---
title: "Geological Maps"
book: "University Earth and Space Sciences — Year 1"
subject: earth
language: en
chapter: 4
exercises: 12
source: https://one-course.com/books/earth/3/en/chapter/4-geological-maps
license: CC-BY-NC-SA-4.0
credit: "One Earth and Space Sciences Book, One Course (one-course.com)"
---

# Chapter 4 — Geological Maps

A geologist crosses a range of low hills with a map board, a notebook and a compass-clinometer. Most of the ground is grass, wood or ploughed field; rock shows only here and there, in a stream bank, a road cutting, a crag. From these scattered glimpses, and from the shape of the land, the geologist must say which rock lies under every square metre of the hills, how the layers are tilted, where they are broken, and in what order all this happened. The result is a [geological map](#def-b1-geological-maps-geological-map): a flat drawing that encodes a three-dimensional body of rock. This chapter explains how such a map is made and how it is read — how the [attitude](#def-b1-geological-maps-attitude) of a surface is measured and computed, why the [outcrop](#def-b1-geological-maps-geological-map) of a layer draws the patterns it does, and how a map tells a history. Its examples come from an imaginary but geologically consistent region, the Varden Hills, which recurs throughout this book.

**You already know.**

A topographic map shows the relief with contour lines, lines of equal elevation; its scale relates distances on the map to distances on the ground (Book 1, grade 9). Relative ages follow from superposition (a layer is younger than the layers below it, unless the sequence has been overturned), from cross-cutting (whatever cuts a rock is younger than it) and from inclusion (Book 1, grade 7). A fault is a fracture along which the two blocks have moved (Book 1, grade 7); the geometry of [folds](#def-b1-geological-maps-fold) and faults is the subject of [Chapter 17](https://one-course.com/books/earth/3/en/chapter/17-faults-folds-and-fabrics#ch-b1-faults-folds-fabrics).

## 4.1 A map is a cut through three dimensions

**Definition 4.1 (Geological map).**

A *geological map* shows, at every point of the ground surface, the rock unit that lies there beneath the soil, with the *contacts* — the surfaces separating two units — and the structures (faults, [fold](#def-b1-geological-maps-fold) axes, [attitudes](#def-b1-geological-maps-attitude) of layers) that affect them. A *map unit* is a body of rock that can be recognised and followed in the field and is drawn in one colour; its *outcrop* is the area where it lies at the surface.

**Definition 4.2 (Digital elevation model).**

A *digital elevation model* is a grid of ground elevations, $z = g(x,y)$, at regularly spaced points. A geographic information system stores such grids and the map’s other layers ([contacts](#def-b1-geological-maps-geological-map), faults, measurements, boreholes) in a common frame of coordinates and computes with them.

**Proposition 4.3 (The outcrop of a contact).**

Let the ground be the surface $z = g(x,y)$ and a contact the surface $z = c(x,y)$. The trace of the contact on the map is the curve

$$
g(x,y) = c(x,y),
$$

the projection on the horizontal plane of the intersection of the two surfaces. The unit lying above the contact crops out where $g > c$.

**Proof.** A point of the map lies on the trace if and only if the ground surface at that point belongs to the contact, that is $g(x,y) = c(x,y)$. Where $g(x,y) > c(x,y)$, the ground is above the contact: the rock at the surface belongs to the unit lying above it. ∎

**Remark 4.4 (How the maps of this book are drawn).**

This is exactly how a geographic information system draws a map from a three-dimensional model: it evaluates $g - c$ on the grid of a [digital elevation model](#def-b1-geological-maps-dem) and traces the curves where it changes sign. The maps of the Varden Hills in this book were drawn that way, from a geological model in which every surface is known.

![The Varden Hills as a digital elevation model: the ground lit from the north-west, with contours every 50\, m (thicker at 250 and 500\, m) and the river, which flows east.](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/fig-60002b7ac8b9.svg)

*The Varden Hills as a [digital elevation model](#def-b1-geological-maps-dem): the ground lit from the north-west, with contours every $50\,\mathrm{m}$ (thicker at 250 and $500\,\mathrm{m}$) and the river, which flows east.*

![The geological map of the Varden Hills, with the contours, the river, two measured attitudes (strike line and dip tick, with the dip in degrees) and the legend. The ages are those of the region’s units; the fault is hidden beneath the younger alluvium of the valley floor.](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/fig-484b41a8097d.svg)

![The geological map of the Varden Hills, with the contours, the river, two measured attitudes (strike line and dip tick, with the dip in degrees) and the legend. The ages are those of the region’s units; the fault is hidden beneath the younger alluvium of the valley floor.](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/fig-6d7b0470bdef.svg)

*The [geological map](#def-b1-geological-maps-geological-map) of the Varden Hills, with the contours, the river, two measured [attitudes](#def-b1-geological-maps-attitude) ([strike](#def-b1-geological-maps-attitude) line and [dip](#def-b1-geological-maps-attitude) tick, with the [dip](#def-b1-geological-maps-attitude) in degrees) and the legend. The ages are those of the region’s units; the fault is hidden beneath the younger alluvium of the valley floor.*

## 4.2 The attitude of a plane

**Definition 4.5 (Strike, dip and dip direction).**

Let a geological surface — a bedding plane, a contact, a fault — be locally a plane. Its *strike* is the direction of the horizontal lines it contains, given as an [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) (degrees clockwise from north). Its *dip* $\delta$ is the angle between the plane and the horizontal, measured in the vertical plane perpendicular to the strike, $0 \le \delta \le 90{}^{\circ}$. Its *dip direction* $\alpha$ is the [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) of the steepest downward line of the plane. Together they form the plane’s *attitude*, written dip direction/dip, for instance $080/12$.

