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About Hillshades and the Swiss Method

Cartography · Hillshade Variations · concept page · by Jeff Jenness

Summary

Hillshades show the shape and texture of the land. I personally love looking at them, and can get almost hypnotized looking at the mountains and canyons that hillshades reveal. Then I often get lost in daydreams of hiking in and exploring those landscapes. I even use them in most of my visual presentations, such as the background images on these tool help pages.

A hillshade is basically a picture of a landscape lit by an imaginary sun. It is usually made for aesthetic purposes rather than for actual analysis, although there are exceptions to that rule. And the ArcGIS Pro “Hillshade” tool makes a perfectly good basic hillshade. The tools offered here add a few aesthetic variations that the ArcGIS tool does not, though. These tools make hillshades that are correct on latitude/longitude DEMs without projecting them first, hillshades lit from several directions at once, hillshades with the low country darkened as if seen through haze, hillshades with cast shadows that keep some detail inside them, and hillshades that have been deliberately softened. The last four are the parts of what cartographers call the Swiss method, the style of relief shading perfected by hand in Switzerland and first automated by Kurt Brassel in 1974. This page explains what a hillshade is, where the Swiss method came from, and which of the five tools does which part of it.

A gray hillshade of the Grand Canyon seen from above: the main gorge winds from the left edge across the lower middle of the image and up the right side, with a dense branching texture of side canyons along both rims, and the low country of the inner canyon darkened toward charcoal while the high plateaus fade to white
A hillshade of the Grand Canyon, with hypsometric shading applied.
Learn more These pages follow the Raster and Surface Analysis module of my GIS course (see the training page), where the Swiss method is discussed in the third lecture (Insolation, Hillshades and Curvature, video) and worked through by hand in ArcGIS Pro in four lab exercises: hypsometric shading, multiple sun positions, the MDOW hillshade and blurring. Each of those exercises builds in ModelBuilder what one of these tools now does in a single step.

What a hillshade is

Put a sun somewhere in the sky, at a compass direction (the azimuth) and a height above the horizon (the altitude or inclination). For every cell of the DEM, work out which way the ground faces and how steeply, and ask how squarely that little patch of ground faces the sun. Ground that faces the sun directly is brightest; ground tilted away from it is darker; ground that faces away from it altogether gets no light at all. That is the whole idea. The standard formula, which is the one Horn (1981) gives and the one Esri's Hillshade tool uses, is

shade= cos⁡Z⋅cos⁡S + sin⁡Z⋅sin⁡S⋅ cos⁡(Asun−Acell)

where Z is the sun's zenith angle (90° minus its altitude), S is the slope of the cell, and the two A's are the azimuth of the sun and the aspect of the cell. The result runs from 1 for ground that faces the sun squarely down through 0 for ground lit exactly edge-on, and it goes negative for ground facing away; the negative values are just set to 0. Esri's tool multiplies that number by 255, to give the familiar hillshade that runs from 0 to 255. The tools here divide the 0-to-1 range into as many equal gray levels as you ask for, numbered from 0, so the default of 256 levels gives the same 0 to 255. Every hillshade tool in this group uses this formula. What they change is where the suns are, how many there are, and what is done to the result afterward.

The slope and aspect of each cell come from the elevations of its eight neighbors, using the weighted-difference equations that Horn (1981) described and that Esri's Slope and Aspect tools also use. The neighbors on the east side, weighted 1-2-1, are compared with the neighbors on the west side to get the east-west gradient, and the north and south rows are compared the same way. A cell on the edge of the raster, or beside a NoData hole, has no complete ring of neighbors and gets NoData in the output.

A block of nine DEM cells drawn as colored columns of different heights, labeled with their elevations: 190, 170 and 180 in the back row; 160, 175 and 168 in the middle row; 170, 150 and 165 in the front row; a dot marks the center of each cell top
A focal cell of elevation 175 and its eight neighbors.
The same nine cells with six broad arrows laid over them: three light arrows run from the back row to the front row, one over each column, and three dark arrows run from the left column to the right column, one over each row; in each set the middle arrow is drawn wider than the two outer ones
How Horn's equations weight them. The light arrows are the three north-to-south differences, one down each column; the dark arrows are the three west-to-east differences, one along each row. In each set the middle difference, through the focal cell, counts twice and the two outer ones count once.

