## 1. Introduction

[2] Argon isotopes are measured extensively in radioisotopic dating experiments on both terrestrial and extraterrestrial samples. In lunar samples, the stable isotopes ^{36,38,40}Ar are derived from several sources, including decay of ^{40}K, solar implantation, and spallation reactions induced by cosmic radiation [*Turner et al.*, 1971]. There may also be contamination from terrestrial atmospheric argon. Short-lived ^{37}Ar and ^{39}Ar are artificially created by neutron irradiation, primarily of ^{40}Ca and ^{39}K [*Merrihue and Turner*, 1966; *Turner*, 1970; *Turner et al.*, 1971]. (Neutron irradiation creates small amounts of the stable argon isotopes as well, but we monitor and correct for these interfering nuclear reactions. Likewise, small amounts of cosmogenic ^{37,39}Ar are created on the Moon, but the saturation concentrations of these isotopes are negligible, and moreover many ^{37}Ar half-lives have elapsed since the lunar samples were brought to Earth.) Ratios of argon isotopes in several partial releases may be used to evaluate the relative importance of these sources, and to calculate the isotopic compositions of some of these components.

[3] *Albarède* [1978] was the first to use isotopic data from stepwise heating experiments to model the initial distribution of each isotope in a sample. If the initial distribution of argon isotopes in lunar samples can be known, it would be of great importance for constraining the energy spectrum of implanted solar particles, identifying terrestrial contamination, deducing thermal histories, or recognizing argon losses due to nuclear recoil during neutron irradiation.

[4] Lunar impact spherules [*Culler et al.*, 2000; *Levine et al.*, 2005] (see Figure 1) are glass droplets that quench from melted or vaporized rock in the aftermath of a meteorite impact on the Moon. To determine spherule formation ages with the ^{40}Ar/^{39}Ar isochron technique, *Culler et al.* [2000] and *Levine et al.* [2005] degassed individual spherules by stepwise heating with infrared lasers. Impact spherules can be distinguished from lunar volcanic spherules by several criteria [e.g., *Delano and Livi*, 1981], and *Culler et al.* [2000] and *Levine et al.* [2005] argued for impact origin of their spherules chiefly on the basis of their chemical compositions, chemical heterogeneity, and young ages. In this paper, we use the argon isotopic data acquired in these geochronology experiments to follow *Albarède* [1978] in asking whether and how stepwise release data can meaningfully constrain the initial distribution of argon isotopes in a sample. Lunar spherules provide a particularly simple geometry in which to attempt this analysis.

[5] Argon is transported within specimens by diffusion [e.g., *Turner et al.*, 1973], and atoms are released when they diffuse through the surface. Each partial release therefore includes mixtures of argon from many different initial locations in the spherule. Forward modeling of the diffusion process consists of calculating how much argon would be released in each step of a specified heating program, given a certain initial distribution of argon isotopes. We review approaches to the forward problem in section 2. Because the diffusion equation is linear in the concentration, we may imagine the initial concentration to be a linear superposition of several functions, the evolutions of which we can calculate separately. In subsequent sections, we follow *Albarède* [1978] in attempting the inverse of this problem, constraining the initial distribution of argon with the measured partial releases in each heating step. The linearity of the diffusion equation makes this a linear inverse problem.

[6] Our work takes advantage of the speed of contemporary digital computers to perform a more complete search for inverse solutions than could have been achieved earlier. We are therefore able to generalize and extend the method of *Albarède* [1978], and develop new algorithms with which to approach the inverse diffusion problem. In the remainder of this paper, we present solutions to the forward and inverse problems, and apply these to the argon data acquired from lunar impact spherules by *Culler et al.* [2000] and *Levine et al.* [2005].