2.1 Background of Cosmic Rays on Earth and Neutron Detection
 Victor Hess received the Nobel Prize in Physics (1936) for his discovery of cosmic rays in 1912. During the midtwentieth century, scientists found, through both theoretical and experimental work, that the intensity of low-energy cosmic ray neutrons depends on the chemical composition of the material, in particular the medium's hydrogen content due to its high moderation power (as summarized with references in Zreda et al. ). Fast neutrons (~1 MeV), a tertiary cosmic ray flux created by high-energy secondary cosmic ray neutrons, exist in a well-mixed reservoir comprising soil and air [Zreda et al., 2012]. During the moderation process, fast neutrons can mix at the scale of hundreds of meters in air and tens of centimeters in soil [Desilets, 2011].
 The principles of neutron detection with proportional counters are well established [Knoll, 2000]. Here we use the moderated or fast neutron detector implemented in the COsmic-ray Soil Moisture Observing System (COSMOS) [Zreda et al., 2012]. The fast neutron detector is shielded by 2.5 cm of plastic, making it most sensitive to neutrons between 1 and 1000 eV [Desilets, 2011]. We note from neutron transport modeling that the relationship between average hydrogen content and neutron flux is nearly identical over these energy ranges (T. E. Franz, unpublished data, 2013).
2.2 Estimation of Biomass Water Equivalent Using Fast Neutron Intensity
 While the fast neutron detector used here was originally designed to measure soil water dynamics in the near surface over large horizontal areas (~28 ha, a circle with radius of ~300 m at sea level in dry air [Desilets and Zreda, 2013], vertical depths of ~10 cm in wet soil and ~70 cm in dry soil [Zreda et al., 2008]), it is sensitive to all hydrogen inside its measurement volume [Zreda et al., 2012]. By independently quantifying soil moisture (and other nonbiomass hydrogen pools) with direct sampling, we are able to isolate the biomass water signal component in the fast neutron intensity measurements following the framework outlined in [Franz et al., 2013]. Because soil water is typically the largest pool of hydrogen present in the near surface, its uncertainty will control the measurement precision of the biomass hydrogen pool. Despite the large horizontal and vertical heterogeneity, we found from extensive soil moisture field sampling at numerous COSMOS sites (108 total samples collected at each site at six depths, 0–30 cm every 5 cm, and 18 horizontal locations, 0–360° every 60°, and radii of 25, 75, and 200 m) that the standard error of the mean soil moisture as a function of mean soil moisture has a parabolic shape with a maximum value of 0.008 m3/m3 equivalent to 2.4 mm of water or 2.4 kg/m2 at a soil water content of 0.275 m3/m3 (Figure S2).
 In order to isolate the biomass water signal from the convoluted fast neutron intensity measurements, we need to make several assumptions and simplifications about the instrument support volume, instrument sensitivity, estimation of various hydrogen pools inside the support volume through direct sampling, and distribution of hydrogen pools within the support volume. Because the various hydrogen pools can be aggregated in a thin layer (i.e., soil pore water, soil mineral water, and vegetation) or dispersed across the entire support volume (i.e., water vapor and forests), our framework either directly accounts for the mass of the hydrogen pool or removes its signal from the convoluted signal using derived relationships from neutron transport simulations [Franz et al., 2013; Zreda et al., 2012].
 Here we assume that the instrument support volume (86% cumulative sensitivity) is a hemisphere above the surface with a constant radius of 300 m as defined by previous work [Desilets and Zreda, 2013; Zreda et al., 2008]. We note that Desilets and Zreda  found that the support radius is reduced by 20 m per additional 10 g of water per kilogram of air but does increase with elevation above sea level. In order to remove the water vapor component from the fast neutron intensity measurements, we use measurements of surface air temperature, air pressure, and relative humidity to determine the absolute humidity [Rosolem et al., 2013]. Rosolem et al.  found the neutron correction factor as
where CWV is the scaling factor for temporal changes in cosmic ray intensity as a function of changes in atmospheric water vapor, (g m−3) is the absolute humidity at the surface, and (g m−3) is the absolute humidity at the surface at a reference condition (here we use dry air, = 0 ).
