystal to crystal, most significantly as a function of self-irradiation dose (Shuster et al., _{a} and _{0} values for both apatite and zircon observed in natural crystals of low to moderate self-irradiation dose, as is appropriate for the U-Th compositions and ca. 120 Ma formation age of the crystals dated here.

There is an apparent positive correlation between available _{a} and _{0} values (_{a} as

where (dimensional) _{1} and _{2} define a power regression to the data in _{a,Ap} of 120–145 kJ/mol with a standard error of 1.414, _{2} = 5.074 in Equation (8). For zircon we explore a range _{a,Zr} of 160–175 kJ/mol with a standard error of 1.288, with _{2} = 2.083 in Equation (8). Standard errors calculated from the _{a} in

Observed variability in activation energy and frequency factor for the diffusion of He in _{a} and ^{4}He (natural) and ^{3}He (doped). Zircon data include the post-high-T results from Reiners et al. (

Ideally we would use the emplacement age of the basalt to constrain _{r}, and _{i} would correspond to the crystallization age of the plutonic host constrained from other methods. However, examination of _{r} and _{i} (dashed lines in _{r} and _{i} from the transects allows us to focus on the pattern of partial resetting, which encodes relative differences in dike-induced wall rock heating.

Emplacement of magmatic dikes induces a transient heating of host rocks such as recorded by the geothermometers described in section 3. Heat transport in host rocks is often assumed to be dominantly conductive, although advection of heated host pore fluid (or magmatic volatiles) also probably contributes. Coupled advection-diffusion of heat may also contribute mechanical impacts. For example, thermal pressurization of host pore fluid has been called upon to explain fracture patterns around some shallow dikes (Delaney,

The Maxwell Lake dike was previously modeled assuming conductive host rock heating in 1D, with either an analytic parameterization of magma emplacement (Petcovic and Grunder,

We assume that temperature varies in the dike-perpendicular dimension only. Small deviations from planar dike geometry are often observed and could contribute to complexities in the thermal field, but are excluded here. We model temperature evolution in a multicomponent 1D system consisting of dike material and host rock, following

where the index _{i} is the density of material _{p} the specific heat capacity, _{i}(_{i}(_{p} and

Melt fraction vs. temperature curves for basalt and tonalitic rocks. Black curves are reproduced from Petcovic and Dufek (

Equation (9) is solved on a spatial domain of length

where _{l}, 1 is the dike interior temperature (taken to be the magma liquidus) and _{BG} is the background temperature of the crust at the paleo depth of the dike. At one end of the domain, a Dirichlet boundary condition _{BG} for all time mimics a background geotherm. We choose a large domain size (

At the other end of the domain, we assume that the dike is actively transporting magma for a certain length of time _{f}, during which temperature in

and imposed only when _{d} ≥ _{bdy}(_{c} and τ_{w} as parameters that control the time variation of temperature within the dike. Parameter τ_{c} scales the dike's overall longevity; for example, if dike flow is modeled as a step function (as in Petcovic and Grunder, _{c} is the total duration of flow. Such a model likely oversimplifies many dike emplacement scenarios, because finite magma supply implies pressure gradients that decline over time, which leads to growing thermal boundary layers and decreasing temperature at the dike-host rock contact (Bruce and Huppert, _{w} controls how rapidly the dike temperature decreases around τ_{c}, as a model for such flow steadiness.

Equation 11 states that temperatures within the dike _{f}, the time at which point the tanh function is within a small threshold δ of unity (we take δ = 0.01). At this time, we assume that the dike has stopped transporting magma and switch to a Neumann boundary condition at the dike center for all subsequent time ∂_{f})/∂_{f}. Because _{l,1} > _{BG}, initially dike emplacement drives diffusion of heat away from the dike. However, if at any time prior to τ_{f} the specified dike temperatures imply _{d}(_{bdy}(_{f}).

_{c}, τ_{w} relate to τ_{f}. Larger values of τ_{w} result in dike temperatures that more gradually transition from the initial dike temperature (the liquidus _{l,1}) to a time-evolving host rock temperature at the dike boundary τ_{bdy}(

Normalized dike temperature from Equation (11), illustrating trade-offs between the scale for dike active flow τ_{c} and the scale for flow unsteadiness τ_{w}. τ_{f} is the total duration of temperatures elevated over background, and may be similar to τ_{c} for small τ_{w}. Equation (11) is enforced in our model for dike contact temperatures _{d} > _{bdy}(_{f}.