Why the sun is always in the northwest

Every hillshade tool defaults to a sun at 315° (northwest) and 45° above the horizon, and the reason why is interesting because this isn't a place the sun normally sits, and especially not that high in the sky (at least in the Northern hemisphere). In the northern hemisphere the sun mostly shines from the south, with short exceptions in the early morning and late evening. A hillshade lit from the south, though, where the sun actually sits, is a well-known optical illusion: most people (though not everyone) see the relief inverted, with canyons rising as ridges and mountains sinking into pits. Light the same ground from the northwest and the illusion goes away. Therefore we often use a fake sun location, situated in the north, to both make the map look more “realistic” and also to avoid confusing our audience. You are free to put the sun anywhere; the tools here accept any azimuth and any altitude. Just look at the result and make sure it reads the way you expect.

A gray hillshade of a winding canyon system with the sun in the northwest: the canyon reads correctly as a gorge cut below the plateau, with its side canyons branching away from the main channel
The sun in the northwest, at an azimuth of 315° and inclination of 45°, where it rarely appears in the real world and never at this actual location. The Grand Canyon actually looks like a canyon.
The same canyon system with the sun in the southeast: to most eyes the relief now appears inverted, the canyon reading as a raised ridge system standing above the surrounding plateau
The same terrain with the sun in the southeast, at an azimuth of 135°, where it really is. To most people the canyon now looks like an oddly-shaped series of blisters, hills and ridgelines.

There is actually an interesting body of literature discussing this phenomenon, which cartographers call relief inversion or the terrain reversal effect. The basic issue probably has something to do with the fact that humans are just used to having light shine from above. When we're looking at a hillshade, our brains just assume light is coming from the top of the image, which is usually oriented north on a map (Ramachandran 1988; Kleffner and Ramachandran 1992; Liu and Todd 2004 review the long history of the idea). So when we see the landscape illuminated from south (from the bottom of the image), we misinterpret the direction that the shadows and bright spots on the image are facing.

Notice that “above” here means the top of the image as it falls on the eye, not north and not even up. Kleffner and Ramachandran (1992) had people lie on their sides, and the assumed light turned with their heads. That gives a simple cure for an image that is lit from the south: turn it upside down. Ramachandran (1988) makes the point with shaded disks, which reverse when the page is turned over. Bernabé-Poveda and Çöltekin (2015) tested it on satellite images, which in the northern hemisphere are lit from the south: rotating the images 180° raised the share of correct answers from 40% to 72%, and rotating southern-hemisphere images, which read correctly to begin with, dropped it from 76% to 28%. The image below is the south-lit hillshade from above, turned upside down and otherwise untouched.

The south-lit hillshade of the canyon system rotated 180 degrees, so that the light now falls from the upper left of the image: the canyon again reads as a gorge cut below the plateau
The same south-lit hillshade rotated 180°, so that south is at the top and the light comes from the upper left of the image. Nothing else was changed, and the canyon is a canyon again.

Of course, if you're going around turning your maps upside down, make sure you add a North arrow so people know which way is North!

It's not just a matter of light from above vs. light from below, either. Why do we default to northwest (315°)? It turns out that “west” in general is not always the best choice. Smith et al. (2015), in a test with shaded spheres rather than maps, show how the direction you normally read influences how you perceive this shading, and placing the sun in the northwest is actually best for people who read left-to-right. People who read right-to-left, on the other hand, tended to perceive the shape a little quicker if the illumination was put in the northeast (the difference for those readers fell just short of statistical significance). And Biland & Çöltekin (2017) take it farther, arguing that an angle of 337.5° actually reduces misinterpretation better than any other sun angle they measured, including the standard 315°: 96% of their participants' answers were correct at 337.5° and 80% at 315°. They don't say what direction their test subjects read, but the experiment was run at the University of Zurich, Switzerland, with local participants working in German or English, so the odds are they read left-to-right.

Morgenstern et al. (2011) and Liu and Todd (2004) argue that there is a lot more to human perception of depth than just an assumed illumination angle, which is not surprising. Morgenstern et al. found that the light-from-above assumption is a weak one, which gives way as soon as the image holds real evidence of where the light is coming from, such as cast shadows. Liu and Todd found that their observers leaned more strongly toward seeing bumps than dents than they did toward any direction of light. But the fact remains that a large proportion of the population experience this optical phenomenon.

