Structural Collocation: Estimating Precipitation Observer Error from Disagreement in Rain-Field Geometry, and Detecting Shared Error Without Ground Truth

1. Introduction

Precipitation analyses from radar, geostationary infrared and passive microwave disagree, and where gauges are sparse there is no direct way to say which is closer to the rain at the ground. Triple collocation (TC; Stoffelen 1998) resolves this in principle: if three observers measure the same signal with mutually independent errors, each observer's error variance follows from the three pairwise covariances. Extended TC adds each observer's correlation with the unknown truth (McColl et al. 2014). TC has been applied to precipitation amounts over the United States with a multiplicative error model (Alemohammad et al. 2015), and in regime-binned form, and its central weakness is well documented: when two observers share an error, TC underestimates their errors and overestimates the third.

Two gaps motivate this work. First, TC has been applied to rain amounts. Much of what makes satellite and radar precipitation fields disagree is geometric: where rain is placed, how it is spread, how many rain objects there are. Spatial verification methods for forecasts (SAL, MODE, FSS, optimal transport; Wernli et al. 2008; Davis et al. 2006; Roberts and Lean 2008; Francis et al. 2025) measure such geometry against a reference, but they need a reference. We found no application of collocation to structural coordinates. Second, the independence assumption is usually argued from instrument physics ("different sensors") rather than tested. We show a case where physically different sensors are not independent observers, how the error appears as a plausible but wrong ranking, and how it can be detected and localised.

Contributions. (i) Structural collocation on centroid and dispersion, validated on a synthetic control with a mass-conserving blur (and a record of how a non-conserving blur created spurious shared errors); threshold counts unvalidated and excluded from interpretation. (ii) A real-data case in which a radar–infrared–microwave triplet produces a confident inverted ranking; four candidate explanations were tested pre-registered and did not account for it within the tested variants (the IMERG calibration chain, smoothness with a close match, the window grid (tested shift and size), convective-core keying) and two exploratory checks found the same (geostationary parallax, radar beam quality). (iii) A ranking reversal when a satellite is replaced by rain gauges, robust to which satellite is replaced, and a four-observer dependence pattern consistent with a shared satellite error, with its identification limit stated, replicated on an unseen month under a frozen pipeline. (iv) A sensitivity analysis bounding the attribution by the radar–gauge error dependence it tolerates, and a five-observer design validated in simulation. (v) Six practical rules, scoped to the tested configuration. A pre-registered negative result on predicting point radar error from structural disagreement is reported in Appendix B.

2. Data

All data cover the contiguous United States (CONUS), hourly accumulations ending 2024-07-01 01:00 UTC to 2024-07-22 00:00 UTC (504 hours; 502 for analyses using GOES, see Appendix A), and, for the replication in Section 4.7, hourly accumulations ending 2024-08-01 01:00 UTC to 2024-08-22 00:00 UTC (504 hours, all with GOES), fetched only after the replication plan was frozen. Every field is placed on the 0.1° MRMS-aligned grid. Table 1 lists the observers.

None of R, GIR, MW uses gauges at the time of observation; the IMERG infrared estimate is calibrated to the microwave estimate through a 30-day rotating accumulator, and the microwave estimates are quantile-matched to the GPM combined radar–radiometer product (Huffman et al. 2023). Figure 2 shows one hour through each observer.

3. Method

3.1 Windows and structural reads

The domain is tiled with non-overlapping 15 × 15-cell (1.5°) windows. A window-hour enters when radar is ≥ 90% valid, the infrared field is complete, the microwave field covers ≥ 80% of the window, and every observer's window sum is ≥ 1. For each observer field A over window coordinates (y, x) centred on the window, we compute the intensity-weighted centroid and the dispersion:

We also tested the wet-area fraction (cells ≥ 0.5 mm h⁻¹, or cold cloud for GIR), the number of rain objects (8-connected components of ≥ 2 cells) and log mean rain as an amount-based control.

3.2 Collocation estimators

Under the additive model x_i = a_i + b_iT + e_i with errors independent of each other and of T, the ETC estimates for a triplet (i, j, k) with covariance matrix C are

ν is written nf in the figures. Positions are in grid cells of 0.1° (about 10 km).