For granitic wall rocks we assume a melt fraction law _{2}(

where _{s,1} and _{l,1} are the basalt solidus and liquidus temperatures and

To solve Equation (9) we rewrite in terms of an effective heat capacity

ξ ∈ [0, 1] is the new spatial coordinate and λ is a stretching factor (Erickson et al.,

The choice of coordinate transform (Equation 13) is motivated by a need for implementing an efficient numerical solution to Equation (14) that retains high resolution near the dike for accurate prediction of partial melt, described in the next section. When discretized in space, a grid of evaluation points with uniform spacing becomes a staggered grid that concentrates points near the dike where more numerical accuracy is desired. Equation (14) is discretized using 2nd order centered finite differences and an adaptive 4th order Runge Kutta method in time. We have tested this code against a benchmarked numerical solution with equally spaced grid points (Karlstrom et al.,

Typical model output is shown in _{c} = 3 years, τ_{w} = 0.1 years in _{BG}. Resetting of thermochronometers is predicted by applying these time-temperature histories to Equations (5) and (7), which result in time dependent partial resetting as plotted in

Temperature as a function of distance from dike center _{BG} (red and blue colors). Dike interior temperature evolution is given by the black curve. Forced heating by magma flow is close to a step function in time (τ_{c} = 3 yr, τ_{w} = 0.1 yr, giving τ_{f} = 3.04 yr,

Relationship between partial resetting of apatite and zircon He thermochronometers as a function of heating and hold time (kinetics of thermal vs. chemical diffusion), with the same parameters as _{BG}. Curves on each panel represent listed distances

Epistemic and aleatoric uncertainties limit our ability to accurately invert for the longevity and steadiness of magma flow through CRFB dikes or the pre-intrusion temperature of host rocks. Aleatoric uncertainties in material parameters such as thermal conductivity, activation energy for Helium diffusion in zircon or apatite may trade off with uncertainties in the precise location of samples in relation to the dike. Such uncertainties are minimizable given sufficiently accurate sampling, calibration of fractional resetting models and experimental petrology, and sufficient resources to carry out the needed experiments. However, for the models described in sections 3–4, additional epistemic uncertainties such as the precise 3D geometry of the dike margin, the functional form of the dike temperature through time that parameterizes magma flow, or the presence of advective heat transport (essentially, whether our model describes the relevant physical problem), are convolved with aleatoric uncertainties.

We therefore approach inverse modeling as an information-gathering exercise specific to our hypotheses. We pose a forward model and then systemically vary unknown parameters to assess trade-offs and find a best fit. A generic data vector containing _{k} = _{k} − _{k}(

We implement a Bayesian inversion utilizing MCMC sampling of the parameter space. An introduction to Bayesian inversion can be found in Mosegaard and Tarantola (

Our goal is to derive a multidimensional posterior PDF that represents the inversion solution in accordance with Bayes' Theorem

Equation (15) states that the posterior PDF

It is worth emphasizing that in the framework of Bayesian statistics, the inverted parameters are given by distributions specified in the posterior PDFs rather than by single best-fitting values. Posterior PDFs reflect both our ability to predict the data with a model and our prior state of knowledge, for which we know some parameters very well and some very poorly. For all parameters in this study we assume a uniform prior distribution with upper and lower bounds. We assume that errors _{k} = diag(σ_{k}), with σ_{k} the variance associated with the error

where _{k} = _{k} − _{k}(_{k}| is the determinant of the covariance matrix. In our application, the number of datasets

We derive posterior PDFs for model parameters that minimize residuals through MCMC sampling. We implement the Metropolis-Hastings algorithm (Metropolis et al.,

Parameters, uniform prior parameter ranges, and MCMC inversion results.

