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A small residual slope can create a meaningful height error when the substrate reference and printed-film measurement lie far apart. The relevant quantity is not tilt alone, but tilt multiplied by the spatial separation used in the height comparison. A thickness uncertainty review should therefore preserve the reference coordinates, fitted plane and covariance, not just a final leveled image.
Measurement purpose
Quantify baseline-plane correction and residual uncertainty for an identified printed-film step using actual measurement coordinates.
Specimens and conditions
- Physical regions
- Establish valid film and substrate boundaries independently of the tilt calculation.
- Support state
- Record mounting, scan direction and reference areas so orientation comparisons remain interpretable.
Equipment and records required
- Surface profiler: Retain raw coordinates and height data rather than only a rendered leveled image.
- Analysis software: Export plane coefficients, selected regions and fit uncertainty or sufficient data to evaluate it.
Method sequence
- Fit
Fit a justified local reference plane using geometrically adequate support.
Record: Coordinates, coefficients and residual map.
- Propagate
Calculate slope-times-separation correction and relevant covariance contributions.
Record: Correction and uncertainty budget.
- Challenge
Compare permitted region changes and controlled orientations without hiding disagreement.
Record: Sensitivity checks and final method choice.
Decision and uncertainty
Accept a corrected step only when the physical reference and baseline model support its stated uncertainty.
Include slope covariance, extrapolation, position and model inadequacy where consequential; repeated pixel count alone does not establish independence.
The metrology owner approves the measurement model and the drawing owner applies the agreed decision rule.
Traceable outputs
| Record | Required contents |
|---|---|
| Tilt contribution record | Raw height, signed separations, fitted slopes, correction and residual uncertainty. |
| Model adequacy evidence | Reference geometry, residual structure, sensitivity and orientation comparisons. |
Method review decisions
- Calculate the slope contribution using the actual separation of compared regions.
- Distinguish a measured correction from the uncertainty that remains after correction.
- Check whether plane fitting removes fixture tilt or incorrectly removes specimen form.
Begin with an already defined physical step
Identify the valid substrate and film regions before evaluating numerical tilt. This analysis assumes those surfaces genuinely represent the required physical boundaries. If the lower interface is buried or the baseline is another material stack, a precise plane correction cannot solve that different problem.
Record the coordinate system, units and direction of positive height. Use the centroids or explicitly defined comparison positions of the regions, not an approximate distance read from a screenshot. Preserve the raw data because a leveled export may no longer contain the information needed to reconstruct the applied correction.
Translate residual slope into a height contribution
For a locally planar baseline z equals a x plus b y plus c, the baseline height difference between the two comparison positions is a times their x separation plus b times their y separation. If this term is not removed correctly, it appears in the reported step. The constant offset cancels in a difference; the slope terms generally do not.
In a one-dimensional illustrative scan, a residual slope of 0.8 micrometers per millimeter and a separation of 2.5 millimeters produce a 2.0 micrometer apparent height contribution. The same slope over 0.25 millimeters contributes 0.2 micrometers. These values describe geometric sensitivity, not an acceptable thickness or instrument specification.
hcorrected = Δz − aΔx − bΔy
- Δz is the measured mean height difference between the defined film and reference regions.
- a and b are fitted baseline slopes in height per lateral distance.
- Δx and Δy are the signed separations of the region comparison positions.
A local plane adequately represents the relevant baseline over the compared regions. Curvature, surface discontinuities and invalid physical references require another model.
Support the plane with geometry that constrains both slopes
A narrow strip of reference pixels can constrain slope strongly along its length and poorly across its width. Thousands of nearly collinear points do not provide the same two-dimensional support as a well-distributed area. Examine the layout of valid baseline regions and the distance over which the plane is extrapolated.
Where accessible, baseline regions on more than one side of the film can expose an unstable extrapolation. Exclude printed features from the baseline by a documented geometry rule, not by selecting only pixels that make the fit flatter. If the ceramic is curved, the residual pattern may show systematic form that a single plane cannot represent.
Carry slope uncertainty and covariance into the result
For the tilt contribution alone, treating coordinate separations as fixed, the variance is the squared x separation times the variance of a, plus the squared y separation times the variance of b, plus twice their product times the covariance of a and b. The covariance can matter when the reference geometry couples the two fitted slopes.
For an illustrative one-dimensional slope standard uncertainty of 0.12 micrometers per millimeter across 2.5 millimeters, the tilt standard-uncertainty contribution is 0.30 micrometers. It is not the complete thickness uncertainty. Add the relevant height measurement, model inadequacy, position and other terms with their actual correlations rather than treating repeated fitted pixels as independent physical measurements.
Do not confuse removing tilt with removing its uncertainty
Applying a two micrometer correction does not imply a two micrometer uncertainty, and obtaining a small correction does not imply negligible uncertainty. The correction estimates a systematic contribution; its residual uncertainty depends on the fit evidence and model. Record both independently so a reviewer can reconstruct the result.
| Finding | Meaning | Next action |
|---|---|---|
| Large slope with well-supported local plane | A sizeable but potentially correctable contribution | Apply and document correction with residual uncertainty |
| Small slope from a narrow reference strip | Small estimate but potentially weak cross-slope knowledge | Improve geometric support or enlarge uncertainty |
| Curved residual pattern after plane removal | Plane may not represent substrate form | Review a physically justified local model |
| Result changes strongly with baseline region | Reference or extrapolation sensitivity | Investigate before thickness disposition |
| Good repeatability with unchanged fitting mask | Stable procedure only | Still assess reference geometry and model bias |
Repeat orientation as a diagnostic, not an automatic averaging rule
Reacquire the same physical feature after a controlled in-plane rotation or scan-direction reversal when the instrument and fixture permit it. Transform coordinates back to specimen axes before comparing. A difference that follows instrument direction can indicate acquisition or leveling effects; a feature that follows the specimen may represent real topography.
Rotation also changes mounting and may alter contact or support. Record those changes rather than attributing every difference to tilt. Do not average disagreeing orientations merely to reduce the reported spread. First determine whether both runs measure the same regions with valid signal and equivalent physical support.
Preserve real topography when changing the baseline model
A higher-order surface can make residuals look smaller while removing real ceramic form or printed-film variation. Model complexity should follow the physical reference and measurement objective, not the desire for a visually flat map. Keep the original plane result when evaluating a proposed alternative so the consequence is visible.
Test modest, predeclared changes in baseline windows and quantify their effect on the corrected step. Treat large sensitivity as a reason to improve the method or report a broader uncertainty. Do not use repeated re-fitting until one region selection happens to pass the drawing limit. The analysis recipe is part of the controlled measurement definition.
Deliver a coordinate-aware uncertainty record
Save raw and corrected height differences, region coordinates, plane coefficients, fit support and the uncertainty calculation. State whether slope covariance and position uncertainty were included, neglected with justification or unavailable. Keep units consistent across micrometers, millimeters and any angular representation; an unconverted slope unit can create a large arithmetic error.
The final result should identify the residual contribution relevant to the actual step separation. Reassess it when the feature moves, the baseline area shrinks, the substrate form changes or the objective captures a different field. A correction validated for a nearby reference is not automatically valid for a distant region on the same ceramic panel.
Provide the raw profile and baseline fit
The uncertainty calculation needs spatial data and region definitions.
- Raw height coordinates and identified physical step boundaries.
- Baseline and film masks with fitted plane coefficients.
- Fit covariance, repeat runs and permitted alternative regions.
- Support state, measurement units and required decision interval.
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