**Notation 4.6 (Strike by the right-hand rule).**

A [strike](#def-b1-geological-maps-attitude) can be read in two opposite directions. The *right-hand rule* chooses the one for which the plane [dips](#def-b1-geological-maps-attitude) to the right of an observer looking along the [strike](#def-b1-geological-maps-attitude): [strike](#def-b1-geological-maps-attitude) $= \alpha - 90{}^{\circ}$ (modulo $360{}^{\circ}$). On a map, the symbol is a [strike](#def-b1-geological-maps-attitude) line with a short tick pointing down the [dip](#def-b1-geological-maps-attitude), labelled with the [dip](#def-b1-geological-maps-attitude).

![A plane striking north and dipping 30 to the east (attitude 090/30), cutting a horizontal plane along its strike line. Not to scale.](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/fig-2c7851542ce2.svg)

![A plane striking north and dipping 30 to the east (attitude 090/30), cutting a horizontal plane along its strike line. Not to scale.](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/fig-2596f5d0020f.svg)

*A plane striking north and dipping $30{}^{\circ}$ to the east ([attitude](#def-b1-geological-maps-attitude) $090/30$), cutting a horizontal plane along its [strike](#def-b1-geological-maps-attitude) line. Not to scale.*

**Proposition 4.7 (The equation of a dipping plane).**

With $x$ to the east, $y$ to the north and $z$ upward, the plane of [dip direction](#def-b1-geological-maps-attitude) $\alpha$ and [dip](#def-b1-geological-maps-attitude) $\delta$ through the point $(x_0, y_0, z_0)$ is

$$
z = z_0 - \tan\delta\,\bigl[(x - x_0)\sin\alpha + (y - y_0)\cos\alpha\bigr].
$$

**Proof.** The horizontal unit vector pointing in the [dip direction](#def-b1-geological-maps-attitude) is $\vec u = (\sin\alpha, \cos\alpha, 0)$ (an [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) is measured from north, the $y$ axis, toward east, the $x$ axis). Along the [strike](#def-b1-geological-maps-attitude), perpendicular to $\vec u$, the plane is horizontal; along $\vec u$ it descends by $\tan\delta$ per unit of horizontal distance. The horizontal distance of a point from the [strike](#def-b1-geological-maps-attitude) line through $(x_0, y_0)$, counted positive [down-dip](#def-b1-geological-maps-attitude), is $(x-x_0)\sin\alpha + (y-y_0)\cos\alpha$, whence the formula. ∎

**Theorem 4.8 (Attitude from three points).**

Let $P_1, P_2, P_3$ be three non-collinear points of a plane, and $\vec n = \overrightarrow{P_1P_2} \times \overrightarrow{P_1P_3} =
(n_x, n_y, n_z)$, chosen with $n_z > 0$ (replace $\vec n$ by $-\vec n$ otherwise). Then the [dip](#def-b1-geological-maps-attitude) and the [dip direction](#def-b1-geological-maps-attitude) of the plane are

$$
\delta = \arccos\frac{n_z}{\|\vec n\|}, \qquad
  \alpha = \text{the azimuth of the horizontal vector } (n_x, n_y),
$$

that is the angle $\alpha \in [0{}^{\circ}, 360{}^{\circ})$ with $\sin\alpha = n_x/\sqrt{n_x^2+n_y^2}$ and $\cos\alpha = n_y/\sqrt{n_x^2+n_y^2}$ (the plane is horizontal if $n_x = n_y = 0$).

**Proof.** $\vec n$ is perpendicular to two independent vectors of the plane, hence normal to it. The [dip](#def-b1-geological-maps-attitude) is the angle between the plane and the horizontal, equal to the angle between their normals, $\vec n$ and the vertical $\vec k$: $\cos\delta = \vec n \cdot \vec k / \|\vec n\| = n_z/\|\vec n\|$. By [Proposition 4.7](#prop-b1-geological-maps-plane), the plane is $z = \text{const} -
\tan\delta\,(x\sin\alpha + y\cos\alpha)$, that is $\tan\delta\sin\alpha\,x + \tan\delta\cos\alpha\,y + z = \text{const}$; its upward normal is proportional to $(\tan\delta\sin\alpha,
\tan\delta\cos\alpha, 1)$, whose horizontal part points in the [dip direction](#def-b1-geological-maps-attitude) $\alpha$. Since $\vec n$ is a positive multiple of this normal, $(n_x, n_y)$ points in the [dip direction](#def-b1-geological-maps-attitude) too. ∎

**Method 4.9 (The three-point problem).**

To find the [attitude](#def-b1-geological-maps-attitude) of a contact met in three boreholes (or seen at three [outcrops](#def-b1-geological-maps-geological-map)):

1. for each borehole, take the elevation of the contact: the ground elevation minus the depth at which the contact was met;
2. write the three points $P_i = (x_i, y_i, z_i)$ in metres, $x$ east, $y$ north;
3. compute $\vec n = \overrightarrow{P_1P_2} \times  \overrightarrow{P_1P_3}$ and make $n_z$ positive;
4. apply [Theorem 4.8](#thm-b1-geological-maps-three-point) ; give the [attitude](#def-b1-geological-maps-attitude) as [dip direction](#def-b1-geological-maps-attitude) / [dip](#def-b1-geological-maps-attitude) and, if asked, the [strike](#def-b1-geological-maps-attitude) by the right-hand rule.

