Audit measurement continuity before interpreting a trend

Define the evidence question

A time series contains physical variation and the history of how observations were made. Instrument replacements, station moves, sampling time and processing changes can introduce discontinuities. A documented artifact should be investigated rather than automatically interpreted as an environmental change or removed merely because it is inconvenient.

Reconstruct the observation or physical model

The original six-year series is 10+a t, with a known instrument offset b added from year 3 onward. An equal-weight least-squares line with intercept uses the covariance of time and observations divided by the time sum of squares. For times 0 through 5, the centered-time sum of squares is 17.5; the step contributes 4.5 b to the numerator, so its trend contribution is 9 b/35.

Separate the competing interpretations

The corrected series subtracts the independently stipulated offset only where the instrument changed. It retains the underlying trend. Both raw and corrected values remain visible, and the correction has a stated provenance. A fitted slope is not the same as the final-minus-initial difference, although both summarize aspects of the record.

Challenge uncertainty and identification

Real homogenization requires station metadata, overlapping measurements, neighboring references and uncertainty checks. Those sources can themselves have limitations. The virtual offset is known by construction; the exercise neither estimates an actual climate trend nor implies that all discontinuities are artificial. It teaches why transparent corrections and independent validation matter.

Worked original packet

At underlying trend 0.2 °C/year and offset +1 °C, the raw fitted trend is 0.2+9/35≈0.45714 °C/year. Correcting the documented step restores 0.2. Raw endpoint change is 2 °C over five years, whose endpoint slope 0.4 differs from the regression slope.

Investigation

Compare offsets −1,0,+1 at the same physical trend. Retain the full six-point sequences and independently reconstruct the centered-time numerator, then specify evidence that would justify a correction in a real record.

Changed-case transfer

At underlying trend 0.4 and instrument offset−1, calculate raw fitted trend and raw endpoint change.

Feedback

Raw trend 0.4−9/35≈0.142857 °C/year; raw endpoint change 5×0.4−1=1 °C. Removing a negative artifact increases the estimated trend; correction direction follows evidence, not a desired result.

Sources and evidence boundary

Original prose, numerical packets and exercises. Primary method/mechanism source, checked 30 October 2026. No current measured climate values, personal risk estimates, exposure limits or regulatory conclusions are inferred from these fictional cases. Retain first reasoning and critique in the course workspace. No physical pollutant exposure or environmental intervention is requested.