Sub-Poisson Drop Arrival Regularity in Convective Rainfall: Evidence from an Acoustic Disdrometer Built from a Cooking Pot

§ ABSTRACT

§ 1 Experimental Setup

The acoustic sensor system consisted of a consumer smartphone microphone placed inside an inverted metal cooking pot outdoors during a rainfall event. This configuration was not designed in advance as a scientific instrument — it emerged opportunistically during an overnight rain event.

Sensor Configuration

§ 2 Methods

Signal Processing Pipeline

The raw audio was bandpass filtered to the pot's resonant frequency band (64–256 Hz) using a 4th-order Butterworth filter implemented via second-order sections (SOS). Standard filtfilt was found to produce NaN outputs on the 91.9M sample file due to numerical instability — sosfiltfilt was substituted as a stable alternative. Individual drop impacts were detected as peaks in the absolute filtered signal exceeding a fixed amplitude threshold (3× the 95th percentile of the noise floor measured from a quiet window at minute 9), with a minimum inter-peak distance of 200 ms to prevent double-counting of pot resonance ringing.

Noise floor (min 9, 95th pct):  0.00130
Fixed threshold (3× floor):     0.00390
Min peak distance (200ms):      8,820 samples @ 44,100 Hz
Single-impact decay time:       ~150–200 ms
Spurious secondary at:          +51.8 ms (inside exclusion zone)

§ 3 Drop Size Distribution Evolution

The spectral centroid — the frequency-weighted center of mass of the acoustic spectrum at each moment — serves as a proxy for median drop size. Larger drops impart more energy at lower frequencies when striking a metal surface, shifting the centroid downward. The 514 Hz decline from the pre-convective phase (mean 2,261 Hz) to the convective core (mean 1,747 Hz) documents a systematic increase in drop size as the event intensified.

Ridge Plot Spectrogram Wavelet Scalogram Centroid vs Piezo

Fig 1. Drop size distribution evolution — ridge plot. Each row is a 2-minute window colored from dark purple (earliest, 20:24) to bright yellow (latest, 20:54). The narrow peaked distributions in early windows broaden and shift rightward as convection develops, showing the transition from small-drop stratiform to large-drop convective DSD. White dashed lines mark GMM component boundaries.

Fig 2. Full-event STFT spectrogram (log frequency axis). The persistent bright band below 128 Hz is the pot resonance being excited by drop impacts. Vertical streaks are individual large-drop impacts exciting the full frequency range. The dramatic broadband brightening after 25 minutes corresponds to the convective core arrival.

Fig 3. Morlet wavelet scalogram of a 60-second window during peak convective rain (20:48–20:49 EDT). The CWT provides better joint time-frequency resolution than STFT. The continuous bright orange band at 14–200 Hz reflects near-continuous pot excitation at high drop rates. Individual drop transients appear as vertical streaks spanning the full frequency range.

Fig 4. Spectral centroid (blue, left axis) vs Ecowitt piezo rain rate (orange, right axis). The centroid begins declining before the piezo registers meaningful rain — acoustic sensing detects DSD changes earlier than the rain gauge responds. The centroid minimum near 20:50 aligns with piezo rate maximum.

Fig 5. Spectral centroid KDE by rain phase. Blue (pre-convective) peaks at ~2,200 Hz; orange (first burst) at ~1,850 Hz; red (convective core) at ~1,600 Hz. The three distributions are well-separated, confirming that the DSD shift is not gradual noise but a genuine phase transition in hydrometeor character.

§ 4 KCCX Radar Integration

Twenty-five KCCX WSR-88D Level II volume scans were downloaded from the IEM NEXRAD archive covering 20:01–22:00 EDT. Dual-polarization variables were extracted at the Backyard PWS gate (beam height 2.48 km AGL, range ~87 km). Radial velocity was NaN throughout — the S/SSW flow direction at this azimuth (128°) produced near-zero-isodop geometry.

Hydrometeor Classification

Cross-correlation coefficient (CC) exceeds 0.97 for the majority of the rain period, confirming pure liquid rain at 2.48 km AGL with no mixed-phase or frozen hydrometeors. The physically impossible CC = 1.042 at 21:37 is a calibration noise-floor artifact as Z collapses to 4.5 dBZ. This validates the acoustic methodology: the pot was recording impacts from pure liquid raindrops throughout.

Z-R Consistency Check

At peak Z = 40.0 dBZ, Marshall-Palmer Z-R (Z = 300R^1.4) yields R ≈ 7.6 mm/hr (0.30 in/hr) at 2.48 km AGL. The Ecowitt piezo registered 0.14 in/hr at the surface — approximately half the radar estimate. This discrepancy is physically expected: the radar samples 2.5 km aloft, and some drop coalescence and evaporation occurs during descent. The agreement is within the expected range.

Fig 6. Four-panel KCCX radar integration overlay. ZDR and acoustic spectral centroid are anti-correlated as expected (larger drops → higher ZDR → lower centroid Hz), providing independent physical validation of the acoustic drop-size proxy.

§ 5 Non-Poisson Drop Arrival Statistics

The standard assumption in rainfall modeling is that drops arrive as a Poisson process — each drop independent of all others, arrivals memoryless. A Poisson process produces an exponential inter-arrival time (IAT) distribution and a Fano factor (variance/mean of counts in fixed windows) equal to 1.0 at all scales.

Monte Carlo Phase Shuffling Results

To distinguish genuine non-Poisson structure from sample-size artifacts, the actual inter-arrival times were randomly permuted 1,000 times to generate a null distribution of KS statistics under the Poisson hypothesis. Only the convective core's observed KS statistic (0.454) exceeded the null 99th percentile (0.369).