On paper, the same problem is solved with structure contours: on the line joining the highest and the lowest point, find by proportion the point at the elevation of the third one; the line through these two points of equal elevation is a [strike](#def-b1-geological-maps-attitude) line; the [dip](#def-b1-geological-maps-attitude) follows from the elevation difference between two parallel [strike](#def-b1-geological-maps-attitude) lines and their spacing.

**Example 4.10 (The east limb of the Varden anticline).**

Three boreholes, E1 at $(4000, 6000)$, E2 at $(4600, 7200)$ and E3 at $(5100, 6400)$ (metres east and north of the map’s south-west corner), meet the base of the limestone at elevations $174$, $4$ and $-74\,\mathrm{m}$. Then $\overrightarrow{P_1P_2} = (600, 1200, -170)$, $\overrightarrow{P_1P_3} = (1100, 400, -248)$; their cross product is $(-229\,600, -38\,200, -1\,080\,000)$, which points downward, so

$$
\vec n = (229\,600,\ 38\,200,\ 1\,080\,000).
$$

Hence $\cos\delta = 1\,080\,000/\|\vec n\| \approx
0.9776$, $\delta \approx 12.2{}^{\circ}$, and the [dip direction](#def-b1-geological-maps-attitude) is the [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) of $(229\,600, 38\,200)$, $\alpha \approx 80.6{}^{\circ}$: the [attitude](#def-b1-geological-maps-attitude) is $081/12$, [strike](#def-b1-geological-maps-attitude) $351{}^{\circ}$ by the right-hand rule. The base of the limestone [dips](#def-b1-geological-maps-attitude) gently east-north-east.

## 4.3 Outcrop patterns

**Proposition 4.11 (Horizontal and vertical contacts).**

The trace of a horizontal contact follows the contour line of its elevation. The trace of a vertical contact is a straight line on the map, whatever the relief.

**Proof.** For $c(x,y) = z_0$, the trace $g(x,y) = z_0$ is the contour $z_0$ ([Proposition 4.3](#prop-b1-geological-maps-intersection)). A vertical plane is the set of points whose horizontal position lies on one straight line, at any elevation; its trace on the map is that line. ∎

**Theorem 4.12 (The V-rule).**

A planar contact crosses a valley whose floor descends downstream with gradient $s > 0$. Let $a$ be the tangent of the contact’s apparent [dip](#def-b1-geological-maps-attitude) in the downstream direction ($a > 0$ if it [dips](#def-b1-geological-maps-attitude) downstream, $a < 0$ if it [dips](#def-b1-geological-maps-attitude) upstream, $a = 0$ if horizontal). Then the [outcrop](#def-b1-geological-maps-geological-map) trace makes a V whose apex lies on the valley floor and

- points *downstream* if $a > s$ : the contact [dips](#def-b1-geological-maps-attitude) downstream more steeply than the valley floor descends;
- points *upstream* if $a < s$ : the contact is horizontal, [dips](#def-b1-geological-maps-attitude) upstream, or [dips](#def-b1-geological-maps-attitude) downstream less steeply than the floor.

A vertical contact makes no V.

**Proof.** Let $\xi$ be the distance measured downstream along the valley floor. The floor is at elevation $z_g(\xi) = z_0 - s\xi$; in the vertical section along the valley, the contact is at $z_c(\xi) = c - a\xi$. On the floor the trace is where $z_g = z_c$:

$$
\xi_{\text{floor}} = \frac{c - z_0}{a - s}.
$$

On a valley side, at a point higher than the floor by $h > 0$ and at the same distance from the valley’s source, the ground is at $z_0 - s\xi + h$, so the trace is where $c - a\xi = z_0 - s\xi + h$:

$$
\xi_{\text{side}} = \frac{c - z_0 - h}{a - s}, \qquad
  \xi_{\text{floor}} - \xi_{\text{side}} = \frac{h}{a - s}.
$$

If $a > s$, the trace lies farther downstream on the floor than on the sides: the V points downstream. If $a < s$, the reverse. A vertical contact crosses the floor and the sides at the same $\xi$. ∎

![The V-rule computed. In each panel a river (blue arrow) flows north down a valley, whose contours (every 10\, m) point upstream; a contact striking east–west is traced in red. Horizontal, it follows the contours; dipping upstream, or downstream less steeply than the valley floor, it makes a V pointing upstream; dipping downstream more steeply (here 10), a V pointing downstream. A vertical contact would cross straight.](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/fig-321a4df10adb.svg)

*The V-rule computed. In each panel a river (blue arrow) flows north down a valley, whose contours (every $10\,\mathrm{m}$) point upstream; a contact striking east–west is traced in red. Horizontal, it follows the contours; dipping upstream, or downstream less steeply than the valley floor, it makes a V pointing upstream; dipping downstream more steeply (here $10{}^{\circ}$), a V pointing downstream. A vertical contact would cross straight.*

**Remark 4.13 (The width of an outcrop).**

The width of a unit’s [outcrop](#def-b1-geological-maps-geological-map) depends on its thickness, on its [dip](#def-b1-geological-maps-attitude) and on the slope of the ground: a thin layer dipping in the same direction as the slope can crop out over a wide area. Thickness is computed from a map and a cross-section in [Chapter 5](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#ch-b1-cross-sections).