With a fourth observer, the error covariance of a pair (p, q) can be estimated using two observers (u, v) assumed independent of everything else as instruments:

A one-factor model with independent errors also implies vanishing tetrads among four observers (Spearman and Holzinger 1924):

with R radar, I the infrared observer, M microwave and G gauges.

A single shared error between I and M inflates r_IM only, so t₃ ≈ 0 while t₁, t₂ > 0. The same pattern arises from a shared error between R and G; four observers cannot separate the two (Section 5.3). By construction t₂ − t₁ = t₃, so the three tetrads carry two independent constraints, not three.

3.3 Uncertainty and pre-registration

Intervals are 95% cluster-bootstrap percentiles, resampling (UTC day × 5° tile) blocks (300–500 resamples, fixed seeds). Every experiment below was specified, with its predictions and decision rule, in a plan file frozen before any value of that experiment was computed; the Appendix B plan received two amendments, both written before any target or predictor value existed. Follow-up splits added after scoring are labelled exploratory.

3.4 Synthetic anchor

We generated 20,000 fields of known truth on a 64 × 64-cell canvas (1 + Poisson(1.5) Gaussian rain cells, widths U(0.8, 2.5), lognormal peak intensities). Simulated observers re-render the same cells with independent per-cell errors: radar-like R (position sd 0.5 cell, no blur, intensity noise lognormal 0.3), microwave-like S1 (sd 0.8, blur 1 cell, 0.4), infrared-like S2 (sd 1.5, blur 2, 0.6) and, for the five-observer benchmark, a gauge-like G_s and a fifth observer. Blur is a mass-conserving Gaussian convolution, which leaves the centroid unchanged and adds 2σ² to the squared dispersion. Reads are taken on the full canvas and in central and shifted 15 × 15 windows. A shared displacement (one per field, N(0, σ_sh) per axis) can be added to S1 and S2, or to R and G_s (the mirror case). True noise fractions and coordinate-error correlations are computed against the known truth reads. A first version of this simulation widened cells with fixed peaks, which does not conserve rain mass; it produced artefacts and is superseded (Section 4.1, Appendix A).

4. Results

4.1 Synthetic validation

The rendering behaved as theory requires (centroid unchanged to 10⁻¹⁵ cell; squared dispersion increased by 2σ² within 0.04%). With nothing planted, the satellite-like observers' centroid errors were uncorrelated on the full canvas (centroid y -0.001), while dispersion errors kept a small correlation (+0.021, within the pre-registered |r| < 0.05 but not zero); windows added a small shared error (centroid up to 0.023, dispersion up to 0.14), growing as the storm sat further from the window centre (Figure 3A). Collocation recovered centroid and dispersion with the correct ranking on the full canvas and in every window (maximum error 0.006 centroid, 0.031 dispersion).

A planted shared satellite error moved the estimates toward inversion: at σ_sh = 0.5 cell, R 0.090 and S1 0.095 (truths 0.046, 0.135), not yet crossed; at σ_sh = 1.0, R 0.199 and S1 0.084 (truths 0.046, 0.230), inverted (Figure 3B). With five observers, the tetrad audit flagged a planted shared satellite pair and the mirror radar–gauge pair in 10 of 10 blocks each, with no false alarms when nothing was planted.

First simulation (superseded). The non-conserving first version reported a baseline correlation between the two blurred observers' errors (0.16 on centroid y), and a dispersion mis-ranking, both artefacts of the blur, and an inversion already at σ_sh = 0.5, which the artefact helped produce. Its findings for the variables the corrected control did not re-test still stand as provisional: the object count failed (errors up to 0.55), wet fraction gave impossible negative error variances in 100% of resamples, and log mean rain was stable but biased by up to 0.063. Threshold counts are therefore not interpreted below.

