Drag to orbit, wheel/pinch to zoom. Drops are drawn ~8x thicker than real so you can see them.

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Rainfall Intensity Lab

How many drops, of what size, fall per second at a given rain rate - and how well can a small gauge measure it?

-WMO-style class

-drops reaching the ground

-median volume diameter D0

-liquid water in the air

-radar reflectivity (from DSD)

-Z = 200 R^1.6 rule of thumb

-slope L (solved so R matches)

-slope L from 4.1 R^-0.21

Virtual rain gauge

0 stime elapsed

0drops caught

0 mmrain depth (volume / funnel area)

0 mLwater volume

...gauge estimate of R

...error vs true R

Falling drops in flight:
Ambient streaks:
Expected noise of a 60 s reading with this funnel:

Independent checks

Each row compares two separate calculations. The closed forms use Gamma-function moments of the exponential spectrum; the other column adds up thousands of size bins by brute force or by random simulation.

The science

Drop spectrum. Marshall and Palmer (1948) found that N(D) = N0 exp(-L D) with N0 = 8000 m^-3 mm^-1 describes many stratiform rain events. Here the slope L is solved numerically so the total rain rate equals your slider; the classic fit L = 4.1 R^-0.21 is shown beside it (the two agree within about 15%, because they come from separate empirical fits).

Fall speed. Large drops fall faster: v(D) = 9.65 - 10.3 exp(-0.6 D) m/s (D in mm), an Atlas-Ulbrich-type fit to Gunn-Kinzer data, floored at 0.3 m/s. Drops reaching a collector are weighted by N(D)·v(D) - that is why the histogram is converted back with the divide-by-v step used by real disdrometers.

Rain rate. R = 3.6e-3 (pi/6) ∫ N(D) v(D) D^3 dD [mm/h] (D in mm). With the exponential N and this v the integral has the closed form R = 3.6e-3 (pi/6) N0 · 6 · [a/L^4 - b/(L+c)^4]. Reflectivity Z = ∫ N D^6 dD = N0 · 720 / L^7 (mm^6 m^-3), dBZ = 10 log10 Z. Liquid water W = 1e-3 (pi/6) ∫ N D^3 dD g/m^3.

Gauge noise. Drops arrive as a Poisson process with rate A·F0; each carries a random volume, so the total is a compound Poisson sum. Its relative standard deviation is sqrt(F6 / (A T)) / F3, where F_k = ∫ N v D^k dD. Halving the funnel diameter makes a 60 s reading twice as noisy. The Monte Carlo experiment checks this.

Simplifications. No wind, evaporation, splashing, drop break-up, or coincident-drop gauge losses; drops are spheres (no oblateness); single exponential spectrum (real spectra vary drop-to-drop and storm-to-storm); sampled sizes 0.2-8 mm. Z = 200 R^1.6 is a statistical rule that varies widely between storms. Class limits (2.5, 7.6, 50 mm/h) are the commonly used ones. The sight glass magnification is funnel area / glass area. The ambient streak field is a thinned picture for atmosphere; the gauge counts come from the random simulation.