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Use a reconnaissance phase to learn where verification is valuable, while keeping inclusion probabilities and survey weights visible.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · Build the design
Route 2 · Allocate the sample
Route 3 · Estimate and transfer
Route 4 · Control the risks
Route 5 · Verify and continue
CHAPTER 1 OF 12 · BUILD THE DESIGN
1. Why two phases can make a soil survey smarter
A soil map or proximal sensor can cover many locations cheaply, while laboratory measurements are slower and more expensive. Two-phase sampling uses that asymmetry deliberately. Phase one collects inexpensive auxiliary information broadly. Phase two verifies the target soil property on a designed subsample.
An adaptive design uses the first verified observations to decide where the remaining laboratory effort is most valuable. It may send more samples to strata with high residual variance, weak map performance or important decision thresholds. The design remains a probability sample only when the selection rules and inclusion probabilities stay visible.
This article extends the Soil Science root, two-phase map correction and spatially balanced map-update design. The goal is not to chase surprising points. It is to learn where verification reduces uncertainty without losing design-based accountability.
CHAPTER 2 OF 12 · BUILD THE DESIGN
2. The design separates a frame, a pilot and an allocation rule
Begin with a finite frame of mapped units or eligible field locations. Partition it into prespecified strata such as soil-landscape units, predicted-value bands or management zones. Draw a small probability pilot in every stratum so no domain begins invisible.
Use the pilot to estimate residual variability after the phase-one map or model. Allocate the remaining phase-two samples using a written rule, for example proportional to N_h s_h/√c_h, where N_h is stratum size, s_h is pilot residual standard deviation and c_h is verification cost.
Record selection probabilities at every phase. If adaptation is deterministic given the pilot, the final inclusion probability can be derived from the design. If field crews make informal substitutions, that calculation breaks and the resulting weights no longer describe the realised sample.
CHAPTER 3 OF 12 · ALLOCATE THE SAMPLE
3. Worked allocation: spend verification where residuals vary
Consider 1,000 mapped units in four strata. Their sizes are 300, 300, 200 and 200. A pilot verifies ten units per stratum and estimates residual standard deviations of 0.4, 0.8, 1.2 and 1.6. Sixty laboratory samples remain.
With equal costs, allocate approximately in proportion to N_hs_h: 120, 240, 240 and 320. Rounding gives additional samples of 8, 16, 16 and 20, for final stratum samples of 18, 26, 26 and 30.
| Stratum | Frame N_h | Pilot residual SD | Added samples | Final n_h | Approximate weight N_h/n_h |
|---|---|---|---|---|---|
| A | 300 | 0.4 | 8 | 18 | 16.67 |
| B | 300 | 0.8 | 16 | 26 | 11.54 |
| C | 200 | 1.2 | 16 | 26 | 7.69 |
| D | 200 | 1.6 | 20 | 30 | 6.67 |
The high-variance strata receive more effort, but the low-variance large stratum is not abandoned. That floor protects domain estimates and prevents a pilot fluctuation from assigning almost no verification to part of the landscape.
CHAPTER 4 OF 12 · ALLOCATE THE SAMPLE
4. Compare adaptive allocation with equal allocation
Ignoring finite-population corrections, a stratified mean has variance proportional to ΣN_h²s_h²/n_h divided by N². For the adaptive final sizes 18, 26, 26 and 30, the approximate standard error is 0.0930 in the target property units.
| Final allocation | Stratum sample sizes | Approximate standard error | Largest/smallest weight |
|---|---|---|---|
| Residual-variance allocation | 18, 26, 26, 30 | 0.0930 | 2.50 |
| Equal allocation | 25, 25, 25, 25 | 0.0963 | 1.50 |
| Cost-adjusted example | 21, 33, 21, 25 | 0.0963 | 1.79 |
The gain over equal allocation is real but modest. That is a useful result: an aggressive design is not automatically justified by a small theoretical precision gain. Weight spread, logistics, domain coverage and robustness to pilot error also matter.
CHAPTER 5 OF 12 · ALLOCATE THE SAMPLE
5. Residual variance is more useful than raw variance
If phase one already predicts broad soil gradients, phase two should target what the map misses. Use residuals from a cross-fitted or otherwise honest phase-one model, not residuals calculated after fitting and evaluating on the same tiny pilot without correction.
Raw variance may send samples to a stratum whose variation is already well explained by covariates. Residual variance sends effort where verification can improve or correct the map. It also aligns the design with estimators that combine the broad auxiliary census with laboratory outcomes.
Inspect residual bias as well as spread. A stratum with low variance but a large systematic bias may deserve a minimum or targeted allocation even when N_hs_h is small.
CHAPTER 6 OF 12 · ESTIMATE AND TRANSFER
6. Weight stability is a design constraint, not an afterthought
Inverse-probability weights grow when a stratum has a small sampling fraction. Highly variable weights reduce effective sample size and make estimates sensitive to a few observations. Set a minimum sample per stratum and a maximum acceptable weight ratio before allocation.
