© Ulf Erlingsson, 1994-2006

Swedish version: Hållbarhetsindex

When studying human history in a perspective of millennia rather than centuries, the preconditions change drastically. Suddenly factors and processes appear that were quite irrelevant in the shorter time perspective. They become not just relevant, but decisive for the development. Two factors have to be considered, the one being very slow and gradual changes that causes a society not to be sustainable in the long run, not “in harmony with nature.” The other factor is catastrophes with very low probability, but extremely large magnitude.

To quote myself: “Environmental protection is only the first step on the way to a sustainable society. On the next level environmental considerations must be brought into the planning process. And still further ahead, the degree of sustainability of the society must be given the priority in the evaluation process, instead of economy or welfare—which is self-evident, since economy and welfare are only temporary in a non-sustainable society.”¹ Welfare of course has the literal meaning ‘wellbeing’ in this quote, not the American meaning of a social service.

In other words, the sustainability of the culture must be a fundamental criterion in society planning, if the intention is for the culture to survive for any significant period of time. But in order to focus on the sustainability it has to be quantifiable.

Therefore I constructed two equations, one to calculate the degree of sustainability in the exploitation of natural resources, and another for calculating the sustainability in the release of pollution. They are here presented in a generalized form, as a unified equation.

Sustainability Index

Resource exploitation index

The basic idea is very simple: Calculate the quotient between how much of a resource that is formed every year, and how much is used up every year. Many resources are not renewed continuously, but with long intervals. Example: Glacifluvial gravel is formed during every ice age, and there is roughly one ice age per hundred thousand years. The amount of gravel that exists now should therefore be divided by 100,000 to calculate how much can be exploited per year—although it is actually adequate that the gravel lasts until the next ice age arrives, and it may only be half as long to that time. It’s a small improvement.

If we use the resource 10 times faster than it is renewed the sustainability index is thus 1/10. If we on the other hand use only a tenth of it the quotient becomes 10/1. If you use exactly what you can, the quotient is 1.

However, this quotient is not very practical to work with, since it is not intuitive, and it does not lend itself for easy classification in map and statistical programs. In order to get more user-friendly numbers one should take the logarithm of the quotient. It leads to that zero signifies a total exploitation (the quotient is one), positive numbers are sustainable, and negative numbers are un-sustainable.

The question is just which logarithm to choose. If the base is 10, the index value –1 would mean an overexploitation of 10 times, -2 would mean 100 times, and –3 would mean 1000 times. It gives small numbers, which means that also the decimals become important.

If, on the other hand, the base is 2, the index –1 means that the resource is overexploited 2 times, –2 signifies 4 times, and –3 calculates to 8 times. Every time the index is increased by 1, the sustainability is doubled, and when it is decreased with one, it is halved. The numbers get larger and the decimals can often be left out.

Although, the most important result to observe is and remains if the index is over or under zero, since (1) all positive numbers are sustainable and all negative unsustainable, and (2) the degree of un-sustainability according to this criterion does not say how long time it will take before things get out of hand. It just says how much we have to change society to reach sustainability.

Here is the equation for sustainable use of resources (since Latin and Greek letters are so overexploited as mathematical symbols I use runes):

[1]

where is sustainability index, is the rate of renewing, and is the rate of exploitation, that is the usage per year (the runes are hagalar, faehu, uruz; they are the initial runes of the Swedish names of the terms). can be estimated as follows:

[2]

where is the size of the reserves, and is the cycle time (the runes are raidho and jara; the latter was chosen since it means ’year’). The cycle time is the time it takes for the resource to form. If the rate of generation is known, for instance the rate of growth of peat, that value can of course be taken directly for .

Incidentally, while is called "faehu" it means 'cattle' and it represents wealth. Equation 2 can therefore be thought of as "wealth = resources per year" and compared to the dividend of an asset, where the asset is our planet and solar system.

A short discussion about the application will follow, but first few words about the pollution index.

Pollution index

If we regard cleanness (absence of pollutions) as a resource, we can use Eq. 1 also for pollutants. A separate pollution index is thus superfluous. The renewing rate is in this case the pollution absorption rate (the rate by which the area can absorb, neutralize, and or adjust to a pollutant), and the exploitation rate is the pollution rate in this case. Positive values of are sustainable in the long term, negative ones are unsustainable, just as before.

Usage

The index is defined from the point of view of natural sciences, and in a strict sense it is only useful as such. In the strict sense it shall be used on a specific and well-delimited resource, for instance oil resources on planet Earth. Oil is slowly renewed, and over the long term (tens of millions of years) one can not take out more than what is being formed. We are of course doing that today. Another example is peat; over a century it is a non-renewable resource, but over 10,000 years it is renewable. A third example is wind energy. It is not stored, as opposed to oil and peat, so it cannot be overexploited in that sense. The fact that a near hundred percent usage of the wind energy may lead to unsustainable side effects is another story, but one that also must be considered in estimating .