_{BG} |
_{c} |
_{w} |
_{a, Zr} |
_{a, Ap} |
_{f} |
|||
---|---|---|---|---|---|---|---|---|

Prior lower bound | 25 | 0.1 | 0.1 | 1 | 160 | 120 | ||

Prior upper bound | 100 | 15 | 15 | 10 | 170 | 145 | ||

Best | 99.6 | 2.6 | 0.4 | 2.4 | 160.8 | 143.8 | 2.7 | 49.2 |

Median | 82.4 | 2.3 | 0.5 | 4.7 | 166.6 | 129.1 | 2.7 | 42.7 |

68% conf. interval | [65.8, 93.7] | [1.3, 4.6] | [0.1, 3.8] | [2.1, 8.1] | [162.3, 172.2] | [122.6, 138.2] | [1.4, 5.4] | |

95% conf. interval | [37.4, 99.0] | [0.9, 9.1] | [0.1, 12.0] | [1.2, 9.7] | [160.4, 174.6] | [120.3, 144.0] | [0.9, 10.2] | |

1.26 | 1.12 | 1.06 | 1.13 | 5.14 | 2.67 | 1.13 | ||

Best | 25.4 | 0.7 | 6.4 | 9.2 | 160.1 | 144.6 | 2.2 | 7.9 |

Median | 27.7 | 1.0 | 6.9 | 5.5 | 162.6 | 142.6 | 2.6 | 9.1 |

68% conf. interval | [25.7, 31.4] | [0.7, 1.6] | [3.9, 11.3] | [3.5, 8.2] | [160.7, 167.2] | [139.6 144.3] | [1.7, 4.1] | |

95% conf. interval | [25.1, 35.8] | [0.6, 2.3] | [2.2 14.4] | [2.0, 9.7] | [160.1, 172.6] | [136.0 144.9] | [1.1, 5.2] | |

1.48 | 1.24 | 1.09 | 1.15 | 7.25 | 6.71 | 1.15 |

_{f} is calculated from Equation (11). The Gelman-Rubin diagnostic

For the 1D model and data presented in sections 3 and 4, there are a minimum of 26 parameters that must be constrained at each dike: for both host rock and intruded basalt, we need solidus and liquidus temperatures, melt fraction exponent (if using Equation 12 as a model for compositional dependence of partial melting), mixture density, latent heat of fusion, and two thermal parameters (conductivity and heat capacity). For the fractional resetting calculation to match thermochronometric ages, we further need the activation energies, diffusivities, and mean grain sizes for He diffusion in both apatite and zircon, as well as un-reset ages (crystallization age of pluton) and reset ages (exact timing of dike emplacement). Finally, we have two parameters associated with the dike flow model, mean dike thickness, and the background temperature _{BG} of the domain.

Our use of solely thermochronologic and structural constraints for models is insufficient to resolve this number of parameters, so we reduce the parameter space dimensionality to focus on particular sensitivities of the inversion. We use experimental values for thermal and Arrhenius parameters in the thermochronologic fractional resetting calculation, then assume known un-reset and reset ages as dictated by the data, approximately the crystallization age and Grande Ronde emplacement age respectively. However, _{r} and _{i} are determined by the thermochronologic transects described in section 3 in order to focus model inversion on the partial resetting pattern. We assume a uniform dike width approximately equal to field measurements.

We can assess much of the inherent trade-offs in thermal models by considering only one unknown thermal parameter—a thermal conductivity _{a} (_{a,Ap}, _{a,Zr}, _{bg}, _{w}, _{c},

The MCMC method is guaranteed to converge given infinite time (e.g., Mosegaard and Tarantola, ^{4} − 10^{6}, e.g., Anderson and Segall,

The forward model described in section 4 is only marginally well-suited to this procedure. In each iteration, we must solve a partial differential equation for long enough time that the transient heat pulse from the dike propagates past our farthest-out partially reset samples ~ 100 m from the dike contact. As

In addition to time stepping, we must use sufficient spatial resolution near to the dike that partial melting of host rocks can be accurately predicted. We find ~ 1 m spatial resolution near the dike to be sufficient. As a final complication, a Dirichlet temperature condition far from the dike is used to enforce a background geotherm in 1D, and we must have a large enough spatial domain that this does not affect the transient heating near to the dike. We find through numerical experimentation that ~ 500 m total domain length is sufficient.

