Corresponding author: Y. Harada, Department of Geophysics, Kyoto University, Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan. (firstname.lastname@example.org)
 The origins of the lunar crustal magnetic fields remain unclear although dozens of magnetic field measurements have been conducted on and above the lunar surface. A major obstacle to resolving this problem is the extreme difficulty of determining a surface distribution of small-scale magnetization. We present a new technique to map small-scale magnetic fields using nonadiabatic scattering of high-energy electrons in the terrestrial plasma sheet. Particle tracing, utilizing three-dimensional lunar magnetic field data synthesized from magnetometer measurements, enables us to separate the contributions to electron motion of small- and large-scale magnetic fields. We map significant kilometer-scale magnetic fields on the southwestern side of the South Pole-Aitken basin that are correlated with larger-scale magnetization. This implies that kilometer-scale magnetization may be ubiquitous over the lunar surface and related to the large-scale magnetization.
 The Moon does not possess a global intrinsic magnetic field, but it has remanent crustal magnetic fields distributed widely and nonuniformly over the lunar surface [Dyal et al., 1974] that are known as magnetic anomalies. Lunar crustal magnetization has been extensively studied for decades, but its origins remain one of the key unsolved problems in the study of lunar history [Collinson, 1993]. Critical hurdles include the difficulty in mapping lunar magnetic fields with a wide range of spatial scales from a few kilometers or less up to hundreds of kilometers.
 Direct measurements of lunar crustal magnetic fields are provided by magnetometers installed on lunar orbiting spacecraft as well as by those deployed on the lunar surface. When a lunar orbiter travels over a magnetic anomaly, the magnetometer detects Moon-related field variations, which are fixed to the selenographic coordinate system. After detrending the external field variations, we can map the global distribution of the three components of the lunar crustal magnetic fields [Hood et al., 2001; Purucker, 2008; Tsunakawa et al., 2010]. In practice, the orbital altitude, usually tens of kilometers or more, limits the detectable spatial scale length of the crustal fields. The strength of small-scale magnetic fields distributed on the surface decreases sharply with altitude and eventually becomes smaller than the magnetometer noise level at the orbital altitude. Lunar surface magnetometers made direct measurements of the surface magnetic fields at the Apollo and Lunokhod-2 landing sites [Dyal et al., 1974; Dolginov et al., 1976]. Strong crustal magnetic fields up to ∼327 nT were measured, and both the field strength and polarity vary significantly over a few kilometers or less. The surface magnetic field data are only available for a few landing sites, and it remains uncertain whether kilometer-scale magnetization exists elsewhere on the lunar surface.
 Another approach is indirect estimation of crustal magnetic fields using charged-particle motion in the vicinity of magnetized regions. The most frequently used remote sensing technique is “electron reflectometry” (ER), which is based on the magnetic mirror effect [Anderson et al., 1975; Halekas et al.2001; Mitchell et al., 2008]. The ER technique assumes adiabatic behavior, which means that the field variation encountered by electrons within a single gyration orbit is small compared with the initial field. This assumption is not appropriate for magnetic fields with spatial scales smaller than the electron gyrodiameter (200 eV electrons have a gyrodiameter of 9.5 km in a 10 nT magnetic field), and crustal-field strengths could be significantly underestimated by the ER technique in the presence of kilometer-scale magnetization [Halekas et al., 2010].