## 4.4 Folds, faults, intrusions and unconformities on a map

**Definition 4.14 (Anticline and syncline).**

A *fold* is a bend of once-planar layers. An *anticline* is a fold whose layers are convex upward (an upfold); a *syncline* is a fold whose layers are concave upward (a downfold). The two sides of a fold are its limbs; its *fold axis* is the line along which the layers bend most, and the axis plunges when it is not horizontal. A line such as a fold axis is located by its trend, the [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) of its horizontal projection taken in the downward direction, and its plunge, the angle it makes with the horizontal ([Chapter 6](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#ch-b1-field-lab-methods-1)). Fold geometry in three dimensions is the subject of [Chapter 17](https://one-course.com/books/earth/3/en/chapter/17-faults-folds-and-fabrics#ch-b1-faults-folds-fabrics).

**Proposition 4.15 (Fold patterns).**

On a map of eroded folded layers, an [anticline](#def-b1-geological-maps-fold) shows the oldest units in its core and younger units outward on both sides; a [syncline](#def-b1-geological-maps-fold) shows the youngest in its core. If the [fold axis](#def-b1-geological-maps-fold) plunges, the [outcrops](#def-b1-geological-maps-geological-map) close around the [fold](#def-b1-geological-maps-fold): the closure of an [anticline](#def-b1-geological-maps-fold) points in the direction of plunge, that of a [syncline](#def-b1-geological-maps-fold) the opposite way.

**Partial proof.** In an [anticline](#def-b1-geological-maps-fold) the layers rise toward the axis, so a roughly level erosion surface cuts deepest into the sequence there and exposes its oldest, lowest units; in a [syncline](#def-b1-geological-maps-fold) the youngest units are preserved lowest, in the core. If the axis plunges, consider a contact of an [anticline](#def-b1-geological-maps-fold): it is an arch whose crest descends along the plunge. Where the crest is above the ground, the ground cuts the arch along two limbs; where the crest passes below the ground, the contact no longer reaches the surface. The two limbs of the trace therefore join round the point where the crest meets the ground, and that closure lies in the direction of plunge. A [syncline](#def-b1-geological-maps-fold) is a trough whose bottom rises the opposite way, so its closure points against the plunge. The argument assumes gentle relief compared with the drop of the axis. ∎

**Proposition 4.16 (Faults, intrusions and unconformities).**

On a map:

- a fault interrupts and offsets the [contacts](#def-b1-geological-maps-geological-map) it crosses; on the block that moved down, younger units come to the surface opposite older ones; the trace of a dipping fault obeys the V-rule like any plane;
- a pluton or a dyke cuts across the [contacts](#def-b1-geological-maps-geological-map) of the rocks it intrudes, and is younger than them;
- at an unconformity, the [contacts](#def-b1-geological-maps-geological-map) of the older rocks are cut off along the base of the younger ones, which may rest on several older units in turn.

**Method 4.17 (Reading the history from a map).**

1. List the units in order of age, from the legend and from superposition.
2. Find what cuts what: faults, intrusions, unconformities; each is younger than everything it cuts and older than everything that seals it.
3. Find the [folds](#def-b1-geological-maps-fold) and their axes; a [fold](#def-b1-geological-maps-fold) is younger than the youngest unit it [folds](#def-b1-geological-maps-fold) .
4. Write the events in order, each with the evidence on the map.

**Example 4.18 (The history of the Varden Hills).**

On the map of the Varden Hills: the schist is the oldest unit; the granite cuts it, so it is younger; the sandstone rests on both, cutting off the granite’s outline: an unconformity, after a long erosion. The sandstone, limestone, shale and chalk follow one another without a break and are folded together into an [anticline](#def-b1-geological-maps-fold) (sandstone and basement in the core, near $x = 3.5\,\mathrm{km}$) and a [syncline](#def-b1-geological-maps-fold) (chalk in the core, east of the map’s centre): the folding came after the chalk was laid down. The fault offsets the folded [contacts](#def-b1-geological-maps-geological-map): it is younger than the folding. Finally the river has cut down and spread its alluvium, which no fault breaks.

**In the field — Measuring an attitude.**

On a clean bedding surface, the geologist lays the edge of a compass-clinometer horizontal and reads the [strike](#def-b1-geological-maps-attitude) on the compass, then sets it down the steepest slope and reads the [dip](#def-b1-geological-maps-attitude) on the clinometer. Every measurement is written in the notebook with its position and plotted on the map at once, so that an inconsistent reading shows up while the [outcrop](#def-b1-geological-maps-geological-map) is still at hand. Today, positions come from satellite receivers, measurements are entered on tablets that run a geographic information system, and [outcrops](#def-b1-geological-maps-geological-map) are photographed and placed on [digital elevation models](#def-b1-geological-maps-dem).

![](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/img-e43a48f12c85.jpg)

**History — The first geological map of a country.**

The surveyor William Smith, who dug canals and drained land in England, noticed that each layer of rock carried its own fossils, in an order that was the same everywhere. Using the fossils to recognise the layers, he followed them across the country for more than fifteen years. His map, *A Delineation of the Strata of England and Wales, with part of Scotland*, dated 1 August 1815, is considered the first [geological map](#def-b1-geological-maps-geological-map) of a complete country.