4.2 Real data: an inverted ranking

On 25,468 window-hours, the triplet (radar, IMERG IR, MW) gave centroid-y noise fractions R 0.254, IR 0.538, MW 0.070: microwave rated least noisy, radar second. Because the IMERG infrared estimate is calibrated to the microwave, we replaced it with raw GOES-16 brightness temperature (14,585 window-hours): R 0.257, GIR 0.505, MW 0.036 (Figure 4). The ranking was unchanged on centroid and dispersion (the variables validated in §4.1; wet fraction and object count are shown in Figure 4 for completeness but are not interpreted). The amount-based control was unstable in both triplets (impossible variances in 100% and 98% of resamples) although it was stable on synthetic fields, so its instability reflects the real observers' amount relationships rather than the estimator. With three observers the ranking follows from the raw correlations (centroid y: r_R,GIR = 0.61, r_R,MW = 0.85, r_GIR,MW = 0.69): microwave is rated best because the infrared field agrees more with microwave than with radar. Clusters: 1,199 (A) and 965 (A2). The merged IMERG product, used as a fourth observer in the IMERG-IR triplet, showed a positive error correlation with its microwave input (0.26 to 0.37 on the stable variables), a dependence that exists by construction; in the first simulation the magnitude of such estimates was unreliable when the instruments were themselves dependent, and the corrected control did not re-test it, so it is not interpreted further.

4.3 Alternative explanations tested

Smoothness. Radar was smoothed with a Gaussian before windowing, over a ladder up to σ = 3 cells. Microwave's median lag-1 autocorrelation was 0.70; radar's was 0.63 unsmoothed and 0.76 at the nearest ladder step, σ = 0.5 cell, so the match was approximate. The gap ν_R − ν_MW remained +0.214 [+0.186, +0.240] at that step and +0.155 at σ = 3 (Figure 5A); the sample also changed along the ladder (14,751 to 15,467 window-hours). A pre-registered repeat on a fixed sample of 14,725 window-hours, with radar blurred to match microwave's smoothness to within 0.013 in lag-1 autocorrelation (σ = 0.4; lag-1 0.687 vs microwave 0.700), gave +0.218 [+0.192, +0.250], essentially unchanged, and +0.145 at σ = 3. The synthetic calibration in Figure 5B comes from the superseded first simulation and is shown for the record only.

Window grid. Re-tiling with a half-shifted grid and with 25 × 25 windows left the gap at +0.199 and +0.181 (baseline +0.224), and the point-support reversal at -0.231 and -0.210 (pre-registered). This shows robustness to the tested grid shift and window size; it does not exclude windowing or representativeness effects in general (the 25 × 25 gap is about a fifth smaller).

Parallax. GOES cold-cloud centroids lie increasingly north of radar rain with latitude (-0.90 cells, about 10 km, at 45–55°N), the expected geostationary parallax; microwave does not follow (-0.12), so parallax is not shared (Figure 5C, exploratory).

Convective-core keying. Reading every observer on its own core (cells at or above its 80th percentile) left the gap at +0.207 (full area +0.224). The gap was largest in the weakest windows (+0.333 in the lowest peak-intensity quintile vs +0.115 in the highest; exploratory).

Radar beam quality. The gap did not decrease with radar quality index; in the best-beam windows (RQI ≈ 0.96) with weak rain it was +0.317 (Figure 6, exploratory).

4.4 A sharp, independent fourth observer

A three-observer ranking cannot reveal a shared error among its legs, so we added a fourth observer, rain gauges, which none of the other observers uses. We restricted to window-hours with ≥ 5 reporting gauges (2,402 window-hours, median 9 gauges) and read every observer at the gauge locations (point support), splitting at the median radar peak. With (R, GIR, MW) at point support the inversion reappeared (+0.195 [+0.116, +0.281], weak half). Replacing GOES with gauges reversed it: (R, MW, G) gave -0.196 [-0.260, -0.134], with noise fractions radar 0.063, gauges 0.225, microwave 0.259 (Figure 7). The GOES–microwave error correlation with radar and gauges as instruments was +0.35 [+0.29, +0.42] (weak) and up to +0.51 (strong, centroid x).