What proportion of the population experiences this optical illusion? For a plain hillshade, the evidence says nearly everyone. Biland and Çöltekin (2017) had 27 people say whether a marked landform was a valley or a ridge on hillshades lit from 16 directions. With the light anywhere from 112.5° to 225°, no more than 10% of the answers were correct, and the participants were as confident of their wrong answers as of their right ones. In a second experiment with 33 hillshades lit from the south-southeast, the mean accuracy was only 2%, and the authors concluded that “it is likely everyone experiences the illusion when photographic cues are not present” (Çöltekin and Biland 2019). Marking the direction of the light on the image did not help. On satellite images people do somewhat better. In an online survey of 535 people, 40% of the answers on south-lit images were correct (Bernabé-Poveda and Çöltekin 2015), and in the laboratory 15% to 18% (Çöltekin and Biland 2019). Those who get the right answer seem to do it by reading the rivers, the snow and the vegetation, and people who work with satellite images often are the ones who benefit most from such clues. A bare hillshade has none of them. None of these studies puts a number on the share of people who are immune to the illusion, if anyone is. Çöltekin and Biland (2019) say more evidence is needed “to establish what portion of the population can indeed bypass the illusion,” and Liu and Todd (2004), who found large differences among their seven observers, warn that it is “dangerous to generalize to the entire population.”

On a personal note, I have shown the hillshade below of Oak Creek Canyon, in northern Arizona, with illumination from the southeast, to my Forestry GIS classes at Northern Arizona University for several years, and consistently roughly 3/4 of my students report seeing this as a ridge instead of a canyon. I include myself in that group, even though I know exactly what this hillshade is showing.

A gray hillshade of Oak Creek Canyon lit from the southeast: to most viewers the canyon and its side canyons appear as bright raised ridges standing above the surrounding plateau
Oak Creek Canyon, in northern Arizona, illuminated from the southeast. Is this a mountain range or a canyon?
The same hillshade of Oak Creek Canyon lit from the northwest: the canyon and its side canyons now read as dark gorges cut down into the plateau
The same landscape illuminated artificially from the northwest.

On a tangential note, people have commented on this optical illusion for centuries. I really love reading the manuscript by D. Rittenhouse from 1786, where he ponders why the pits and cracks on the bricks on his chimney-hearth appear inverted when he looks at them through a pair of lenses that turn the image upside down, and depending on how he directs the light onto them. Somehow manuscripts from that time are just warmer and friendlier than today. For example, “Though I was well satisfied of the truth of this explanation, I resolved nevertheless to bring it to the test of experiment...”. It feels more like reading Charles Dickens or J. R. R. Tolkien than it does a scientific manuscript. His explanation is essentially the modern one: we judge relief by rules “imperceptibly formed in the mind, and confirmed by long experience,” rules we rely on “even without knowing that we do so,” and they mislead us when the light is not where we take it to be.

Rittenhouse also noticed that the illusion can be broken, and that once broken it tends to stay broken. No effort of the mind would do it; but if he touched a finger or a pen to the bricks while looking, “the deception vanishes in a moment,” and “after the mind has been undeceived by these means once or twice, it does not readily admit of the imposition again.” My own experience with hillshades is a little different. I can overcome the illusion while I am looking at an image, but it is not broken forever. It returns on a fresh look. The modern studies touch on this only in passing. Ramachandran (1988) remarks that shaded shapes “can sometimes be perceptually reversed” by the viewer. About 70% of the participants of Çöltekin and Biland (2019) said the terrain flipped between valley and ridge for them at least once, though only about 4% of the time, and the experienced users of satellite images were the ones who most often got past the illusion, which the authors take as a sign that the skill is learned. None of these papers tested whether a person who has once seen a particular image correctly keeps seeing it that way.

A poor-man's insolation

A hillshade can also be a poor-man's map of sun exposure. Since you can set the sun to any position, you can set it where the sun really is at a particular place, date and hour (NOAA's solar calculator, in the references, will tell you), and the hillshade then shows roughly how much direct sun each slope receives at that moment. It is a rough estimate, and working out the sun position for a season rather than a moment is tedious; Pro's solar-radiation tools do that job far better. But for a quick look at which slopes catch the winter afternoon sun, a hillshade with a real sun position is a reasonable start.

Hillshades from latitude/longitude DEMs

Esri's Hillshade tool assumes the cells are square and the same size everywhere, in the same units as the elevations. On a DEM in geographic coordinates, where a cell is a fraction of a degree wide and its ground width shrinks toward the poles, neither is true, and the usual advice is either to project the DEM first or to fudge a Z factor that converts meters of elevation into something like degrees so the result at least looks plausible. The four hillshade-producing tools here take a different route: for a geographic DEM they compute the true ground dimensions of the cells for every row of the raster from the spheroid, so the gradients, and therefore the shading, are correct at every latitude with no projection and no fudge factor. Projected DEMs are handled with the ordinary constant-cell formulas, which are faster. The tools also read the elevation units from the DEM's vertical coordinate system when it has one, assume meters for a geographic DEM that does not say, and ask you to choose between meters and feet only when the DEM gives them nothing to go on. If you would rather project the DEM anyway, the Projecting Rasters page explains why the resampling method matters.