Fano Factor

The Fano factor at the 500 ms window remains deeply sub-Poisson across all three detection threshold settings, confirming the finding is not an artifact of the 200 ms minimum inter-peak distance.

Phase-Resolved Fano — Convective Core

Within the convective core, every 2-minute sub-window has a Fano factor below 1.0. The correlation between sub-window Fano and drop rate is r = −0.939: as rain intensifies, drop spacing becomes more regular.

Fig 7. Seven statistical tests for the non-Poisson finding. From top: (1) Fano factor vs window size — rises steeply above 1.0 at large scales indicating long-range dependence; (2) Pair correlation function — peak at 214ms lag; (3) Allan variance — slopes near −1.3 indicating quasi-periodic structure; (4) Monte Carlo phase shuffling — only convective core diamond lies above the null box; (5) Multiscale entropy — monotonically increasing with scale, characteristic of long-range correlated process; (6) Conditional intensity function; (7) Phase-resolved Fano in convective core.

§ 5.5 Matched-Detector Poisson Null Test

Threshold sensitivity shows the finding persists across floor settings — but a deeper objection remains: any minimum inter-peak distance imposed on any arrival process, including a pure Poisson one, will produce some regularity. The matched-detector null directly tests whether the observed Fano falls outside what the detector itself can manufacture on pure Poisson input.

Fig 7. Matched-detector Poisson null test. Blue histogram: 1,000 synthetic Poisson sequences processed through the identical 200 ms detection floor (null range 0.459–0.531, median 0.500). Red line: observed Fano = 0.387. The gap of 0.072 Fano units between the observed value and the nearest surrogate represents atmospheric signal the detector cannot produce artificially. Empirical p = 0.000.

§ 6 Convergent Evidence Summary

Six independent lines of evidence point to the same conclusion: drop arrivals in the convective core are sub-Poisson, with the degree of regularity increasing with rain rate.

Fig 8. Threshold sensitivity analysis across 100/200/400 ms minimum inter-peak distances. Top: Fano factor by phase — convective core remains deeply sub-Poisson at all settings. Middle: KS statistic — increases with threshold in convective core (opposite of artifact behavior). Bottom-left: Phase-resolved Fano curves track each other closely. The finding is not an artifact of the detection floor.

§ 7 Time Alignment & Sensor Comparison

Cross-correlation of the acoustic drop count series with the Ecowitt piezo rain rate returned a best lag of +10 minutes (acoustic leads piezo), peak normalized correlation 0.819. Naively this might suggest a 10-minute error in the assumed 20:22 recording start time. However, KCCX radar provides an independent timing reference.

The acoustic onset at 20:25 agrees exactly with KCCX radar at lag = 0. The piezo lags by 9–24 minutes depending on threshold. The +10 minute cross-correlation lag reflects piezo sensor latency, not a recording clock error. Audio start time 20:22 EDT is confirmed. All phase boundaries stand.

Fig 9. Time alignment verification. The cross-correlation peak at +10 minutes reflects piezo sensor latency, confirmed by independent KCCX radar onset timing at 20:25 EDT agreeing with acoustic onset at the same time.

§ 8 Discussion

Physical Interpretation

The most parsimonious physical explanation for sub-Poisson drop arrival regularity that strengthens with rain rate is aerodynamic drop–drop interaction at high drop density. At convective core rates of 160–205 drops per minute, the mean inter-arrival time is 290–375 ms. At a terminal velocity of 6–8 m/s for 2–3 mm drops, this corresponds to a vertical column spacing of roughly 1.7–3.0 meters between successive drops. At this spacing, the turbulent wake of a leading drop can influence the terminal velocity and trajectory of a trailing drop, creating a weak hydrodynamic repulsion that regularizes the spacing. This mechanism would predict exactly what is observed: more regular spacing at higher rain rates (more drops per unit volume, more interactions) and sub-Poisson rather than super-Poisson statistics (repulsion pushes toward regularity, not clustering).

Open Questions

Limitations

§ 9 Data & Reproducibility

Drop Count Time Series (Interactive)

Full Drop Count Table

File Inventory

Analysis Pipeline Summary

Environment:     conda 'weather', Python 3.x
Key packages:    librosa, scipy, numpy, pyart (arm-pyart 2.1.1),
                 sklearn, antropy, boto3, matplotlib

Audio loading:   librosa.load(sr=None, mono=True)
Bandpass:        sosfiltfilt, Butterworth order-4, 64–256 Hz
                 (NOT filtfilt — blows up on 92M samples)
Detection:       scipy.signal.find_peaks
                 height=fixed_threshold (0.00390)
                 distance=int(sr * 0.2)  # 200ms
Noise floor:     np.percentile(|y_band[min9_window]|, 95) * 3
Fano factor:     custom compute_fano() — histogram variance/mean
Entropy:         antropy.sample_entropy(order=2)
Radar:           pyart.io.read() → get_gate_lat_lon_alt()
                 → nearest-gate extraction at 40.2645N, 76.8835W

Fig 10. Full event overlay: acoustic RMS and Ecowitt piezo rain rate (top); acoustic drop count per minute (middle); mean impact amplitude as drop size proxy (bottom). The acoustic sensor captured the full intensification curve; recording ended before the event peaked.

Fig 11. Single drop impact analysis at 20:25:12.228277 EDT. Raw waveform (top), bandpass filtered 64–256 Hz (middle), signal envelope (bottom). Pot resonance decays within 150–200 ms. The only spurious secondary detection at +51.8 ms is inside the 200 ms exclusion zone, confirming the detector is not double-counting pot ringing.