A pre-registered robustness check replaced the other satellite: in (R, GIR, G) radar remained least noisy (0.033 vs GOES 0.625), and radar's noise fraction did not depend on which satellite was dropped (difference -0.029 [-0.072, +0.006]). The tetrads (Eq. 4) gave t₁ = +0.120 [+0.089, +0.153], t₂ = +0.134 [+0.099, +0.173] and t₃ = +0.014 [-0.003, +0.033] (weak, centroid y). Since t₃ = t₂ − t₁, this is two independent constraints, and it is compatible with a single two-group split; an interval containing zero for t₃ does not establish that no other dependence exists. All pre-registered predictions held in both strata and both axes. The estimated GOES–microwave correlations above use radar and gauges as instruments and are conditional on their errors being independent (Section 5.3).

4.5 Where the coupling grows

We tested whether the coupling grows with the share of cold cloud lying over no surface rain (anvil fraction). No change was detected: the normalised GOES–microwave error correlation was 0.37 in the lowest and 0.39 in the highest anvil tercile (difference +0.014 [-0.097, +0.111]). The effect of the coupling on the three-observer ranking did grow, from +0.054 to +0.362. In absolute terms the shared covariance roughly doubled (1.31 → 2.64 cells²), but so did every observer's error variance, gauges included (2.31 → 4.70); anvil fraction correlates -0.34 with radar peak intensity (Figure 8).

4.6 Sensitivity to radar–gauge dependence

We fitted a one-signal model to the four observers at gauge points with an assumed radar–gauge error correlation ρ_RG, swept from 0 to 0.5, and a free GOES–microwave error correlation, by maximum likelihood. The fit is identical at every ρ_RG, so the data cannot fix it; the sweep traces the equivalence class. We report two outcomes separately, by interval. The inferred satellite dependence (GOES–microwave error correlation, interval excluding zero) stayed detected up to ρ_RG = 0.475–0.5 across strata and axes in July, and through 0.5, the top of the tested range, in August. Radar's ranking as least noisy (interval of ν_R − ν_MW excluding zero) held up to 0.275–0.4 in July and 0.375–0.4 in August; beyond that the interval includes reversal. The point estimate of the ranking reversed at 0.425–0.50 in July and had not reversed by 0.5 in August (gap -0.017 at 0.5, interval spanning zero) (Figure 3C). Neither month tests values above 0.5. In both months this bounds how far the assumption can move; it does not measure the actual correlation.

4.7 Replication on an unseen period

Every real-data result above is in-sample (July). We froze the July pipeline (script hashes, inclusion rules, strata, seeds, six primary predictions and a scoring rule) before fetching any data for August 2024, then ran it unchanged. Before the August run, the July outputs were regenerated with the period switch in place and reproduced byte-for-byte; an independent re-implementation written from the plan files alone also reproduced the July quantities to within 4 × 10⁻⁹. August gave 14,738 area and 1,933 point-support window-hours. All six predictions held on both axes and in every required stratum (Table 2): the area inversion (+0.259 on centroid y), the reversal when gauges replace GOES (-0.286, weak stratum), the mirror triplet, the GOES–microwave error correlation (+0.425) and the tetrad pattern. Magnitudes were similar or somewhat larger than in July. The effect is therefore not specific to July; the replication does not change what four observers can identify, and both months are adjacent and warm-season. Replication across adjacent months with the same instruments, processing and season shows repeatability, not attribution: a persistent measurement or representativeness effect would replicate too.

5. Discussion

5.1 What is new and what is not

Triple collocation, its sensitivity to shared error, and vanishing tetrads are established. Satellite precipitation products are known not to be fully independent (e.g. Tian and Peters-Lidard 2010 use inter-product spread as uncertainty, acknowledging shared features). The contributions here are narrower: collocation on structural coordinates, with a synthetic validation, on a mass-conserving control, of which structural variables are admissible; a worked case in which physically different sensors fail the independence premise, where the failure surfaces only as a plausible ranking; and a detection and localisation procedure (sharp independent observer, mirror triplet, tetrads) that turns the failure into a measurement. Read most directly, the nonzero tetrads in both months reject a single common signal with mutually independent errors for these four observers; the satellite attribution is one reading of that failure. Two things were not tested: whether geometric coordinates reveal dependence that amount-based or representativeness analyses miss, and whether the procedure, used as a warning that a collocation ranking is not trustworthy, improves on existing diagnostics.

