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I am trying to customize an Albers and a Hotine Oblique Mercator (HOM) projection to minimuze distortion in the region I am analyzing. The region extends from about 51 to 62 degrees latitude, covering an area about the size of the Ukraine. The region is oriented NW - SE.

I want to make sure I am using correct methods for determining the two projection parameters: lat/long of the projection center, and centerline azimuth. I am using ArcMap v10. Here's the procedure I've followed so far:

  1. Created a single polygon that defines the analysis region (by, generally, creating a convex hull around the extent of the watersheds covering the region). This polygon is the area I am customizing the projection for.
  2. Projected the polygon to Geographic/NAD 83.
  3. Used Jeff Jenness' Tools for Graphics and Shapes
    (http://www.jennessent.com/arcgis/shapes_graphics.htm) to determine the polygon's center of mass on the GRS80 spheroid. The resulting coordinates are what I used for the "projection center" parameter.
  4. To determine the centerline azimuth, I first projected the polygon to an azimuthal equidistant projection, specifying the projection center at the coordinates determined in Step 3.
  5. Then I drew a polyline (in the azimuthal equidistant projection), snapped to the projection center point, representing the directional trend of the region polygon. To get the azimuth at the projection's center, I used Jeff Jenness' Tools for Graphics and Shapes to determine the beginning azimuth of the geodesic curve at the central point.
  6. For the Albers projection I am using the longitude for the projection center, as determined in step 3. Am also using the awesome spreadsheet created by Bill Huber (http://forums.esri.com/Attachments/34278.xls) to determine where to place the standard parallels to minimize the scale distortion within the polygon region.

If needed to know, I am using ArcMap's version of the HOM that uses a central line defined by one point at the projection's center and its angle of azimuth. ESRI calls this the "Hotine_Oblique_Mercator_Azimuth_Center". In EPSG, I believe this is the Oblique Mercator, Hotine Variant B, EPSG method code 9815.

I am hoping there are some projection experts out there that can tell me if the above procedure, especially Steps 3 and 4, is a correct way to determine the needed projection parameters. Am I on the right track? Is it correct to be determining the center on the spheroid and the angle of the geodesic from the center point (instead of a "2d" geometric center and azimuth)?

I hope the problem description was clear. I'm eagerly looking forward to any answers, tips, discussion, etc.! Thank you.

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Somewhat related, focusing on more strategic issues: gis.stackexchange.com/questions/2769/… –  whuber Aug 24 '12 at 14:54

2 Answers 2

up vote 8 down vote accepted

The approach described in the question exhibits exceptional care in the selection of projections for a given study area. This answer aims only to make a more direct connection between the objective (of minimizing distortion) and the steps that were and can be taken, so that we can be confident such an approach will be successful (both here and in future applications).

Type of distortion

It helps to frame the problem a little more clearly and quantitatively. When we say "distortion," we can be referring to several related but different things:

  • At each point where the projection is smooth (that is, it's not part of a "fold" or join of two different projections and is not on its boundary or a "tear"), there is a scale distortion which generally varies with the bearing away from the point. There will be two opposite directions in which the distortion is greatest. The distortion will be least in the perpendicular directions. These are called the principal directions. We can summarize the scale distortion in terms of the distortions in the principal directions.

  • The distortion in area is the product of the principal scale distortions.

  • Directions and angles can also be distorted. A projection is conformal when any two paths on the earth which meet at an angle are mapped to lines guaranteed to meet at the same angle: conformal projects preserve angles. Otherwise, there will be a distortion of angles. This can be measured.

Although we would like to minimize all of these distortions, in practice this is never possible: all projections are compromises. So one of the first things to do is prioritize: what kind of distortion needs to be controlled?

Measuring overall distortion

These distortions vary from point to point and, at each point, often vary by direction. In some cases we anticipate performing calculations that cover the entire region of interest: for them, a good measure of overall distortion is the value averaged over all points, in all directions. In other cases it is more important to keep the distortions within stated bounds, no matter what. For them, a more appropriate measure of overall distortion is the range of distortions encountered throughout the region, accounting for all possible directions. These two measures can be substantially different, so some thought is needed to decide which is better.

Choosing a projection is an optimization problem

Once we have selected a way to measure distortion and to express its value for the entire region of interest, the problem becomes relatively straightforward: to select a projection among those supported by one's software and to find allowable parameters for that projection (such as its central meridian, scale factor, and so on) which minimize the overall measure of distortion.

In application, this is not easy to carry out, because there are many projections possible, each typically has many parameters that can be set, and if average distortions over the region are to be minimized, we also need to compute those averages (which amounts to performing a two or three dimensional integration each time any projection parameter is varied). In practice, then, people usually employ heuristics to obtain an approximate optimum solution:

  • Identify a class of projections suitable for the task. E.g., if correct evaluation of angles will be important, restrict to conformal projections (like the HOM). When computation of areas or densities is important, restrict to equal-area projections (like the Albers). When it's important to map meridians to parallel up-and-down lines, choose a cylindrical projection. Etc., etc.

  • Within that class, focus on a small number known--through experience--to be appropriate for one's region of interest. This choice is typically made based on what aspect of the projection may be needed (for the HOM, this is an "oblique" or rotated aspect) and the size of the region (world wide, a hemisphere, a continent, or a smaller one). The larger the region, the more distortion you have to put up with. With country-sized or smaller regions, careful selection of a projection becomes less and less important, because the distortions just don't get that great.

