## 1. Introduction

There is considerable demand for reliable and accurate measurements of surface fluxes over different land surfaces to develop better understanding of land–atmosphere interactions in order to improve models and prediction capability. A key component in both water and energy balances is the evaporation, or latent heat flux. Catchment-scale information on the water balance is essential for assessing flood risk and, especially important where supplies may be limited, for managing fresh water provision (e.g. irrigation scheduling).

Two-wavelength scintillometry offers the ability to estimate the latent heat flux over large areas (e.g. 2–5 km^{2}) (Kohsiek, 1982; Hill, 1997; Green *et al.*, 2001; Meijninger *et al.*, 2002, 2006). Studies have shown that measurements of turbulent heat fluxes integrated over an area of this order offer suitable comparison data for evaluating land surface schemes in numerical weather prediction models (Beyrich *et al.*, 2002; Beyrich and Mengelkamp, 2006), hydro-meteorological models (Samain *et al.*, 2011) or satellite retrievals of similar scales (Hoedjes *et al.*, 2007; Kleissl *et al.*, 2009).

Two-wavelength scintillometry requires one visible or near-infrared instrument (here referred to as optical) and another at millimetre or radio wavelengths (Andreas, 1989), positioned so that their beams are close together (Lüdi *et al.*, 2005). The intensity fluctuations of each beam are converted into a measure of the refractive index fluctuations of air at the corresponding wavelength using turbulence and wave propagation theory (Tatarski, 1961; Wheelon, 2006). The quantity retrieved is the path-averaged refractive index structure parameter, (Wang *et al.*, 1978). At optical wavelengths the refractive index fluctuations are almost entirely due to temperature fluctuations, whereas longer wavelengths have greater sensitivity to humidity fluctuations.

For each wavelength can be written in terms of temperature () and humidity () structure parameters and the cross-structure parameter (*C*_{Tq}). Solving simultaneously yields the contributions from temperature and humidity, using the two measured values of plus an estimate of the temperature–humidity correlation coefficient *r*_{Tq} (Hill *et al.*, 1988). Alternatively the cross-structure parameter can be found directly by correlating the signals at each wavelength: this bichromatic method enables the correlation between temperature and humidity to be measured and thus removes the need to assume a value for *r*_{Tq} (Beyrich *et al.*, 2005; Lüdi *et al.*, 2005). Monin–Obukhov Similarity Theory (MOST) is then used to calculate the scaling variables of temperature and humidity, from which sensible and latent heat fluxes are found (Kohsiek, 1982; Hill, 1997; Green *et al.*, 2001; Meijninger *et al.*, 2002, 2006).

Previously, the absolute humidity (*Q*, i.e. the mass of water vapour per volume of moist air, kg m^{−3}) has been used to represent the water vapour content of the atmosphere. The original derivations (Hill *et al.*, 1980; Andreas, 1988) partitioned the refractive index fluctuations into temperature fluctuations and absolute humidity fluctuations. Hill (1989) uses absolute humidity to derive structure parameters and scaling variables; Kohsiek (1982), Kohsiek and Herben (1983), Hill *et al.* (1988), Green *et al.* (2001), Meijninger *et al.* (2006) and Evans (2009) all employ absolute humidity in the calculation of the latent heat flux, denoted *L*_{v}*E*, where *L*_{v} is the latent heat of vaporization and *E* is the evaporation. Possibly the use of *Q* in these studies is because Wyngaard and Clifford (1978) stated that (where *w*′ represents fluctuations in vertical wind speed). This is not always correct, especially for high Bowen ratio (*β*) conditions. Wesely (1976), one of the first to suggest deriving fluxes from scintillometry, used water vapour pressure (*e*, Pa).