5.2 Implications for satellite-only evaluation

Collocation of satellite products against each other is attractive exactly where ground observations are absent. In this sample, the four-observer pattern is consistent with an error shared by an infrared and a microwave observer relative to surface rain, with an estimated normalised strength near 0.4 and no detectable change across anvil extent; its effect on rankings was largest where all observers were noisiest. If that reading holds, collocation using two satellites as two of three legs can flatter both in such conditions. We did not establish the physical source. Both satellites respond to cloud and hydrometeors aloft rather than to rain at the surface; a coupling that does not vary detectably with anvil extent is compatible with both observing the cloud system, and the simple anvil form was not supported. That the two groups respond to different physical quantities (surface rain versus aspects of the cloud system) is a hypothesis; processing, resolution and sampling are alternative explanations.

5.3 Identification

With four observers, a shared error between GOES and microwave relative to surface rain is observationally equivalent to a shared error between radar and gauges relative to a cloud-system signal. Taking surface rain as the target gives the analysis its purpose but does not show that radar and gauge errors are uncorrelated, so the satellite attribution and the estimated correlations are conditional on that assumption. Vogelzang and Stoffelen (2021) make the same points for wind collocation: resolution (representativeness) differences generate error covariance, and a covariance solution that fits does not by itself establish a correct error model. A fifth observer would help only under explicit assumptions about its own signal and errors; different physics alone does not guarantee identification. The sensitivity analysis (Section 4.6) quantifies this: the satellite dependence stays detected for radar–gauge error correlations up to 0.475–0.5 (through the tested maximum of 0.5 in August), and radar's ranking is clear up to 0.275–0.4. It shows how far the assumption can move before the ranking changes, not where the actual correlation lies. In simulation, five observers resolved the ambiguity in either direction (Section 4.1), under that simulation's assumptions about the fifth observer's signal and errors; a real fifth observer would need those assumptions justified.

5.4 Practical rules

These are findings for the tested variables, simulation and observing configuration, not general laws.

1. Prefer geometric coordinates. With a correct blur, centroid and dispersion were recovered and ranked correctly; threshold counts failed in the superseded simulation, were not re-tested, and are excluded from interpretation.
2. A shared error among the legs is invisible to the estimator; it appears only as a plausible but wrong ranking.
3. Check independence after feature extraction and windowing, and audit the simulation itself: a blur that did not conserve mass created shared centroid errors that a correct blur did not; a small dispersion correlation remained under the correct blur, and windows add a small shared error. Validating the rendering and validating the measurements' error assumptions are separate checks.
4. Judge independence by the quantity each observer responds to, and check derived products for calibration chains.
5. The bias from a shared error is largest where all observers are noisy.
6. Validate on synthetic fields with known truth before reading a real-data ranking.

5.5 Limitations

Two adjacent months (warm-season convection; July, and the August replication) over CONUS, hourly, 0.1°; the replication shows repeatability under the same instruments, processing and season, not attribution. Several splits are exploratory and uncorrected for multiple comparisons. Microwave coverage ≥ 80% selects hours near overpasses; the gauge sample requires rain at the gauge points and is wetter than the full sample. The synthetic model is a simple cell model (corrected version in Section 4.1). Point-support centroids from about nine gauges per window are coarse. Collocation assumes each observer is linear in a common signal; for the cold-cloud field, whose centroid tracks cloud rather than rain, this is an approximation, and the four-observer checks are the evidence that it does not drive the results.