To accomplish the required number of simulations in a reasonable amount of time, we have minimized inversion complexity in two ways. First, we use a parallelized MCMC code package (MCMC Hammer, Anderson and Poland,

We run 70–75 Markov chains in parallel with randomized initial guesses until 2−4 × 10^{4} kept MCMC steps per chain are achieved. In total 2.4 × 10^{6} kept samples from all chains are incorporated into a posterior PDFs for Maxwell Lake, with a mean accepted fraction of 0.55 across all chains (meaning that roughly twice as many simulations were performed as kept). For the Lee dike, 3.0 × 10^{6} kept samples are incorporated into a posterior PDF and the MCMC algorithm exhibited a mean accepted fraction of 0.23. Three diagnostics of MCMC convergence are discussed in

Marginal posterior PDFs for the Maxwell Lake and Lee dikes are shown in _{f}) that were sampled during MCMC inversion in

Marginal posterior PDFs for _{f} is derived using Equation 11). Black horizontal bars correspond to 68% confidence intervals associated with expectation values from the posterior PDF, while black symbols are the median of the distribution (

MCMC marginal posterior PDFs as in _{BG}, τ_{c}, τ_{w}, _{a,Ap}, _{a,Zr} are estimated parameters, while total flow duration τ_{f} is derived using Equation (11). Covariance plots show how parameters trade off with one another within the posterior PDF, warmer colors indicate higher probability.

MCMC marginal posterior PDFs as in _{BG}, τ_{c}, τ_{w}, _{a,Ap}, _{a,Zr} are estimated parameters, while total flow duration τ_{f} is derived using Equation (11).

In general, we find that the overall pattern of partial resetting is well-fit by our model. Although total residual errors for the Maxwell Lake dike are lower than for the Lee dike (

Example fits of Maxwell Lake and Lee dike thermochronology data, plotting curves associated with the median of MCMC-derived PDFs (

Posterior PDFs for each dike exhibit the well-known trade-offs between parameters in a heat conduction model (

The Maxwell dike is best explained by an emplacement scenario in which sustained dike heating (and thus magma flow) occurred over τ_{f} ~ 1.4–5.4 years, based on the 68% confidence intervals for the posterior PDF (_{w} in the range of 0.1−3.8 years implies that unsteady flow is not likely, so τ_{f} as derived through Equation (11) is largely a function of the timescale for active flow τ_{c}. Background temperature is constrained to be between 65.8−93.7°C, and is relatively sharply peaked. On the other hand thermal conductivity is predicted to be between 2.1 and 8.1 W/mC, but is not sharply peaked implying poor constraint overall. Activation energies for apatite and zircon are even more poorly constrained. Good fits to thermochronometric ages can be attained for any value of _{aAp} and _{a,Zr}, and the Gelman-Rubin diagnostic

Further refinements might be made if we assume particular parameter values, due to covariance between some parameters. For example, _{a,Ap} is seen to be well-correlated to _{BG}, with higher background temperatures implies higher activation energy parameters. Likewise, a longer lived dike (larger _{f}) implies lower background temperature _{bg} and lower wallrock conductivity. Tonalite rock conductivity of 3 W/mC as assumed by Petcovic and Dufek (

For the Lee dike, predicted total heating duration is similar to Maxwell Lake with τ_{f} ~ 1.7–4.1 years. However, in contrast to Maxwell Lake significant time variation in heating is predicted, with large flow unsteadiness scale τ_{w} ~ 3.9–11.3 years implying an immediate and extended decrease in boundary temperature over the active duration τ_{f}. This is illustrated in _{c} and τ_{w} for each dike. Thermal conductivity is predicted to be in the range of _{BG} ~ 25.7–31.5 C. The Lee dike inversion appears to place somewhat better constraints on activation energies, predicting _{a,Zr} ~ 160.7−167.2 kJ/mol and _{a,Ap} ~ 139.6−144.3 kJ/mol. However, MCMC convergence is questionable for these parameters (

_{bdy} taking parameters τ_{c}, τ_{w}, _{BG} from the median of MCMC posterior distributions (_{d} from Equation (11). _{bdy} diverges from _{d} in time due to thermal buffering from heated wall rocks.

