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<br />6. DETERMINATION OF Ze-S RELATIONSHIPS <br /> <br />6.1 General <br /> <br />Two fundamentally different approaches have been used by investigators over the past five decades in <br />developing relations between "reflectivity" and snowfall. In one approach the radar reflectivity factor, Z, <br />is calculated from particle size spectra observations. Melted drop diameters of snow particles are usually <br />used in calculating Z. Size-mass relations and particle fall speed relations are also used, generally based <br />on some of the many such relationships report.ed in the literature (e.g., Mitchell et aI., 1990). Radar <br />observations are not involved in this approach. <br /> <br />The same snow particle measurements can be used to calculate S, which represents the snowfall rate. <br />Alternatively, nearby gage or snow board measurements of sufficient resolution can provide observations <br />of S. In this approach a power law with theor~:tical basis (Atlas et aI., 1995) is applied with the form: <br /> <br />Z = aSP <br /> <br />(1) <br /> <br />where Z is expressed in mm6 m-3 and S in mm h-l. The same metric (SI) units will be used in all similar <br />equations in this report although Sand SD are often reported in the conventional English units used in the <br />United States. The coefficients a and p are empirically determined from available data sets. Many early <br />studies provided a and p values for rain using this approach, and a limited number for snow, as <br />summarized by Battan (1973). Perhaps the best known relationship for snow was provided by the <br />reanalysis of four earlier studies done by Sekhon and Srivastava (1970). They recommended use of: <br /> <br />Z= 1780S.2l <br /> <br />(2) <br /> <br />In the other fundamental approach the quantity Ze, measured by radar, is related to S by the same power <br />law relation but using Ze rather than Z. <br /> <br />Z =aSJ <br />e <br /> <br />(3) <br /> <br />Equation (3) provides a practical means of relating what a radar actually observes to snowfall rate. <br />Observations of S are typically made by ground-based gages or snow boards at some limited range from <br />the radar and at some limited vertical distance below the radar beam, both to minimize known range <br />effects. The coefficients a and p are again empirically determined coefficients but the a value will differ <br />between the two approaches. While Z and Ze are essentially equal for rain, that is not the case for snow. <br />As discussed by Smith (1984) the relationship is: <br /> <br />Ze = 0.224Z <br /> <br />(4) <br /> <br />so a values derived from equation (3) will be about a factor of 4.5 less than those derived from equation <br />(1). Perhaps because Z and Ze are equal for rain, a values for snow from the two approaches have <br />sometimes been inappropriately grouped together in published summaries. Such grouping suggests a <br />much greater range of a than actually exists from experimental results. <br /> <br />Both the above approaches have their strengths and weaknesses. The approach using calculations of Z <br />will not be discussed in much detail here because it was not chosen for use with the SAA development. <br />But this common approach, discussed in the classic textbook by Battan (1973), has been used by many <br />investigators over decades for both rain and snow. It is a physically based approach which has <br />considerable appeal because it addresses the fundamentals of the Z-S relationship. If sufficient care is <br /> <br />15 <br />