EQUIVALENT REFLECTIVITY (Ze) the concentration of uniformly distributed, small (diameter of one-sixteenth  or less) water particles which would return the amount of power received.

Precipitation targets produce a range of dBZ values from 0 in very light rain to in excess of 60 in extremely heavy rain and/or hail. Without knowing the precise nature of the targets at the far end of the radar beam, the radar system uses the measured average returned power to estimate the equivalent reflectivity (Ze) of all targets. Traditionally, Ze values have been grouped together in six (6) distinct ranges. These are shown in the table below. The six ranges of reflectivity correspond to the digital processor (DVIP) threshold levels which are used in the WSR-74 and WSR-57 radar systems.

The DVIP levels of equivalent reflectivity are shown in mm⁶ (millimeters raised to the _6th power) and in dBZ. The values in parentheses indicate thresholds for the next higher level (45.7 dBZ would be a Level 3; 46.0 dBZ would be a Level

4). The decibel representations are much easier to use than are the Ze values.
 

The logarithmic (base 10) functional notations of reflectivity values are very convenient for use because the physical power relationships of electronic signals found in a radar system's receiver follow the very same relationship.

In WSR-74 and WSR-57 systems, the reflected power is amplified in logarithmic receiver circuitry, and processed in digital equipment (DVIP systems) which integrate the average power returns from each desired atmospheric volume. Thus, the reflectivity (Ze) values may be represented in "ranges" of values specified by the DVIP levels in the table on page 12. The radar display system is driven by video voltages which represent each of the six DVIP levels. This allows the radar operator to easily "quantize" the reflectivity of the meteorological targets being displayed on the radar scope(s).
 

- Reflectivity / Rainfall Relationships

The relationship between equivalent radar reflectivity (Ze) and rainfall rate has been widely investigated over the years, and there are many refinements which are yet to be made. With established NWS radar systems, estimates of rainfall frequently require subjective modification by the operator in order that they fall within a range of reasonable accuracy. The basic relationships between reflectivity and rainfall are based upon empirical studies, where rainfall rate is measured at the earth's surface while equivalent reflectivity is simultaneously being estimated by the radar looking at precipitation targets located directly above the rain gage. Refer to the figure below

Numerous variations of the relationship between reflectivity and rainfall (Z-R) rates have been developed from past studies of the subject. Many of these studies have centered on liquid (as opposed to ice or ice-covered) precipitation. If the particle size distribution were a unique function of the precipitation rate, a universal Z-R relationship would exist. However, these studies have abundantly demonstrated that there is no such unique particle size distribution for any given rainfall rate. Based upon many studies, the NWS has adopted the following two Z-R relationship values

· Ze = 200R^1.6 (for stratiform precipitation)

· Ze = 55R^1.6 (for convective precipitation) where

Ze is equivalent reflectivity in mm⁶/cm³

· R is the rainfall rate in millimeters per hour.

Equations like those shown above are called reflectivity/rainfall equations, or simply "Z-R Equations". They are defined as follows

Z-R Equations --- empirically- derived relationships between equivalent reflectivity (Ze) and rainfall rate (R) for various precipitation types and synoptic situations.

For the moment, we'll note here that a significant number of these Z-R Equations have been proposed (and developed) for use in various weather situations, and for various types of precipitation (thunderstorms, snow, drizzle, freezing rain, hail, etc.). Some examples are

.Ze = 140R^1.5 (drizzle)

· Ze = 500R^1.5 (thundershowers)

· Ze = 2000R^2.0 (snow)

Any of these could be used in mathematical systems such as a DVIP processor.
 

When rainfall rates are converted to inches per hour, the Z-R relationships used by NWS yield the values shown in the table below

The table depicts rainfall rates as a function of both Z-R relationships /used by the NWS. Notice that convective targets generally tend to produce higher precipitation rates than do stratiform targets at the same DVIP level.