Our forward models predict not only the temperature distribution but also the extent and degree of partial melt in the dike and tonalitic host rock (

At Maxwell Lake, partial melt is readily observed both in handsample and thin section, and was mapped petrographically by Petcovic and Grunder (

Large Igneous Provinces such as the Columbia River Flood Basalt Group represent an end member of volcanic activity in which large volumes of magma are rapidly emplaced into the Earth's crust and surface environment, with consequences for tectonics, climate, and life. Constraints on the tempo of these events are diverse but always indirect as there are no historical examples of similar magnitude eruptions. This study introduces new constraints on emplacement histories of two 8–10 m wide CRFB dikes from low-temperature thermochronology, using a Bayesian inversion framework that provides estimation of parameters governing heat transport into wall rocks. Previous work by Petcovic and Grunder (

There is considerable uncertainty in physical parameters—and to some extent the physical processes themselves—involved in dike emplacement. We utilize a forward model that is relatively simple in the spectrum of magma transport modeling: a parameterization of unsteady but monotonically varying magma advection within a dike that governs 1D heat conduction in host rocks. We also simplify the description of He atomic diffusion in modeling partial resetting of thermochronometric ages, by parameterizing the role of grain size as a function of activation energy in zircon and apatite. Such simplifications permit a Bayesian Markov Chain Monte Carlo inversion approach for exploring a six dimensional parameter space with ~ 10^{6} forward calculations to resolve posterior PDFs that outline the most likely parameter values.

Conductive heating models fit the apatite and zircon partial resetting patterns in granitic host rocks well, except for regions near to the dike contact. The manner of this misfit (overpredicted ages near to the dike and thus underprediction of heat pulse propagation) is likely evidence for either pulsed non-monotonic flow, or advection of heat by fluids. Petcovic (

The Maxwell Lake dike has been geochemically linked to the Wapshilla Ridge member of the Grande Ronde formation (Petcovic and Dufek, ^{3} km^{3} erupted in multiple events contained within Wapshilla Ridge). However, macroscale structural characteristics of the Maxwell Lake dike are similar to thousands of other Chief Joseph Dike Swarm segments in the region as illustrated in

Although the total duration of active heating (and thus magma advection) from the Lee dike is not predicted to be significantly different than the Maxwell Lake dike by our inversion, the time variation of dike-host contact temperature and thus magma flux rate is resolvably different. Our model for unsteady heating at the dike/host rock contact (Equation 11) parameterizes the growth of thermal boundary layers separating flowing magma initially at its liquidus temperature from host rocks with parameters τ_{c} and τ_{w} (

We leave explicit consideration of flow mechanics, and thus the assessment of dike-transported flow volumes, for a future study. However, a simple model illustrates some of the expected implications of our results for relative flux between the two dikes. Volumetric flux of unidirectional viscous flow through a slot of length ℓ and halfwidth

where _{bdy}(_{M}, _{L} cubed. Due to the assumption that boundary temperature is proportional both to vertical pressure gradient and dike thickness, the flux ratio is proportional to boundary temperatures _{bdy,M}, _{bdy,L} raised to the fourth power, giving

Equation (18) is plotted in _{bdy,L}(

MCMC inversion predicts a cooler background temperature (by some 50–70 degrees) for the Lee dike relative to Maxwell. Distinct background temperatures would be expected based on the paleodepth differences between Maxwell and Lee dikes. But partial resetting of apatites out to >100 m away from the Maxwell dike contact indicate temperatures that are hotter than any reasonable geotherm at ~ 2 km depths. We speculate that CRFB magmatism transiently increased the regional geothermal gradient (e.g., Murray et al., ^{2} (Morriss and Karlstrom, ^{3}/yr. Crustal magma transport of this magnitude is likely sufficient not only to induce regional heating but also to modulate the bulk rheology of the crustal column participating in magmatism (Karlstrom et al.,

Marginal posterior PDFs for thermal conductivity of wall rocks are somewhat similar for each dike, in both cases suggesting conductivities larger than typical for intact granitic rocks. We view this result as an indication that our physical model is incomplete. Near-surface advection of hydrothermal fluids are an obvious missing ingredient, as this would result in more efficient heat transport away from the dike contact and a larger apparant thermal conductivity. Such results might also be consistent with the lesser degree of partial melt in the vicinity of Maxwell Lake and Lee dikes than predicted by our model (

Our inversions do not constrain activation energies for the He diffusion model at either Maxwell Lake or Lee dikes, as indicated by the shapes of the PDFs (95% confidence intervals for _{a,Zr}, _{a,Ap} overlap for both dikes) and our metrics of inversion convergence (the Gelmin-Rubin diagnostic