 At present, the difficulty in mapping kilometer-scale magnetic fields hinders our ability to interpret the surface distribution of crustal magnetic fields in a comprehensive way. Here we present a new technique to map the small-scale magnetic fields from electron and magnetic field observations in the terrestrial magnetotail, using data from instruments on the Kaguya (SELENE, SELenological and ENgineering Explorer) spacecraft. In contrast to the ER technique, we use nonadiabatic scattering of plasma sheet electrons with energies greater than 1 keV. The combination of the electron observations by Kaguya MAP-PACE (Magnetic field and Plasma experiment-Plasma energy Angle and Composition Experiment) [Saito et al., 2008, 2010] and particle-tracing calculations using lunar magnetic field data synthesized from the Kaguya MAP-LMAG (Lunar Magnetometer) measurements [Shimizu et al., 2008; Takahashi et al., 2009; Tsunakawa et al., 2010] allows us to extract valuable information about the small-scale crustal magnetic fields of the Moon. In this paper, we use “large-” and “small-”scale magnetic fields to describe crustal magnetic fields with spatial scales larger and smaller, respectively, than the lowest orbital altitude of Kaguya at which the magnetometer measurements are used for producing the synthesized lunar magnetic field data (i.e., ∼20 km).
2 Mapping Small-Scale Magnetic Fields
 When the Moon is located in the terrestrial plasma sheet, the lunar surface is exposed to hot electrons. These relatively high-energy electrons are usually absorbed when they strike the lunar surface. This results in a partial loss in the electron velocity distribution function [Harada et al., 2010, 2012]. However, Kaguya sometimes observed high-energy electrons with energies >1 keV traveling upward from a particular local area on the lunar surface. Figures 1a–1b show electron angular distributions from one of these observations at a low altitude of 23 km, whereas Figure 1c shows electron trajectories derived from these electron distributions. Here we trace back the electrons from the spacecraft in a spatially uniform magnetic field using a fourth-order Runge-Kutta integration method. These electrons seem to come from a local area on the lunar surface within a single gyration, implying nonadiabatic scattering by the crustal magnetic fields.
 We now investigate the effect of the large-scale magnetic field on high-energy electron trajectories by introducing three-dimensional lunar magnetic field data synthesized from magnetometer measurements into our particle-tracing calculations. We use the Equivalent Pole Reduction (EPR) method, one of the equivalent source models for mapping large-scale crustal magnetic fields [Toyoshima et al., 2008; Tsunakawa et al., 2010]. This method solves the inverse problem of a surface density distribution of equivalent monopoles (“magnetic charges”) to fit the magnetic field measurements. The three components of the lunar magnetic fields are calculated at arbitrary positions above the surface from positive and negative magnetic charges distributed on the surface. We applied the EPR method to the LMAG measurements obtained over 12 consecutive orbits on 7 May 2009 at low altitudes of ∼20–30 km (Figure 2a) in the terrestrial magnetotail (plasma densities <0.3 cm−3derived from the moment calculation). Figure 2b shows the resulting magnetic charge distribution and the radial component at the Kaguya altitudes calculated from the magnetic charge distribution. These synthesized magnetic field data include lunar crustal fields with length scales greater than 20 km.
 Figure 1d shows the electron trajectories obtained from tracing calculations that include the synthesized lunar magnetic field data. Compared with the uniform magnetic field case shown in Figure 1c, some of the electrons are traced back to outer space, suggesting that these electrons are scattered by the large-scale crustal fields. However, most of the electrons still originate from near the surface even if large-scale magnetic fields are taken into account. This suggests a significant contribution to the electron trajectories from crustal fields that are absent from the synthesized large-scale magnetic field data, i.e., the small-scale magnetic fields (see Figure 3a).
 We can extract information about the small-scale magnetic fields on the lunar surface from these observations of high-energy electron scattering. If we assume that the scale length of the surface magnetic field is much smaller than the electron gyrodiameter (i.e., the nonadiabatic limit), the electron trajectory will be bent sharply as shown in Figure 3b. Since the magnetic force is always perpendicular to the electron velocity, a horizontal component of the surface magnetic field is necessary to modify the electron trajectory upward from the surface. For a given scattered velocity at the surface (cf. v1in Figure 3b), we can derive a lower limit of electron momentum change caused by magnetic scattering, by considering the electron trajectory where the deflection angle, Δψ, is the smallest (cf. the dashed trajectory in Figure 3b). From the equation of motion of an electron with charge e and mass me in a magnetic field B, medv/dt=ev×B, the velocity change of the electron along this trajectory, Δv, satisfies the equation
where Bh⟂is the horizontal component of the surface magnetic field perpendicular to the electron velocity, Lh⟂is its horizontal scale length, and v is the electron speed. The product Bh⟂Lh⟂ can be derived as
A single electron-trace calculation gives a vector of v1, and we can easily calculate Δv and convert it to Bh⟂Lh⟂. Thus, a number of electron-tracing calculations can constrain the lower limit of the product of the horizontal magnetic field component and its horizontal scale length, BhLh, at a particular position on the lunar surface.
