Missing points on a BMP curve: when interpolation is honest and when it is not

Missing points on a BMP curve can be filled by interpolation only within the measured range and only as estimated values: interpolation does not create new measurements, does not change the final BMP result if a further reading was taken after the gap, and does not allow days to be "written in" after the last measurement. Gaps often occur in practice: a weekend, a sensor failure, a broken gas sample, a break in manometric readings. Below we show how to proceed reliably.

What we actually measure

In a BMP test the laboratory records, on successive dates, the amount of methane from individual bottles: with the substrate, from the blank and from the cellulose control. The net result is the difference between the substrate series and the mean of the blanks, expressed per gram of volatile solids (VS). We describe the protocol in the article BMP test protocol step by step, and the method of calculation in Calculating methane production in the BMP test (Hafner 2020). What matters: the cumulative curve of a single bottle is the sum of all increments since the start of the test, so it does not decrease.

Why a gap does not change the final result

If the reading after the gap is correct, it contains everything that was produced since the previous reading. We only do not know the distribution of production over time within the gap, but the total is known. Interpolation spreads this total over the missing days, so it matters for the shape of the curve (for example the fit of a kinetic model), but not for the final BMP. It is quite different when the last measurements are missing: then it is not known whether production has ended, and the result must be regarded as incomplete.

What to interpolate and with what method

1. Interpolate the cumulative curve, not the daily increments. Daily increments change from day to day, and after smoothing their sum might cease to agree with the measurement.
2. Choose a monotonicity-preserving method. Linear interpolation is the simplest and most transparent, but the curve is usually concave (the rate decreases), so the chord lies below the true curve. A cubic spline can "overshoot" and produce decreasing values, which does not occur physically; in such cases monotone variants (for example PCHIP) are used.
3. Approximate the uncertainty. The error grows with the length of the gap and with the curvature of the curve: in the phase of rapid production (the first days) it is greater than in the plateau phase.
4. Do not use interpolated points for the end-of-test criterion. The criterion from the literature (daily methane production below 1% of the cumulative net increment for 3 days) is based on actual readings; Hafner et al. (2020) recommend it among the validation criteria. With readings less frequent than daily, the rate between actual readings is assessed. Details: BMP test validation criteria according to Holliger et al. (2016).

An illustrative example

The numbers below are hypothetical and serve only to show the mechanism. Suppose the readings of cumulative net methane (NmL CH₄/g VS) were: day 0 — 0; day 1 — 40; day 2 — 85; day 3 — 125; day 6 — 205; day 7 — 225. Readings are missing on days 4 and 5. Linear interpolation gives an increment of (205 − 125) / 3 ≈ 26.7 NmL/g VS per day, that is about 152 on day 4 and about 178 on day 5. Because the curve is concave, the real values were probably slightly higher, and the difference may reach several NmL/g VS; this is why the points 152 and 178 must be flagged as estimated. The result on day 6 (205) and subsequent readings remain unchanged.

Extrapolation: what not to do

Extrapolation beyond the range of the study requires stronger assumptions than interpolation, because the model must predict behaviour that has not been measured. The final yield should not be written in merely so that all curves have the same length. If one series ends earlier, then we either shorten the comparison to the common range, or we state that the series was interrupted. Fitting a model (first-order or Gompertz) makes it possible to estimate the plateau, but this is a model result, not a measurement — see First-order kinetics in the BMP test and The modified Gompertz model in BMP curves. In the Substraty.pl BMP database a record's curve is marked as measured (measurement points) or modelled (a curve with parameter k).

Mean of several series

When replicates were measured on different dates, the mean is calculated on an agreed time grid: either only on the days on which all series were measured, or after interpolating each series onto a common grid, with an explicitly described rule for gaps. The standard deviation at interpolated points is not an independent measurement and underestimates the real uncertainty. The source data should remain available alongside the calculations.

Checklist

• Does the report distinguish measured points from estimated ones?
• Was the cumulative curve interpolated with a monotonicity-preserving method?
• Are the last readings missing? If so, the result is incomplete.
• Was the end-of-test criterion assessed on actual readings?
• Were the raw data retained? Hafner et al. (2020) indicated that processing and data entry errors were among the important sources of scatter between laboratories.

The full list of questions for a report can be found in the article How to read a BMP test report.