The Swiss method

Various authors have suggested improvements to the basic hillshade. The most influential set of ideas is usually called the Swiss method, after Eduard Imhof and the Swiss school of relief cartographers, whose hand-painted shaded relief set the standard for the twentieth century. Strictly, the Swiss method is a manual method: the cartographer paints the shading by hand, adjusting the light locally to bring out each landform, softening the tones in the lowlands to mimic haze, and reserving the strongest contrasts for the high country. Imhof (2007) devotes a whole book to it, and he was frankly skeptical that a computer could do the job; in his chapter on early computer shading he lists the difficulties, and concludes that the machine's output should be treated as a working draft for a competent cartographer to finish.

Kurt Brassel, working in Zurich with a line printer, was the first to automate the parts that could be automated (Brassel 1974). Three of his suggestions have become the recognized components of a computer Swiss-style hillshade, and each of them corresponds to one or two of the tools in this group.

Light from more than one direction. A single sun shows the shape of the land well only where its light strikes the slopes from the side, and washes out everything else: a ridge running straight toward the sun shows as an even gray on both sides, and a slope facing squarely away from it is a featureless black. Brassel's remedy was to let the light direction vary from cell to cell, swinging it a certain number of degrees to either side of the main direction according to the aspect of the cell, and to let the sun's height vary as well. Imhof (2007, p. 211) suggested a simpler version: make a second shading with the light moved 10° or 15° to one side of the first, and combine the two images photographically. In this toolbox the Multi-Sun Hillshade does the direct thing, averaging hillshades from two to eight suns you choose.

Three hillshades of the same canyon country. Two small ones at the top are labeled Hillshade at 360 degrees and Hillshade at 270 degrees, each with an arrow pointing down to a large hillshade below that is the sum of the two and shows relief on slopes of every orientation
Two ordinary hillshades of the same ground, one lit from the north (360°) and one from the west (270°), and the two simply added together. Each sun shows slopes that the other leaves flat.

The MDOW Hillshade does a cleverer thing, devised by Robert Mark of the USGS (Mark 1992). It computes four hillshades, with the sun fanned around a primary direction, and blends them cell by cell, giving each sun a weight that depends on the aspect of the cell: no weight when the cell faces straight toward that sun or straight away from it, and full weight when the sun is at right angles to the way the cell faces. Every slope is therefore lit mostly from the side, which is the direction that shows its shape best.

Three panels: a raster DEM in a yellow-to-green elevation ramp, with an arrow to a standard gray hillshade of it, and an arrow down to a larger multidirectional oblique-weighted hillshade of the same ground, which is paler overall and shows fine ridges and drainages picked out on slopes of every orientation
A DEM, the standard hillshade made from it, and the multidirectional, oblique-weighted hillshade of the same ground.

Atmospheric perspective. In a real landscape seen from above, the low country is farther away and seen through more air, so its relief looks softer and less contrasty than the peaks. Brassel built that into his model: he strengthened the contrast of the shading on the heights, reduced it in the lowlands, and added a slight general darkening of the low country, which he called “obscuring.” Here the Apply hypsometric shading option in each of the four hillshade tools darkens the low elevations inside the output raster, and the Simple Hypsometric Shading tool gets a similar effect with layers alone, draping a half-transparent black-to-white elevation layer over any hillshade you already have.

Three panels: a raster DEM in a yellow-to-green elevation ramp, with an arrow to a standard gray hillshade of it, and an arrow down to a larger hillshade of the same ground with hypsometric shading, in which the tone grades from darker ground to lighter ground with elevation
A DEM, the standard hillshade made from it, and the same hillshade with hypsometric shading applied.

Softening. Brassel argued that the smallest terrain features carry too little information for their shading to be trusted; their light and shade, he wrote, tend to be “fictive” or “creative,” and so they should be blurred to de-emphasize them. He was also printing on a line printer, and blurring the output photographically made it a good deal easier on the eye. The USGS uses a softened hillshade under its topographic maps for a related reason: a crisp hillshade competes with the contours and the roads, while a soft one stays in the background, and Buckley and Barnes (2004) recommend smoothing on the same grounds. The Blurred Hillshade tool is this component.