![William Smith’s map of 1815 (public domain).](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/img-a529e4760eee.jpg)

*William Smith’s map of 1815 (public domain).*

## 4.5 Exercises

**Exercise 4.1 ★.**

Define the [strike](#def-b1-geological-maps-attitude), the [dip](#def-b1-geological-maps-attitude) and the [dip direction](#def-b1-geological-maps-attitude) of a plane. A bed has [attitude](#def-b1-geological-maps-attitude) $135/40$: give its [strike](#def-b1-geological-maps-attitude) by the right-hand rule and draw its map symbol.

**Solution of Exercise 4.1.**

[Strike](#def-b1-geological-maps-attitude): [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) of the horizontal lines of the plane; [dip](#def-b1-geological-maps-attitude): angle between the plane and the horizontal, measured perpendicular to the [strike](#def-b1-geological-maps-attitude); [dip direction](#def-b1-geological-maps-attitude): [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) of the steepest downward line. For $135/40$, the [strike](#def-b1-geological-maps-attitude) by the right-hand rule is $135 - 90 = 045{}^{\circ}$. Symbol: a [strike](#def-b1-geological-maps-attitude) line oriented N 45 E, with a tick on its south-east side (toward $135{}^{\circ}$) labelled 40.

**Exercise 4.2 ★.**

Write the equation of the plane of [attitude](#def-b1-geological-maps-attitude) $090/30$ through the point $(0, 0, 500)$, coordinates in metres. At what elevation is it at $(1000, 0)$ and at $(0, 1000)$?

**Solution of Exercise 4.2.**

$z = 500 - \tan30{}^{\circ}\,(x\sin90{}^{\circ} + y\cos90{}^{\circ}) = 500 - 0.577\,x$. At $(1000, 0)$: $500 - 577 = -77\,\mathrm{m}$. At $(0, 1000)$: $500\,\mathrm{m}$, the plane being horizontal along its north–south [strike](#def-b1-geological-maps-attitude).

**Exercise 4.3 ★.**

On a map, the trace of a contact follows the contour line $350\,\mathrm{m}$ exactly, over several kilometres of hills and valleys. What can you say about the contact? And if its trace is a straight line across hills and valleys?

**Solution of Exercise 4.3.**

A trace that follows a contour is a horizontal contact, at $350\,\mathrm{m}$. A trace that runs straight across hills and valleys belongs to a vertical contact.

**Exercise 4.4 ★.**

On the map of the Varden Hills, which unit crops out in the core of the [anticline](#def-b1-geological-maps-fold) along the river, and which unit occupies the core of the [syncline](#def-b1-geological-maps-fold)? Explain why with the principle of superposition.

**Solution of Exercise 4.4.**

Along the river, the core of the [anticline](#def-b1-geological-maps-fold) shows the schist and the granite (beneath the sandstone); the core of the [syncline](#def-b1-geological-maps-fold) is occupied by the chalk. The oldest units lie lowest in the sequence: where the layers arch upward and erosion cuts deepest into them, the oldest appear; where they sag, the youngest are preserved in the middle.

**Exercise 4.5 ★★.**

A contact is met at elevations $300\,\mathrm{m}$ at $(0, 0)$, $300\,\mathrm{m}$ at $(1000, 0)$ and $200\,\mathrm{m}$ at $(0, 500)$ (metres, $x$ east, $y$ north). Find its [strike](#def-b1-geological-maps-attitude), [dip direction](#def-b1-geological-maps-attitude) and [dip](#def-b1-geological-maps-attitude), first by reasoning on structure contours, then with [Theorem 4.8](#thm-b1-geological-maps-three-point).

**Solution of Exercise 4.5.**

The first two points are at the same elevation: the line $y = 0$ is a [strike](#def-b1-geological-maps-attitude) line, so the [strike](#def-b1-geological-maps-attitude) is east–west. The contact is $100\,\mathrm{m}$ lower $500\,\mathrm{m}$ to the north: it [dips](#def-b1-geological-maps-attitude) north, $\tan\delta = 100/500 = 0.2$, $\delta \approx 11.3{}^{\circ}$; [dip direction](#def-b1-geological-maps-attitude) $000{}^{\circ}$, [strike](#def-b1-geological-maps-attitude) $270{}^{\circ}$ by the right-hand rule. With the theorem: $\overrightarrow{P_1P_2} = (1000, 0, 0)$, $\overrightarrow{P_1P_3} = (0, 500, -100)$, $\vec n = (0, 100\,000,
500\,000)$, $\cos\delta = 500\,000/509\,902 \approx 0.981$, $\delta \approx
11.3{}^{\circ}$, and $(n_x, n_y) = (0, 100\,000)$ points north: the same result.

**Exercise 4.6 ★★.**

A valley floor descends northward at $30\,\mathrm{m}$ per km. Which way does the V of the [outcrop](#def-b1-geological-maps-geological-map) of a contact point if the contact (a) is horizontal; (b) [dips](#def-b1-geological-maps-attitude) $20{}^{\circ}$ south; (c) [dips](#def-b1-geological-maps-attitude) $1{}^{\circ}$ north; (d) [dips](#def-b1-geological-maps-attitude) $20{}^{\circ}$ north? Justify with [Theorem 4.12](#thm-b1-geological-maps-v-rule).

**Solution of Exercise 4.6.**

Downstream is north, $s = 0.03$ ($1.72{}^{\circ}$). (a) $a = 0 < s$: V upstream (south), along the contours. (b) $a = -\tan20{}^{\circ} < 0 < s$: V upstream. (c) $a = \tan1{}^{\circ} \approx 0.017 < s$: V upstream. (d) $a = \tan20{}^{\circ}
\approx 0.364 > s$: V downstream (north).

