A Modified Degree-Day Method for Volume and Timing Estimation of Snowmelt and Refreezing

The standard degree-day (SDD) method, for the estimation of snowmelt water equivalent (SWE), relies on daily air temperature mean as a single input variable to approximate the diurnal snowpack energy exchange. Although convenient and popularly used, such simplified approach can seriously miscalculate the volumes of water that change phase when daily air temperatures fluctuate either side of the freezing point. Additionally, the SDD method does not recognize the physical process of diurnal snowmelt water refreezing and neither observes the duration of snow melting or water refreezing events.

A modified degree-day (MDD) approach overcomes the common SDD method deficiencies by removing dependence on the daily air temperature mean, estimated over a 24-h period beginning at midnight, and replacing it with the “true” or mathematical diurnal air temperature extrema pair at known times of occurrence. Mathematical extrema, or points on the temperature-time curve in which the daily air temperature trend changes its sign, are identified using Climatological Observing Window Night and Day (COW_N-D). The discrete search method of the COW_N-D algorithm detects daily minimum and maximum temperature, using separate nighttime and daytime search periods, in accordance with the radiative sequence.

The MDD concept recognizes the fact that the information on temporal variation around the air temperature threshold point is of critical value to the degree-day method accuracy. To achieve that, instead of using daily average temperature, the MDD method relies on prior knowledge of mathematical temperature-time extrema pairs for capturing the sequential minima (T_n, t_n) and maxima (T_x, t_x) of the diurnal air temperature function. In that way, the modified formula retains the simplicity of a standard degree-day approach while targeting the exact segments of the diurnal temperature curve that directly relate to periods of melting and refreezing. Reproduction of air temperature variability is performed by means of analytical air temperature approximations, with the precondition that a continuous piece-wise function of choice connects exactly in mathematical temperature-time extrema points. Corresponding synthetic air temperature data enable estimation of time duration of temperature oscillation around the threshold and a proper separation of positive and negative segments on the temperature-time curve. Reproduced air temperature distribution further serves as a basis for derivation of the newly introduced degree-day parameters. In addition to volumetric estimation of daily snow melting and water refreezing volumes, the “effective temperature” and "effective time” parameters enable estimation of the timing aspect of these processes, such as the onset and time duration of melting and refreezing events. While the effective time parameter (t_EFF) specifies the intervals between “time intercepts” (points at which the approximating curve intersects the threshold), the effective temperature (T_EFF) indicates a representative temperature of a section enveloped by the temperature-time curve. The product of these two parameters corresponds to the area of a potential melting or refreezing event, expressed in SWE units (mm/°C∙day).

The MDD method was evaluated for multiple sites in the snow-belt region of Canada where the availability of hourly records of daily temperatures allowed the required MDD input parameters to be calculated reliably and thus used for comparative purposes. During testing, the MDD input parameters were derived from mathematical air temperature extrema and their times of occurrence with alternative approaches to air temperature approximation using linear, trigonometric and exponential functions (Fig.1).

Very good agreement was obtained for all sites against reference benchmarks confirming the validity of the MDD approach for both, diurnal snow melting and water refreezing estimation. From a practical perspective, it is suggested that there is no significant benefit to be gained by using approximating functions more complex than the linear method for supplementing the missing continuous air temperature measurements. Additionally, an event-based time discretization is recommended as oppose to a fixed 24-hour discretization that commonly fragments melting and refreezing events (Table 1). Finally, the MDD approach is not seen as a replacement for the regular SDD method, so much as a tool that can be applied when the SDD methodology is likely to become unreliable. This is best achieved by using the Hybrid SDD-MDD algorithm that invokes the MDD approach only when the necessary conditions arise (Fig.2).