Another example of how to use the index is to divide up the oil resources per country, and calculate a sustainability index for each country. Since environmental policy is a responsibility of sovereign states, this is rather logical.

When it comes to pollution some exploitation is local (e.g., of the soil), other is regional (e.g., of the Baltic Sea), and still other is global (e.g., of the atmosphere). The index should be used on an area that is delimited by the natural processes, for instance water pollution reaching the Baltic Sea should be compared to what the Baltic Sea can tolerate. This can also be done for each basin of the Baltic Sea (Bothnian Bay, Gulf of Finland, Riga Bay, etc.; see examples).

Since the pollutants are coming from land, the sustainability index can also be mapped to the drainage area. So far things are simple, but now it gets political. The reason is that the index ought to be normalized based on what is going on where in the drainage area. A heavily populated part of the drainage area will of course have a different pollution profile than a large forest area with paper pulp industries. We are not helped in our analysis if we assign the same sustainability index to the whole region, but the question is how to compare it to the geography of the area.

Should the pollution be normalized based on population density, area, or some other factor or combination of factors? Or should one take status quo as the point of origin (which is a political choice with no foundation in science)?

One can make many maps for analyzing the data, and many patterns will emerge that may be interesting to analyze. In the case I have tested, the Baltic Sea after the collapse of the iron curtain, many unexpected effects appeared. It was so completely different economical systems, and the steering effect the economical system had on the activities in the drainage area led to radically different environmental effects.

My personal conclusion was the following: The economical system in the society has a powerful, albeit perhaps unpredictable, influence on the degree of sustainability. To get a new method, such as agriculture in the Stone Age, to find a long-term sustainable form, is something that might take millennia of fine-tuning of society’s rules and methods of governance.

Everything is cyclical

The index is founded on a way of viewing things that postulates that everything is cyclical, that everything is recreated sooner or later. When dealing with things in the universe that are not recreated, e.g. that the sun burns out, one can use the lifetime of the star as the measure. There is always a value that can be used for —and the maximal value is the lifetime of the universe.

Fortunately, many common resources can be assigned a renewing rate of between a few years and the length of a glaciation cycle. If a soil profile has been destroyed due to inappropriate cultivation methods, it may heal by itself within one or a few hundred thousand years. Mountains are eroded, rivers migrate, the climate varies, and all of this together leads to changes that can bring a fresh start.

There are a couple of special cases. When it comes to pollutions, for example, if there is a pollutant that can not be accepted in any amount whatsoever (for example a molecule that would multiply itself and destroy everything else), then the pollution absorption rate must be set to zero (0). The sustainability will become - (negative eternity) if > 0, or 0 if = 0. The other special case is if > 0 and = 0, then = (that is, if nature can tolerate some pollution but there is none, or there is a renewable resource that we are not using at all, then the sustainability is eternally large).

Examples

Three maps with sustainability indexes for nitrogen (N) and phosphorus (P) releases to the Baltic Sea are presented below. They were based on preliminary reserch results when made in 1995—notably, the value of how much the sea can tolerate is a rather wild guess, while the release values are more reliable.

Strict Usage

The release rate per sub-basin of the Baltic Sea is compared to what that sub-basin can tolerate, . Rather unsurprising, the two densily populated gulfs are un-sustainable, while the sparsely populated north is sustainable.

Normalized to Land Area

In this map the release rate per square kilometer sub-drainage area is compared to what the entire sea can tolerate per square kilometer drainage area. Note that all areas draining to, e.g., Baltic Proper, are merged to one zone, both east and west of the sea. In this map the sparsely populated north again stands out as more sustainable.

Normalized to Population Density

In this map the release rate per inhabitant in each sub-drainage area is compared to what the entire sea can tolerate per inhabitant in the entire drainage area. Here the densily populated Poland is seen as polluting disproportionately little, while the sparsely populated forest regions in the north appear to pollute disproportinately much.

Remember now that these are examples, and not to be taken for the truth. What they intend to illustrate is simply that the sustainability index can be used in various ways in geography, and that each application can have something to contribute in terms of understanding patters and—hopefully—finding ways to improve our overall sustainability.

¹ Contribution to "Expert Seminar on the Exchange of Spatial Data for Physical Planning and Natural Resource Management in the Baltic Region". Gävle, 5 - 7 sept., 1994, Lantmäteriverket.