Bayesian Markov-Chain Monte Carlo inversion of low temperature thermochronology around two CRFB dikes near the spatial locus of the Chief Joseph Dike Swarm in NE Oregon suggests distinct emplacement histories. By combining physics-based forward models with a probabilistic approach to inversion, we can identify differences in likely patterns of magma flow and in the background temperatures far from the dike contact at the time of emplacement. Although both dikes are of similar spatial dimensions, the Maxwell Lake dike likely transported a larger volume of magma for a longer time than the Lee dike, consistent with near-dike host rock petrography and the correlation of Maxwell Lake dike to Wapshilla Ridge flows. The Lee dike has composition similar to Wanapum lavas but is not yet linked to any particular surface expression. Our inversion suggests that application of thermochronology to other CRFB dikes in granitic host rocks that are common in the Wallowa Mountains region could resolve feeder vs. non-feeder dikes and magma transport mechanics on a LIP scale.

More broadly, this study suggests that Bayesian inversion methods have utility in inverting volcanologic data. We have focused on thermochronology, jointly inverting for partial resetting of two different chronometers by combining models for dike-scale heat conduction with grain-scale models for Helium diffusion in zircon and apatite. Other data, for example from high temperature geothermometry or paleomagnetic partial resetting, could easily be incorporated into this framework and might contribute to a more well-constrained posterior PDF. Model complexity does represent a significant challenge, due to the requirement that many thousands of forward models are needed for the MCMC inversion to converge. More effort is needed to develop reduced-order models for magma transport processes that are faithful parameterizations of computationally intensive multiphysics simulations. The payoff is a more robust predictive understanding of magmatic processes at a level commensurate with data granularity, which is a needed step toward connecting observations of active processes with the geologic record of magmatism.

LK performed MCMC inversions, wrote the forward modeling codes, and the manuscript. KM performed Maxwell Lake dike measurements. PR performed Lee dike measurements. LK and KM devised the study. All authors contributed to manuscript editing.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Matthew Morriss for field assistance during sampling of the Maxwell Lake dike and for discussions surrounding the Chief Joseph Dike Swarm, Heather Petcovic for helpful information about prior work at Maxwell Lake, and Victoria E. Lee for assistance during sampling of the Lee dike. LK acknowledges field work support from the National Science Foundation EAR 1547594. Peter Zeitler and George Gehrels are acknowledged for providing dates on the Cornucopia pluton at the Lee Dike. We thank Kyle Anderson for developing the MCMC Hammer code used for data inversion, Stefan Nicolescu for analytical assistance with the Lee dike samples, and Amanda Maslyn for analytical support at the University of Michigan HeliUM lab. Two reviewers and editor Mattia Pistone provided constructive comments that improved the manuscript.

The Supplementary Material for this article can be found online at:

Table S1 contains thermochronology data for each grain analyzed, sample locations, ages, and errors for Maxwell Lake and Lee dikes.

^{40}Ar/

^{39}Ar thermochronometry of Martian meteorites

Markov Chain Monte Carlo involves the production of a dependent sequence or chain of values which, if run for long enough time, will converge to the underlying posterior distribution (Mosegaard and Tarantola,

We perform three tests to assess convergence of our MCMC inversions. As a first test, we computed the posterior distribution with different kept sample populations, experimenting with the number of ‘burn in' samples discarded and whether or not to “thin” the distribution (i.e., discard every ^{4}, for the tested range of thinning and step size, shapes of the posterior PDFs do not change much. This qualitative test provides some confidence that we are adequately sampling the true posterior.

A second, more quantitative test involves the autocorrelation for each parameter in each chain. ^{3}, the mean autocorrelation for every parameter and both dikes has dropped to near zero and fluctuates about zero as lag increases further. This suggests randomized sampling and is in agreement with the qualitative threshold found in the first test.

As a third test, we compute the Gelman and Rubin (_{a,Zr} and _{a,Ap} are far in excess of 1.2 and the background temperature _{BG} is close to but not under 1.2 for both dikes.

This could indicate that the MCMC has not converged for these parameters and longer chains are required. However, there are also some reasons to be wary of _{BG} below 25 C, and above 100 C is implausible in the upper few km of crust. That the predicted posterior PDF for _{BG} is skewed toward the prior lower limit for the Lee dike, or that the flow unsteadiness parameter τ_{w} is screwed toward its lower limit for the Maxwell Lake dike, is likely a physical result. We thus have reason to suspect that