 We now present the result of mapping small-scale horizontal magnetic fields in the southwestern side of the South Pole-Aitken (SPA) basin. We use electron data from Kaguya MAP-PACE during the same 12 orbits. We trace the electrons with energies >1 keV if the observed counts are greater than 2. The particle-tracing calculation includes both the lunar magnetic field data synthesized by the EPR method and the ambient magnetic field vector obtained by subtracting the synthesized magnetic field data from LMAG measurements at the time when the electron is observed. If the electron traced from the spacecraft strikes the lunar surface, its location and velocity at the surface are recorded. After analyzing all data from the 12 orbits, we divide the surface into 3 km bins and take the maximum value of Δv (or equivalently Bh⟂Lh⟂) in each bin. Since each surface velocity gives a lower limit for BhLhat the point where the electron is scattered, we can constrain BhLh for each bin by taking the maximum value. Figure 2d shows the mapping obtained with this procedure. The colors represent the lower limit of BhLhthat needs to be added to the synthesized large-scale field data to account for the electron observations.
 Here we briefly discuss the mapping of the small-scale magnetic fields on the lunar surface. The inferred BhLh is up to 200 nT km. Since the synthesized lunar magnetic field data already include the large-scale magnetization (>20 km), we can reasonably assume Lh<20 km. This yields the strongest Bh>10 nT. Although this value is not surprisingly strong compared with the Apollo surface measurements of up to hundreds of nanotesla, our result suggests the existence of kilometer-scale magnetization with strength comparable to that of the large-scale magnetization (<20 nT as shown in Figure 2b). While the Apollo and Lunokhod-2 surface measurements were all made on the Moon's nearside, we now find significant small-scale magnetization in the southwestern side of the SPA basin on the farside. This implies that kilometer-scale magnetization is not a unique characteristic of the nearside landing sites but may be ubiquitous over the lunar surface.
 Note that the small-scale horizontal magnetic fields inferred from the high-energy electron data seem to be correlated with the large-scale horizontal fields. The strong small-scale horizontal field regions indicated by the bright red colors in Figure 2d tend to be found in regions where a strong large-scale horizontal magnetic field exists, as denoted by the warmer colors in Figure 2c. This correlation suggests that the small-scale magnetization is not randomly distributed but related to the large-scale magnetization. Our result also implies that the observed keV electrons from the surface are crustal-field related, not caused by random transient features of the ambient field.
 We note that high-energy electrons are expected to be less sensitive to near-surface electric fields than low-energy electrons. In fact, the negative surface potentials estimated from the upward electron-beam energies observed during this time period are a few hundred volts, insufficient to reflect the keV electrons. In addition to the well-established adiabatic reflection of low-energy electrons, nonadiabatic scattering of high-energy electrons can be used as a complementary method to investigate the small-scale component of the crustal magnetization.
 The authors express their sincere thanks to the MAP-PACE and MAP-LMAG team members for their great support in processing and analyzing the MAP data. The authors also express their gratitude to the system members of the SELENE project. This work was supported by a Research Fellowship for Young Scientists awarded by the Japan Society for the Promotion of Science.
 The Editor thanks Jasper Halekas and Erika Harnett for their assistance in evaluating this paper.