A normal gray hillshade of gently rolling country with two small buttes at the bottom edge; the low relief shows as a fine pattern of drainages and scarps
A normal hillshade.
The same area on the USGS topographic map layer: brown contours, blue drainages, black roads, a red section grid and place names, drawn over a very soft, pale shaded relief in which only the broadest forms of the terrain can be made out
The same area in the USGS Topo map layer. The relief beneath the contours, roads and section lines is a softened hillshade, which stays in the background.

Brassel's point about blurring is easy to illustrate. The first image below is a hillshade drawn the way Brassel's modified line printer would have printed it: one dot per square cell, in a 13-step scale from a pinpoint to solid black (remember we're talking about 1974; this was what a university computing center could offer back then). The second is the same image blurred, much as he blurred his own prints photographically. (These two images were made for this page to illustrate the idea. They are not Brassel's own figures.)

A hillshade of a volcano with radiating drainages, printed as a grid of dark-gray dots on white paper, one dot per cell; the dots grow larger in shade and merge into horizontal bars in the darkest areas, with thin white lines between the printed rows
A hillshade printed the way Brassel's line printer would have printed it, one dot per cell.
The same line-printer hillshade after blurring: the dots and bars have merged into smooth gray tones, and the volcano and its drainages read as continuous shaded relief
The same image after blurring, as Brassel blurred his prints. The dots merge into continuous tones.

Choosing among the five tools

ToolWhat it adds to the basic hillshadeReach for it when
Enhanced Hillshade One sun, as in Pro's tool, but correct on geographic DEMs; any number of gray levels; hypsometric darkening; vertical exaggeration; cast shadows that keep detail inside them. You want an ordinary hillshade from a latitude/longitude DEM, or you want shadows that are not solid black.
MDOW Hillshade Four suns fanned around a main direction, blended cell by cell so that each slope is lit by the suns most oblique to it (Mark 1992). A single sun is hiding the structure of ridges and valleys that run toward it; you want the fullest detail in every orientation.
Multi-Sun Hillshade A weighted average of ordinary hillshades from two to eight suns you position yourself, with optional per-sun shadows. You want direct control over where the light comes from, or the classic two-sun look of the lab exercise.
Blurred Hillshade Everything the Enhanced Hillshade does, plus a circular blur of the DEM before shading or of the hillshade after it. The hillshade is competing with the map on top of it, or the fine texture of the DEM is noise you would rather not draw. Or, you just like the look of gentle, soft hillshades.
Simple Hypsometric Shading No new raster at all: a group layer with the DEM, black at the bottom and white at the top, laid half-transparent over an existing hillshade. You already have a hillshade you like and want the atmospheric-perspective effect as an adjustable layer rather than baked into a file.

The tools combine. A common recipe is an MDOW or Multi-Sun hillshade with hypsometric shading turned on, and if the result is still too busy, the same settings in the Blurred Hillshade, which accepts every Enhanced Hillshade option.

The gallery below shows the same stretch of the Grand Canyon through each of the tools. The basic hillshade is at the top. The second row is the Enhanced Hillshade with hypsometric shading, at 256 gray levels and again as a four-tone poster. The third row is the MDOW hillshade without hypsometric shading, and a three-sun Multi-Sun hillshade with it. The fourth row is the Simple Hypsometric Shading group layer with its blend mode set to Multiply, and a Gaussian Blurred Hillshade with hypsometric shading.

Seven labeled hillshade panels of the same stretch of the Grand Canyon. Top: Basic Hillshade. Second row: Enhanced with Hypsometric Shading, with the inner canyon darkened, and Enhanced 4-Color Poster style, drawn in black, two grays and white. Third row: MDOW with no hypsometric shading, pale with detail on slopes of every orientation, and 3 Sun Positions plus Hypsometric Shading. Fourth row: Simple Hypsometric Shading with the Multiply blend mode, the darkest panel, and Gaussian Blur plus Hypsometric Shading, a soft out-of-focus version
One DEM, seven ways: the basic hillshade; the Enhanced Hillshade with hypsometric shading, smooth and posterized; the MDOW and Multi-Sun hillshades; the Simple Hypsometric Shading group layer with the Multiply blend mode; and the Blurred Hillshade.

What the four raster tools share

The Enhanced, MDOW, Multi-Sun and Blurred hillshades are built on one engine, so they behave alike.

Credits and references

By Jeff Jenness, Jenness Enterprises (www.jennessent.com). The Enhanced and MDOW hillshades are ports of the hillshade functions in the author's DEM Surface Tools for ArcGIS (Jenness 2013); the Multi-Sun and Blurred hillshades grew out of the lab exercises in the references. The gradient in all four is Horn's (1981).