6. Conclusions

Structural collocation produces observer rankings that depend on which observers form the triplet. With radar, GOES infrared and passive microwave it rated microwave least noisy; replacing either satellite with gauges reversed that, at the same point support. The four-observer results show a dependence pattern consistent with an error shared by the two satellites relative to surface rain, under the assumption that radar and gauge errors are not strongly correlated (satellite dependence detected up to 0.475–0.5 (July) and through 0.5, the top of the tested range (August); radar ranked least noisy with intervals excluding reversal up to 0.275–0.4 (July) and 0.375–0.4 (August); the actual value is not measured). The results replicated on an unseen month under a frozen pipeline. A corrected synthetic control supports the variable choices and the mechanism, and shows that five observers can resolve the remaining ambiguity under its assumptions. The practical lesson is to test the independence premise of a collocation, after feature extraction, before reading its ranking.

7. Data and Resources

MRMS: NOAA Open Data Dissemination, s3://noaa-mrms-pds. GOES-16 ABI L2 CMIPC band 13: s3://noaa-goes16. IMERG Late V07 half-hourly (GPM_3IMERGHHL.07): NASA GES DISC (Earthdata login). USCRN hourly: NCEI. ASOS hourly precipitation: Iowa Environmental Mesonet. Code, frozen plans, results and the full chronological lab book (LABBOOK.html, including dead ends, amendments and file hashes) are in the project folder precip-disagreement/; the four-observer tetrad audit is tetrad_audit in independence_audit.py. These are available from the author on request; [a public repository link to be added before submission].

Acknowledgements

Analysis code, figures and a first draft of this text were produced with the assistance of an AI model (Claude, ChatGPT, Gemini) working under the author's direction; the author is responsible for the content.

Appendix A. Transparency and Reproducibility

Each experiment has a plan file frozen before computation (FEASIBILITY_PLAN.md, PROBE_A_PLAN.md, PROBE_A2_PLAN.md, T1_PLAN.md, T5_PLAN.md, T6_PLAN.md, T7_PLAN.md, T7R_PLAN.md, T8_PLAN.md, T1BIS_PLAN.md, T7S_PLAN.md, T9_PLAN.md, T10_PLAN.md) and a results file. Deviations: the Appendix B experiment excluded gauges within 1° of the domain edge; two GOES hours had fewer than three scans and were dropped per plan. One prediction premise was wrong and is reported as such (the synthetic S1–S2 error correlation was not zero with nothing planted; the corrected control showed this was an artefact of the non-conserving blur, not of smoothing, and the claim built on it was withdrawn). An interpretation was later corrected: the tetrad pattern was first described as "one shared GOES–microwave error"; the identification limit in §5.3 was found afterwards. An external review identified that the synthetic blur did not conserve rain mass, that the synthetic dispersion ranking was wrong with nothing planted, and that the three tetrads are not independent; the text was corrected accordingly and the corrected synthetic control (T1-bis) and a radar–gauge sensitivity analysis (T7-S) were pre-registered and run; their results replace the provisional statements. Before the replication, an independent re-implementation from the plan files reproduced the July quantities and recorded four details fixed by the code rather than the plans (≥ at the wet threshold, float32 window sums, T7 not requiring the merged product, the median tie rule). The replication's deviations were mechanical (data root chosen by working directory, a missing stations-table producer added and checked byte-identical on July, the Δ interval computed from the existing bootstrap draws, a single-download IMERG fetcher checked identical on July) and are listed in T10_RESULTS.md. The first simulation's results are kept in the lab book for the record; the paper reports them only where the corrected control did not re-test a variable (Section 4.1). All raw files were streamed and reduced in memory; reduced arrays and their digests are listed in the lab book.

Appendix B. A negative result: structural disagreement does not predict point radar error

The work began with a different question: does structural disagreement between radar and satellite predict where radar is wrong at a rain gauge? On 288,690 gauge-hours, fourteen structural reads (location, pattern, overlap, object count, dispersion, void, multiscale, edge, persistence, spatial and temporal variance of disagreement, and an infrared third observer) were added to a baseline (MRMS quality index, plain differences, magnitudes). The validation increment was Δρ = +0.0014 [+0.0008, +0.0021] against a pre-registered threshold of 0.02 (baseline ρ = 0.685), and a spatial-scramble null gave a larger increment (+0.0031). Hourly point error is dominated by rain amount and by the area–point representativeness term, as radar-verification studies predict (Ciach and Krajewski 1999; Ciach et al. 2003).

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