**Exercise 4.7 ★★.**

In the situation of the V-rule, a contact [dips](#def-b1-geological-maps-attitude) downstream with $a = 0.364$ ($20{}^{\circ}$) and the valley floor descends with $s = 0.03$. A point on the valley side is $80\,\mathrm{m}$ higher than the floor. How far upstream of the apex is the trace at that point?

**Solution of Exercise 4.7.**

$\xi_{\text{floor}} - \xi_{\text{side}} = h/(a - s) = 80/(0.364 - 0.03)
\approx 240\,\mathrm{m}$: the trace on the side is about $240\,\mathrm{m}$ upstream of the apex.

**Exercise 4.8 ★★.**

The [outcrops](#def-b1-geological-maps-geological-map) of the Varden [anticline](#def-b1-geological-maps-fold) close around the north end of its core. In which direction does the [fold axis](#def-b1-geological-maps-fold) plunge? Would the [outcrops](#def-b1-geological-maps-geological-map) of the [syncline](#def-b1-geological-maps-fold) close to the north or to the south?

**Solution of Exercise 4.8.**

The [anticline](#def-b1-geological-maps-fold)’s closure points in the direction of plunge: the axis plunges north. With an axis plunging north, a [syncline](#def-b1-geological-maps-fold)’s [outcrops](#def-b1-geological-maps-geological-map) close to the south.

**Exercise 4.9 ★★.**

On the [geological map](#def-b1-geological-maps-geological-map), along the fault, which side shows younger units at the surface? Which block moved down? Check against the map symbol.

**Solution of Exercise 4.9.**

The east-south-east side shows younger units (chalk and shale against older units opposite them): that block moved down. The ticks of the fault symbol are on that side, the downthrown side.

**Exercise 4.10 ★★★.**

Show that two planes of [attitudes](#def-b1-geological-maps-attitude) $(\alpha_1, \delta_1)$ and $(\alpha_2, \delta_2)$ intersect along a line parallel to $\vec n_1 \times \vec n_2$, where $\vec n_i = (\tan\delta_i\sin\alpha_i,
\tan\delta_i\cos\alpha_i, 1)$. Apply it to the planes $090/30$ and $270/30$: what is the line, and what kind of [fold](#def-b1-geological-maps-fold) do the two planes make?

**Solution of Exercise 4.10.**

A line common to both planes is perpendicular to both normals, hence parallel to $\vec n_1 \times \vec n_2$. For $090/30$: $\vec n_1 = (0.577, 0,
1)$; for $270/30$: $\vec n_2 = (-0.577, 0, 1)$. Then $\vec n_1 \times \vec n_2
= (0\cdot 1 - 1\cdot 0,\ 1\cdot(-0.577) - 0.577\cdot 1,\ 0) = (0, -1.155,
0)$: a horizontal north–south line. The first plane [dips](#def-b1-geological-maps-attitude) east, the second west: they [dip](#def-b1-geological-maps-attitude) away from the line, which is the crest of an [anticline](#def-b1-geological-maps-fold) with a horizontal axis.

**Exercise 4.11 ★★★.**

A vertical dyke [strikes](#def-b1-geological-maps-attitude) $045{}^{\circ}$ and crops out at the origin, on a valley floor at elevation $0\,\mathrm{m}$. Where can its trace lie on a hill at $300\,\mathrm{m}$? A second dyke has the same [strike](#def-b1-geological-maps-attitude), [dips](#def-b1-geological-maps-attitude) $60{}^{\circ}$ to the south-east and also crops out at the origin. How far from the origin, and in which direction, does its trace lie on ground at $300\,\mathrm{m}$, measured perpendicular to the [strike](#def-b1-geological-maps-attitude)?

**Solution of Exercise 4.11.**

The vertical dyke’s trace is the straight line through the origin trending $045{}^{\circ}$, whatever the elevation: on the hill it lies somewhere on that same line. The dipping dyke is the plane $z = -\tan60{}^{\circ}\,d$, where $d$ is the horizontal distance from the [strike](#def-b1-geological-maps-attitude) line through the origin, counted positive toward the south-east. At $z = 300\,\mathrm{m}$: $d = -300/\tan60{}^{\circ}
\approx -173\,\mathrm{m}$: the trace lies $173\,\mathrm{m}$ north-west of the [strike](#def-b1-geological-maps-attitude) line through the origin, on the [up-dip](#def-b1-geological-maps-attitude) side.

**Exercise 4.12 ★★★.**

In a [digital elevation model](#def-b1-geological-maps-dem) with a $100\,\mathrm{m}$ grid, the ground at $(x, y)$ and at $(x + 100, y)$ is at 420 and $405\,\mathrm{m}$, and at $(x, y + 100)$ at $412\,\mathrm{m}$. Treating the ground in that cell as a plane, compute its slope (in degrees) and the [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) toward which it faces (down-slope direction).

**Solution of Exercise 4.12.**

$$
\frac{\partial z}{\partial x} \approx \frac{405 - 420}{100} = -0.15, \qquad
  \frac{\partial z}{\partial y} \approx \frac{412 - 420}{100} = -0.08,
$$

so the slope is $\arctan\sqrt{0.15^2 + 0.08^2} = \arctan 0.17 \approx
9.6{}^{\circ}$. The ground descends along $-\nabla z = (0.15, 0.08)$, whose [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) is $\arctan(0.15/0.08) \approx 62{}^{\circ}$: the cell faces east-north-east.