Report the range, coefficient of variation and effective sample size of the final weights. If weights are trimmed, state the trimming rule and quantify the bias–variance trade-off. Calibration to known phase-one totals can improve precision, but it does not erase unknown selection bias.
Keep a ledger of planned and realised inclusion probabilities. A location that was inaccessible, substituted or added because it looked unusual must be represented honestly in the response and selection model.
CHAPTER 7 OF 12 · ESTIMATE AND TRANSFER
7. Changed context: high-variance strata cost four times more
Suppose laboratory access in strata C and D costs four times as much as in A and B. A cost-aware allocation proportional to N_hs_h/√c_h uses scores 120, 240, 120 and 160. Rounding sixty added samples gives 11, 23, 11 and 15.
The final samples become 21, 33, 21 and 25. The approximate standard error rises to 0.0963, but the expensive strata require fewer added visits. The design question is no longer pure precision; it is precision per budget under coverage constraints.
The transfer check is to recompute expected cost, weight spread and domain-specific uncertainty. Do not copy the equal-cost allocation into a new access regime and call it adaptive.
CHAPTER 8 OF 12 · ESTIMATE AND TRANSFER
8. Estimation must follow the realised design
For a stratified mean, estimate each stratum mean from its verified phase-two sample and combine it with known N_h weights. Regression or generalized-regression estimators may use phase-one auxiliary values to improve precision. Doubly robust estimators can offer protection under stated model and response assumptions.
Use the two-phase verification and double-robust route when map correction is the main objective. Keep design variables, selection probabilities and verification outcomes in the analysis data.
Variance estimation must reflect strata, phases, adaptive rules and any spatial dependence. A naive simple-random-sample formula usually understates uncertainty when nearby observations share soil-forming processes.
Spatial balance and adaptation solve different problems
Residual-variance allocation decides how much effort each stratum receives. Spatial balance decides where within a stratum that effort is placed. Both are needed when soil properties are spatially structured.
Within each adaptive allocation, use a documented spatially balanced or probability-based selection method. Avoid letting crews choose the easiest nearby point. Compare the realised sample with the frame on coordinates, terrain, land use and predicted soil values.
Use buffered spatial cross-validation to assess transfer. Random folds can make a map appear accurate because test locations sit next to training locations.
CHAPTER 9 OF 12 · CONTROL THE RISKS
9. Limits and common misconceptions
Adaptive sampling is not permission to sample only anomalies. If inclusion probabilities are unknown, design-based estimates and weights lose their foundation. A model-based analysis may still be possible, but it answers under stronger assumptions.
Pilot residual standard deviations are noisy. Optimising too closely to them can be worse than a restrained allocation with floors and caps. Rare domains, regulatory thresholds and accessibility risks may deserve explicit constraints beyond variance minimisation.
An area-of-applicability mask remains necessary. More verification in difficult regions does not prove that the final model transfers to environmental combinations absent from both phases.
CHAPTER 10 OF 12 · CONTROL THE RISKS
10. Evidence, field records and reproducibility
Marchant and Lark developed a phased adaptive approach for soil mapping in Adaptive sampling and reconnaissance surveys for geostatistical mapping of the soil. Salehi and colleagues studied two-phase adaptive allocation with variance information in Efficient estimators for adaptive stratified sequential sampling.
Archive the frame version, stratum definitions, pilot seed, selection algorithm, cost values, allocation formula, rounding rule, reserve list, accessibility outcomes and every realised inclusion probability. Preserve laboratory quality-control records and the phase-one model version used to compute residuals.
Reproduce the allocation from a frozen pilot table before releasing the field list. A second implementation should return the same selected sample and weight ledger.
CHAPTER 11 OF 12 · VERIFY AND CONTINUE
11. Frequently asked questions
Is phase one always a field sample?
No. It can be wall-to-wall remote sensing, proximal sensing or an existing map, provided its coverage and measurement meaning are documented.
Why keep a minimum sample in every stratum?
It protects domain estimates, limits extreme weights and reduces the chance that noisy pilot variances erase a landscape unit from verification.
Can weights be ignored if a flexible model is used?
Not safely by default. Selection can depend on variables related to the outcome, so the model and design should be reconciled explicitly.
Does adaptive allocation guarantee a better map?
No. Gains depend on pilot quality, residual structure, costs, spatial placement and whether the final model uses the added information effectively.
CHAPTER 12 OF 12 · VERIFY AND CONTINUE
12. Final transfer check and connected reading
Recreate the N_hs_h scores, allocate sixty added samples and compute the final weights. Then introduce fourfold costs in strata C and D and explain why the cost-adjusted design changes both precision and representation.
Continue through the Soil Science root, two-phase map correction, spatially balanced sampling, informative nonresponse, spatial cross-validation and area-of-applicability masks. Together they make the map-update design auditable from field selection to transfer.
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