Apparent vs True Dip Conversion

The Trigonometric Relationship

The relationship between apparent dip and true dip is:

tan(δ) = tan(α) × cos(β)

• δ = apparent dip angle
• α = true dip angle
• β = angle between the cross-section direction and the true dip direction (perpendicular to strike)

When the cross-section is oriented exactly perpendicular to strike (β = 0°), cos(β) = 1 and apparent dip equals true dip. When the section is parallel to strike (β = 90°), cos(β) = 0 and apparent dip is zero, meaning the beds appear horizontal regardless of their actual dip. For all intermediate angles, apparent dip is less than true dip.

To find apparent dip when you know true dip and section orientation:
δ = arctan(tan(α) × cos(β))

To find true dip from apparent dip and section orientation:
α = arctan(tan(δ) / cos(β))

Determining True Dip from Two Apparent Dips

When true dip and strike are unknown, they can be determined from two apparent dip measurements taken in different directions. This is a common field problem. You can measure apparent dips on two non-parallel outcrop faces or road cuts, but need to know the true dip and strike for mapping.

The graphical solution uses the alignment diagram (nomogram) or the stereographic projection: plot both apparent dip measurements as lines on the stereonet, and the great circle passing through both points gives the true dip (maximum dip of the great circle) and strike (the horizontal line along the great circle).

The analytical solution requires solving two simultaneous equations of the form tan(δ) = tan(α) × cos(β) for each measurement, where α (true dip) and the dip direction are the two unknowns. This can be solved trigonometrically or by iterative methods. Most structural geology software (Stereonet, DIPS, GeoCalculator) includes this calculation.

Important: The two apparent dip measurements must be in different directions (not parallel). The greater the angle between the two measurement directions, the more accurate the result. Measurements 60–120° apart give the best results. Measurements less than 20° apart give poorly constrained solutions.

Applications in Cross-Sections and Boreholes

Cross-section construction: When drawing geological cross-sections, the section line is rarely perpendicular to strike for all structures. Each formation contact, fault, or unconformity must be plotted at its apparent dip angle for that section direction. Using true dip on an oblique section exaggerates the steepness of the structure and distorts the geometry. For sections with vertical exaggeration, the apparent dip must be further adjusted: tan(δ_exaggerated) = VE × tan(δ), where VE is the vertical exaggeration factor.

Borehole interpretation: When a borehole intersects a dipping surface, the angle seen in the core or on image logs depends on both the formation dip and the borehole trajectory. A vertical borehole measures the true dip directly (assuming the borehole is truly vertical). A deviated borehole sees an apparent dip that depends on the angle between the borehole azimuth and the dip direction. Dipmeter and image log processing software corrects for borehole deviation to compute true formation dip.

Mining and tunneling: In underground excavation design, knowing the true dip and dip direction of discontinuities relative to the tunnel axis is critical for stability analysis. A joint set dipping into the tunnel face is more favorable than one dipping with the face. Converting between apparent dip (visible on the tunnel face) and true dip (needed for stability calculations) is a routine task in underground engineering.