When the sampling volume contains hail, the Z-R equations used for the standard NWS table above do not accurately represent the precipitation rate. The variability of the size of hailstones as well as the extent and thickness of water coating (if any) have a large effect on the returned power. In addition to being highly variable, reflections from hail targets are usually much stronger than those from liquid precipitation. The result is that rainfall rates will be underestimated unless the radar operator can determine (by other means) that hail is present in the storm. Other factors which can cause underestimates of rainfall are distant targets, small targets, wide radar beams, subrefraction of the beam, attenuation by intervening targets, and wet radomes. The first three of these are a direct result of a target not filling the radar beam. Subrefraction is an effect in which the radar beam may be "bent", causing the energy to propagate over a target. The attenuation factors will depend upon the extent (and intensity) of precipitation between the radar and the desired target. Radome wetness attenuation is usually small, but may become significant if the exterior of the dome is not well maintained (clean, painted, etc.).

All of these possible factors must be considered by the radar operator, based on a knowledge of the radar equipment and the ongoing weather situation.
 

- WSR-88D Reflectivity Measurement

Considering the previous Z-R discussion (many possible reflectivity-rainfall rate relationships), the WSR-88D software has been designed to utilize a set of multiple Z-R equations. The figure below depicts various Z-R relationships, one of which has been selected as the WSR-88D default equation

The rationale for the "Algorithm" Z-R equation is that it should provide a good average for different precipitation types. The design of the WSR-88D system was conceived with the idea that a mean bias adiustment will be applied by the hydrologic software to rainfall estimates, based upon data collected from an "umbrella" of precipitation gages. The gages will report data to a central computer system, which will relay the measured precipitation to the WSR-88D.

This method of verification of rainfall (and subsequent real-time modification of the WSR-88D algorithms) requires a substantial number of gaging stations located within the radar's range. There is some question as to whether an ample number of gages will ever be installed. The WSR-88D default Z-R equation (Ze = 300R^1.4) will be utilized in any case, and will probably be subject to change during the life span of the WSR-88D radar system.

The WSR-88D hydrologic software contains sequential sub-functions which are used to compute derived products of 1-hour, 3-hour, and storm total precipitation accumulation. The five processing functions are

· Precipitation Preprocessing

This function corrects the received data for beam blockage, checks sample volumes against its "neighbors" to eliminate possible isolated (single-bin) reflections, performs vertical continuity checks, and compares adjacent "elevation cuts". The output of the preprocessing function is fed to the Precipitation Rate function.

· Precipitation Rate Determination

This function uses a specified Z-R relationship (usually the default equation) to determine the precipitation rate indicated by the reflectivity measured for each sampling volume. Based on the continuity of the total field volumetric rate, the function determines whether the current rate scan should be sent on to the Precipitation Accumulation function or be discarded. When the rate has been accepted, a range correction is applied.

· Precipitation Accumulation

This function uses the precipitation rate data to estimate the accumulation of precipitation during all or parts of the scan-to-scan period. Accumulation estimates for a given current scan, plus those produced for the past hour, are used to estimate an hourly running total. The hourly total is sent on to the Precipitation Adjustment function

· Precipitation Adjustment

The purpose of this function is to compare the radar precipitation hourly data to actual rain gage measurements collected, and then apply an appropriate bias. The data which results is sent to the Precipitation Products function.

· Precipitation Products

The Precipitation Products algorithm uses the outputs of the Precipitation Adjustment function, along with other data, to generate a set of precipitation products. These are

· An hourly running total (clock-hour) accumulation 2 km x 2 km resolution display product.

· An hourly running total (click-hour) accumulation on a 1/4 LFM grid used for internal numerical use as well as transmission to external computers.

· A three-hour total accumulation 2 km x 2 km resolution display product.

· A storm-total accumulation 2 km x 2 km resolution display.

More information regarding these (and other) WSR-88D algorithms may be found in FMH #11, Part C, Chapter 3.