Unfortunately, the symbols used in the literature are not consistent. Nevertheless, specific humidity *q* (= *Q/ρ*, kg kg^{−1}, where *ρ* is the density of moist air, kg m^{−3}) has been little used in the two-wavelength literature. Tatarski (1961, p 55) first expressed the refractive index in terms of the potential temperature and specific humidity, because these are conserved additives. Hill (1997) and Moene (2003) remark that conserved quantities (i.e. potential temperature and specific humidity) should be used in MOST. In Hill's (1997) detailed discussion of two-wavelength algorithms, the structure parameters and *C*_{TQ} are obtained from and then converted to and *C*_{Tq} for use in MOST, but he gives the evaporation in terms of the absolute humidity scaling variable, *Q*_{∗}, as *E* = −*u*_{∗}*Q*_{∗}, where *u*_{∗} is the friction velocity. Hill (1978) provides a good discussion on the conservation of different humidity variables.

Here we argue that specific humidity should be used in two-wavelength scintillometry: firstly, it is independent of temperature; secondly, it is conserved, thus suitable for use in MOST; and thirdly, it is an appropriate variable to use to estimate the latent heat flux. Whilst *Q* is defined with respect to a volume, which changes with temperature (due to thermal expansion) or pressure fluctuations of vertical motions, *q* uses relative densities (mass of water vapour per mass of moist air). Thus specific humidity is considered a conserved quantity (Lee and Massman, 2011) in the lower part of the atmosphere (for dry adiabatic processes) whereas *Q* is not. Since the latent heat flux is concerned with the transport of water vapour it is necessary to avoid contamination through changes in temperature.

Figure 1 provides an overview of the processing stages discussed. The surface moisture flux can be obtained from measured via two routes, using either specific humidity (route a) or absolute humidity (route b), although as will be shown here, route b is not recommended. Previously, step 1b has been used to find , usually followed by 2b and 3b. The main source of error is from following the b route and taking *u*_{∗}*Q*_{∗} as the evaporation (i.e. stopping after 3b) which is incorrect. Additionally, step 2b is not strictly valid. Errors also arise from inconsistency, such as mistakenly arriving at after applying 1b.

As *Q* contains temperature information, it is not an ideal measure of humidity. is a useful statistic to describe fluctuations in water content, whereas is contaminated by temperature fluctuations. The requirements for MOST scaling are satisfied by *q*_{∗}, the specific humidity scaling variable, but not necessarily *Q*_{∗}. Thus *q*_{∗} is a much more appropriate variable to use to estimate the latent heat flux and properly account for density effects.

Since *ρ* depends on the water vapour content (the relative molecular mass of moist air decreases with water vapour content), the resulting *L*_{v}*E* is usually slightly underestimated when the mass of water vapour is considered relative to moist air. That is, even when using *q*, a small density effect occurs due to the latent heat flux itself. Bakan (1978) concluded that the most suitable quantity to determine the latent heat flux is the mass mixing ratio relative to the density of dry air (*r* = *Q/ρ*_{d}, kg kg^{−1}, where *ρ*_{d} is the density of dry air). Scintillometry data could be processed using *r*, yielding the structure parameter of mixing ratio () and *L*_{v}*E* found from a scaling variable of mixing ratio (*r*_{∗}). In practice other uncertainties in the measurements, processing and assumptions in the derivation of equations outweigh the difference between using *r* or *q*, for example instrumental noise, absorption and limitations of MOST (Medeiros Filho *et al.*, 1983; Meijninger *et al.*, 2006; Beyrich *et al.*, 2012). Mixing ratio formulations are given in the appendix. Note that only the real part of the refractive index fluctuations is considered here (i.e. no absorption).

The objective of this paper is to present the key scintillometry equations in terms of the specific humidity, giving the formulations necessary to calculate specific humidity structure parameters and correctly estimate latent heat fluxes. Section 2 outlines the theory, comparing the use of *q* and *Q*. Equations to correctly calculate the latent heat flux from two-wavelength scintillometry are given and the analogy with other measurement techniques is made in Section 3. In light of these findings, recommendations are given for processing and analysis (section 4), implications are discussed (section 5) and conclusions drawn (section 6).