## 4.6 Problem: Solving the Varden Hills

**Problem 4.1.**

Weekend problem — from boreholes and a map to the fold axis, the fault and the history of a region

A company drilling for water in the Varden Hills records where its boreholes meet the base of the limestone; its geologists also locate the fault where it crosses the ground. Coordinates are in metres east and north of the map’s south-west corner; elevations in metres above sea level. Part I uses the coloured [geological map](#def-b1-geological-maps-geological-map) of [Section 4.1](#sec-4-1).

**Part I — Reading the map.**

1. List the units of the region from the oldest to the youngest.
2. Which units does the river cut through, from west to east?
3. Which unit occupies the core of the [fold](#def-b1-geological-maps-fold) near $x = 3.5\,\mathrm{km}$ ? What kind of [fold](#def-b1-geological-maps-fold) is it?
4. Which unit occupies the core of the [fold](#def-b1-geological-maps-fold) near $x = 7.5\,\mathrm{km}$ ? What kind of [fold](#def-b1-geological-maps-fold) is it?
5. Why are the schist and the granite visible only near the river, in the core of the first [fold](#def-b1-geological-maps-fold) ?
6. What kind of contact separates the sandstone from the granite? Justify from the map.

The boreholes (depth $d$ to the base of the limestone, ground elevation $z_g$) and the three points F1–F3 where the fault crosses the top of the bedrock (F2, in the valley, lies under the thin alluvium, which the fault does not cut):

|  | east | north | $z_g$ | $d$ |
| --- | --- | --- | --- | --- |
| E1 | 4000 | 6000 | 435 | 261 |
| E2 | 4600 | 7200 | 432 | 428 |
| E3 | 5100 | 6400 | 512 | 586 |
| W1 | 1500 | 6000 | 524 | 669 |
| W2 | 2200 | 7200 | 421 | 459 |
| W3 | 2600 | 5400 | 319 | 209 |
|  | east | north | $z_g$ |  |
| F1 | 4197 | 1000 | 405 |  |
| F2 | 5824 | 3500 | 129 |  |
| F3 | 7321 | 6500 | 482 |  |

![](https://one-course.com/images/onecourse/chapters/earth-3/b1-geological-maps/fig-2a261f8ceec4.svg)

**Part II — The east limb.**

7. Compute the elevation of the base of the limestone in E1, E2 and E3.
8. Write the vectors $\overrightarrow{E_1E_2}$ and $\overrightarrow{E_1E_3}$ .
9. Compute their cross product and orient it upward.
10. Deduce the [dip](#def-b1-geological-maps-attitude) of the base of the limestone on the east limb.
11. Deduce its [dip direction](#def-b1-geological-maps-attitude) , then its [strike](#def-b1-geological-maps-attitude) by the right-hand rule.
12. Is the result consistent with the strike-and-dip symbol of the east limb on the map?

**Part III — The fold.**

13. Compute the elevations of the base of the limestone in W1, W2 and W3.
14. Compute an upward normal $\vec n_W$ to the base of the limestone on the west limb.
15. Deduce the [attitude](#def-b1-geological-maps-attitude) of the west limb.
16. Compute a direction of the intersection of the two limbs, the [fold axis](#def-b1-geological-maps-fold) , as $\vec n_E \times \vec n_W$ (with $\vec n_E$ from question 9).
17. Give the trend (the [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) of the axis’s horizontal projection, taken downward) and the plunge (its angle below the horizontal) of the [fold axis](#def-b1-geological-maps-fold) .
18. The base of the sandstone lies on the axis at $230\,\mathrm{m}$ where the river crosses it, near $y = 4\,\mathrm{km}$ . How much lower is it on the axis $8\,\mathrm{km}$ farther north? Is this consistent with question 3 of Part I and with the closure of the [outcrops](#def-b1-geological-maps-geological-map) ?

**Part IV — The fault and the history.**

19. From F1, F2 and F3, compute the [attitude](#def-b1-geological-maps-attitude) of the fault.
20. Along the fault, the units at the surface on the east-south-east side are younger than those opposite them. Which block moved down? Is the fault normal or reverse?
21. The straight line from F1 to F3 passes about $200\,\mathrm{m}$ west of F2. Why does the trace of the fault bend toward the east-south-east where it crosses the river valley, at F2?
22. Put in order: folding; deposition of the chalk; faulting; intrusion of the granite; erosion before the sandstone; deposition of the alluvium.
23. The chalk is 90 million years old and is folded; the alluvium was laid down in the last 2.58 million years and is not faulted. Between what ages did the folding and the faulting take place?
24. To reach the fault at $1000\,\mathrm{m}$ below sea level beneath F1, how far from F1, and in which direction, must a vertical well be drilled?

**Solution of Problem 4.1.**

**1.** Schist, granite, sandstone, limestone, shale, chalk, alluvium.

**2.** From west to east: chalk and shale, limestone, sandstone, then schist and granite near $x = 3.5\,\mathrm{km}$, sandstone, limestone and shale again, then chalk; alluvium along the whole floor.

**3.** The sandstone, with the schist and the granite beneath it along the river: the oldest units in the core, an [anticline](#def-b1-geological-maps-fold).

**4.** The chalk, the youngest unit: a [syncline](#def-b1-geological-maps-fold).

**5.** The base of the sandstone is highest on the [anticline](#def-b1-geological-maps-fold)’s axis, and only along the river has erosion cut down below it.