  • This brings us to the current question: having selected a few projections, how to choose their parameters? This is where the earlier effort to frame it as an optimization problem comes to the fore. Select the parameters to minimize the chosen overall distortion measure. This is frequently done by trial and error, using intuitively reasonable starting values.

Practical appplication

Let's examine the steps in the question from this perspective.

1) (Definition of the region of interest.) It's a simplification to use the convex hull. There's nothing the matter with that, but why not use exactly the region of interest? The GIS can handle this.

2 & 3) (Finding a projection center.) This is fine way to obtain an initial estimate of the center, but--anticipating subsequent stages where we will vary the projection parameters--there is not need to be fussy about this. Any kind of "eyeballed" center will be fine to start with.

4 & 5) (Choosing the aspect.) For the HOM projection, the issue concerns how to orient it. Recall that the standard Mercator projection, in its equatorial aspect, accurately maps the Equator and its vicinity, but then increases its distortion exponentially with distance away from the Equator. The HOM uses essentially the same projection, but moves the "Equator" over the region of interest and rotates it. The purpose is to place the low-distortion equatorial region over most of the region of interest. Due to the exponential growth in distortion away from the Equator, minimizing overall distortion requires us to pay attention to the parts of our region of interest that lie furthest from the centerline. Thus, the name of this game is to find a line (a spherical geodesic) transecting the region in such a way that either (a) the bulk of the area is as close as possible to that line (this minimizes average distortion) or (b) the parts of the region that are furthest from that line are as close as possible (this minimizes maximum distortion).

A great way to carry this procedure out by trial and error is to guess a solution and then quickly explore it with an interactive Tissot Indicatrix application. (Please refer to this example on our site. For the needed calculations, see http://gis.stackexchange.com/a/5075.) The exploration typically focuses on the points where the projection is going to have the most distortion. The TI will not only measure the various kinds of distortion--scale, area, angle, bearing--but will also graphically depict that distortion. The picture is worth a thousand words (and a half dozen numbers).

6) (Choosing parameters) This step is very well done: the question describes a quantitative way to assess the distortion in the Albers (Conic Equal Area) projection. With the spreadsheet in hand, it is straightforward to adjust the two parallels in such a way that the maximum distortion is minimized. It is a little more difficult to adjust them to minimize average distortion across the region, so this is rarely done.

Summary

By framing the choice of projection as an optimization problem, we establish practical criteria for making that choice wisely and defensibly. The procedure can effectively be carried out by trial and error, implying that special care is not needed for initial selection of parameters: experience and intuition are usually enough to get a good start, and then interactive tools like a Tissot Indicatrix app and associated software to compute distortions can help finish the job.

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Thank you. Have found v. little practical details of how to choose the “projection center” (graphical center? spherical center? does it have to be exactly at the center? how to determine the implications of where it is placed?) and how to choose the centerline azimuth (what would be a workflow in ArcMap to correctly determine the azimuth? how to evaluate the distortion associated with the placement? are there strict rules to follow for these decisions? how do other people do this?). I think you answered my question with “this is frequently done by trial and error ..." –  fbiles Sep 3 '12 at 5:17
1  
Definition of region of interest – That is a good suggestion, using “exactly the region of interest.” It does turn the project area from looking like a coffin to looking like a lobster…which is nice. I used the convex hull because the entire western half of the project area is an archipelago. I wanted to be sure the project boundary captured all the outlying small islands and water area in between. –  fbiles Sep 3 '12 at 5:18
    
Thanks for the explanation of your use of a convex hull. In fact, my experience has been that many spatial analyses eventually extend somewhat beyond the original borders of the study area (if only to help avoid edge effects in statistical analyses), so I find usually it's a good idea to define the region of interest as a buffer around the original study area. –  whuber Sep 3 '12 at 14:37

Sorry, I'm posting this under "Answer". Not sure if it's appropriate (it is too long for comments). Am new to this site ... maybe I should have started a related question about evaluating distortion? But, I was working on an idea as a result of this post last week, for evaluating scale distortion associated with choosing different projection center, azimuth, and scale factor values for the HOM. Decided to post the idea here because 1) maybe it will be a useful tool that can be used to help answer parts of the original question and 2) I was hoping for feedback on whether this sounds like a reasonable approach.

Using the same concept as the spreadsheet whuber created for evaluating Albers scale distortion, create a spreadsheet filled with Snyder’s equations for the HOM (ellipsoid formula, “alternative B”, page 74 of “Map Projections – A Working Manual”). The user enters the chosen ellipsoid parameters (a and e), and the "customized" projection parameters (lat/long of projection center, centerline azimuth, scale factor, and false easting/northing). The rest of the projection constants are then automatically calculated. The spreadsheet also contains cells for each lat/long pair (in half-degree increments, or whatever increments are desired) across the projection area. The scale factor and rectified coordinates at each lat/long pair are automatically calculated when changing any of the projection parameters. Now, the scale factor can be numerically evaluated 1) by computing an overall average and range of scale distortion across the projection region, and 2) the point coordinates and their associated scale factors can easily be imported into ArcMap to create a visual picture of how the scale distortion is distributed. Obviously the results are just a sample and will vary depending on how many lat/long locations are evaluated, but does this sound like a reasonable approach?

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+1 It is indeed a reasonable approach: it automates what one does when creating a set of Tissot indicatrices across a region to evaluate a potential choice of projection. –  whuber Jan 23 at 4:38

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