 Because vegetation in forests may be distributed in clumps across the support volume, we introduce an additional correction factor relating the efficiency of neutron moderation from discrete objects versus an equivalent layer of water on the surface (CBWE). Neutron transport simulations indicate that the equivalent layer efficiency factor depends on both the total volume and surface area of tree trunks (Figure S3). Therefore, a priori information is needed about the size and distribution of trunks inside the measurement volume. We note that the support radius and depth will be only slightly reduced by the presence of aboveground biomass given the large open space for neutrons to travel unimpeded. In order to correct for dispersed hydrogen pools above the surface, we define the dry atmosphere neutron counting rate N (counts per hour, cph) as
where NL2 is the level 2 neutron counting rate (http://cosmos.hwr.arizona.edu/) previously corrected for variations in incoming high-energy particles and absolute pressure deviations [Zreda et al., 2012].
 Below the surface, we assume that the support volume is a cylinder with a fixed radius of 300 m and a depth that varies with surface water, soil pore water, mineral water, and bulk density. Assuming no surface water and uniform distributions of soil pore water, mineral water, and bulk density, Franz et al.  found the effective sensing depth z* (cm) to be approximated by the following equation [Franz et al., 2012]:
where 5.8 (cm) represents the 86% cumulative sensitivity depth of low-energy neutrons in liquid water; 0.0829 is controlled by the nuclear cross sections of SiO2; ρbd is the dry bulk density of soil (g cm−3); ρw is the density of liquid water assumed to be 1 (g cm−3); τ is the weight fraction of lattice water in the mineral grains and bound water, defined as the amount of water released at 1000°C preceded by drying at 105°C (gram of water per gram of dry minerals); and SOC is the soil organic carbon (gram of water per gram of dry minerals, estimated from stoichiometry using measurements of total soil carbon, TC, and soil CO2, ). Here measurements of lattice water, total soil carbon, and soil CO2 were made on an ~100 g composite sample (subsampled from the 108 soil moisture samples) collected at the study site and analyzed at Actlabs Inc. of Ontario, Canada.
 With the estimates of sensor support volume, we can compute the mass and molar mass of each element in the system. We assume that the atmosphere is composed of only nitrogen (79% by mass) and oxygen (21% by mass) and follows a standard lapse rate. We further assume that the subsurface is composed of solid grains (pure quartz, SiO2, lattice water, and SOC water equivalent) and soil pore water. Here we assume that the wet aboveground biomass, AGB (kg/m2), is composed of only water (50% by mass) and cellulose (C6H10O5, 50% by mass) but note that this is dependent on plant species and time and should be quantified directly. With the estimates of volume, mass, and chemical composition, we can calculate the hydrogen molar fraction, hmf (mol mol−1), in the cosmic ray probe support volume as
where ∑ Hi is the sum of hydrogen moles from lattice water Hτ, soil organic carbon water equivalent HSOC, pore water Hθ, and vegetation HAGB inside the support volume, and ∑ Ai is the sum of all moles from air NO, soil SiO2, lattice water H2Oτ, soil organic carbon water equivalent H2OSOC, pore water H2Oθ, and aboveground biomass C6H10O5 + H2OAGB inside the support volume.
 Using neutron transport modeling simulations of various soil chemistries, Franz et al.  derived a single relationship between hmf and relative neutron counts:
where Ns represents the site- and instrument-specific fast neutron count rate at saturation (i.e., over liquid water, where the count rates approach a constant value). We note that in this work, we used a single known value of hmf and the corresponding N to specify the free parameter NS in order to minimize site, instrument, and sampling uncertainties. By measuring (either at one snapshot in time or continuously) fast neutron intensity, water vapor, soil pore water, and soil mineral water, estimates of BWE (=0.556*C6H10O5 + H20ABG) can be calculated using equations (1) to (5).