**6.** An unconformity: the sandstone rests on two different older units, schist and granite, and cuts off the outline of the granite.

**7.** $435 - 261 = 174$, $432 - 428 = 4$, $512 - 586 = -74\,\mathrm{m}$.

**8.** $\overrightarrow{E_1E_2} = (600, 1200, -170)$, $\overrightarrow{E_1E_3} = (1100, 400, -248)$.

**9.** The cross product is

$$
(-229\,600,\ -38\,200,\ -1\,080\,000);
$$

oriented upward, $\vec n_E = (229\,600,\ 38\,200,\ 1\,080\,000)$.

**10.** $\|\vec n_E\| \approx 1\,104\,800$; $\cos\delta \approx 0.9776$, $\delta \approx 12.2{}^{\circ}$.

**11.** [Azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) of $(229\,600, 38\,200)$: $\arctan(229\,600/38\,200)
\approx 80.6{}^{\circ}$; [attitude](#def-b1-geological-maps-attitude) $081/12$, [strike](#def-b1-geological-maps-attitude) $\approx 351{}^{\circ}$.

**12.** Yes: the symbol on the east limb has a [strike](#def-b1-geological-maps-attitude) line near north–south and a [dip](#def-b1-geological-maps-attitude) tick to the east, labelled 12.

**13.** $524 - 669 = -145$, $421 - 459 = -38$, $319 - 209 =
110\,\mathrm{m}$.

**14.** $\overrightarrow{W_1W_2} = (700, 1200, 107)$ and $\overrightarrow{W_1W_3} = (1100, -600, 255)$. Their cross product, oriented upward, is

$$
(370\,200,\ -60\,800,\ -1\,740\,000) \quad\longrightarrow\quad
  \vec n_W = (-370\,200,\ 60\,800,\ 1\,740\,000).
$$

**15.** $\cos\delta = 1\,740\,000/\|\vec n_W\| \approx 0.9776$, $\delta
\approx 12.2{}^{\circ}$; [dip direction](#def-b1-geological-maps-attitude): [azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) of $(-370\,200, 60\,800)$, $\approx 279.3{}^{\circ}$. [Attitude](#def-b1-geological-maps-attitude) $279/12$, [strike](#def-b1-geological-maps-attitude) $\approx 189{}^{\circ}$: the west limb [dips](#def-b1-geological-maps-attitude) west-north-west, the east limb east-north-east.

**16.** $\vec n_E \times \vec n_W = (804 \times 10^6,\ -799\,320 \times
10^6,\ 28\,101 \times 10^6)$ (rounded).

**17.** The downward direction is $(-804, 799\,320, -28\,101) \times
10^6$: trend $\arctan(-804/799\,320) \approx 0{}^{\circ}$, that is north; plunge $\arctan\bigl(28\,101/\sqrt{804^2 + 799\,320^2}\bigr) \approx 2.0{}^{\circ}$. The axis plunges $2{}^{\circ}$ toward the north.

**18.** $8000 \times \tan2.0{}^{\circ} \approx 280\,\mathrm{m}$ lower, near $-50\,\mathrm{m}$. The [anticline](#def-b1-geological-maps-fold)’s core is highest near the river and sinks to the north, so its [outcrop](#def-b1-geological-maps-geological-map) closes to the north, as the map shows.

**19.** $\overrightarrow{F_1F_2} = (1627, 2500, -276)$ and $\overrightarrow{F_1F_3} = (3124, 5500, 77)$ give the upward normal

$$
(1\,710\,500,\ -987\,503,\ 1\,138\,500), \qquad
  \delta = \arccos\frac{1\,138\,500}{2\,279\,700} \approx 60{}^{\circ},
$$

and a [dip direction](#def-b1-geological-maps-attitude) $\approx 120{}^{\circ}$. [Attitude](#def-b1-geological-maps-attitude) $120/60$, [strike](#def-b1-geological-maps-attitude) $030{}^{\circ}$.

**20.** The fault [dips](#def-b1-geological-maps-attitude) toward the east-south-east, so that side is the hanging wall, the block lying above the dipping fault plane; it moved down (younger units at the surface): a normal fault.

**21.** The fault plane [dips](#def-b1-geological-maps-attitude) east-south-east, downstream and far more steeply than the river’s floor descends: lower ground meets the plane farther [down-dip](#def-b1-geological-maps-attitude), so the trace swings east-south-east in the valley — the V-rule applied to the fault plane.

**22.** Intrusion of the granite; erosion before the sandstone; deposition of the chalk; folding; faulting; deposition of the alluvium.

**23.** After 90 million years ago (the chalk is folded) and before 2.58 million years ago (the alluvium is not faulted); the folding came first, the faulting after it.

**24.** The fault descends $1405\,\mathrm{m}$ from F1 ($405\,\mathrm{m}$) to $-1000\,\mathrm{m}$; at a [dip](#def-b1-geological-maps-attitude) of $60{}^{\circ}$ that takes a horizontal distance of $1405/\tan60{}^{\circ} \approx 811\,\mathrm{m}$ [down-dip](#def-b1-geological-maps-attitude). The well must be drilled about **$810\,\mathrm{m}$ east-south-east of F1** ([azimuth](https://one-course.com/books/earth/3/en/chapter/1-spherical-astronomy#def-b1-spherical-astronomy-horizontal) $120{}^{\circ}$).