This article presents ways of reasoning and communicating about reducing GHG emissions and energy consumption, which at the time of writing are not in common use. Starting from the observation that many common models for GHG emissions are essentially multiplicative, we describe how to use logarithmic representations for approaching various emissions reduction problems in a straightforward way.

To make things concrete, we propose to use centibels (a unit of measure for base-10 logarithms, similar in nature to the decibels used in acoustics and eletrical engineering) as a measure of variation in GHG emissions or related quantities, as an alternative to physical units or percentages.

Despite their abstract definition, we suggest that centibels might in fact improve the accessibility of environmental transition problems to a broad audience, being a conceptually tangible unit of measure that delivers the practical benefits of logarithms while dispensing with the need to actually teach logarithms. With this in mind, the companion broad-audience article endeavours to provide an intuitive introduction to applying centibels with no more prerequisites than elementary math.

## Introduction: quantifying reductions in centibels

When modeling the GHG emissions or energy consumption of some activity, we are usually trying to assess how they vary under various (reduction) actions. When a positive quantity $q$ varies from an reference level $q_0$ to a new level $q_1$, we might commonly represent that variation by various numbers:

• a scaling factor: $s_\times := \frac{q_1}{q_0}$

• Examples: ×1.25, ×0.25, ×0.5, ×1, ×0.1

• a relative percentage: $p_\% := 100 \cdot (\frac{q_1}{q_0} - 1)$

• Examples: +25%, -75%, -50%, +0%, -90%

In this article, we introduce an alternative way of representing such a variation, lesser-known but arguably useful:

Definition

The centibel variation of a positive quantity varying from $q_0$ to $q_1$ is defined as:

• centibel variation: $c_b := 100 \cdot \log_{10} \frac{q_1}{q_0}$

• Examples: +10 cB, -60 cB, -30 cB, +0 cB, -100 cB

As hinted by the name, such a quantification is measured in centibels (a centibel is one 100th of a bel; bels measure base-10 logarithms. In formulas, centibels go with the constant $\frac{100}{\ln 10}$. There is nothing novel about this.) Figure 1. A range of reductions quantified both in centibels and relative percentages. 

In the previous formula, the $\log_{10}$ function is the base-10 logarithm, which we’ll simply denote $\log$ in the rest of the article. The involvement of this function is why we call the centibel variation a 'logarithmic representation'. This function can be defined as:

$\forall r \gt 0, 10^{\log r} = r$
 We recall here some relevant properties of the base-10 logarithm: for any positive numbers $x, y$ and real number $a$, $\log xy = \log x + \log y$ $\log \frac{1}{x} = -\log x$ $\log x^a = a \cdot \log x$ $\log 10 = 1$ $\log x \lt \log y \iff x < y$ $\log x = \frac{\ln x}{\ln 10}$ $\frac{d \log x}{d x} = \frac{1}{(\ln 10) x}$

Because of these properties, logarithmic representations become interesting when the quantity of interest can be decomposed multiplicatively into factors $q^{(1)} , q^{(2)} , \dots q^{(L)}$:

$q = q^{(1)} \times q^{(2)} \times \cdots \times q^{(L)}$
 For example, such multiplicative decompositions are very common in "emissions factors" databases, in which emissions models are generally of the form: $\text{GHG emissions} = \text{GHG intensity} \times \text{consumed quantity}$ The first factor is often called the emission factor.

When $q$ varies from $q_0$ to $q_1$, it follows from such a decomposition that:

$100 \cdot \log \frac{q_1}{q_0} = 100 \cdot \log \frac{q_1^{(1)}}{q_0^{(1)}} + 100 \cdot \log \frac{q_1^{(2)}}{q_0^{(2)}} + \cdots + 100 \cdot \log \frac{q_1^{(L)}}{q_0^{(L)}}$

This is interesting, because it means that expressed in centibels, the variations of individual factors add up to the total variation. (Such is not the case with relative percentages, which is a common cause of error when applying percentages to multiplicative models.)

This additive decomposition of compounded variations is valuable, because humans have a good intuition for quantities that add, but a poor intuition for quantities that multiply: additive quantities can be easily visualized by depicting them as lengths or displacements (this is the principle underlying most data visualization methods), and can be intuitively manipulated like ordinary extensive quantities (such as dollars, gallons, kilograms, megabytes, etc.)

As an example, consider the following emissions model for cement production:

Example: reducing CO₂ emissions from cement

Assume that we want to reduce CO₂ emissions from cement in a given sector of civil engineering, planning to divide them by 10 (-100 cB).

We decompose these emissions into the following factors:

• CO₂ intensity (tonCO₂e/ton): how much CO₂ is emitted per unit mass of cement.

• Construction density (ton/m²): how much cement is used per unit constructed area.

• Usage (m²): how much area is constructed.

This decomposition corresponds to the following formula

$\text{CO₂ emissions} = \text{CO₂ intensity} \times \text{Construction density} \times \text{Usage}$

Given our reduction objective of -100 cB, we can then allocate reductions on each factor, as illustrated by the following chart: Figure 2. How various reduction actions might be combined to lower CO₂ emissions from cement (numbers chosen arbitrarily).

Notice how this problem, when expressed in centibels, turns into a "budget problem": each factor must contribute an "income" of reduction in centibels, so as to achieve the reduction objective.

We see other potential benefits to centibels, detailed in the next sections:

## Power laws

Power laws generalize over multiplicative models via decompositions of the form:

$q = q_1^{e_1} \times q_2^{e_2} \times \dots \times q_L^{e_L}$

in which each exponent $e_i$ is a real constant, called the elasticity of $q$ in factor $q_i$.

Observe that taking the logarithm turns such a decomposition into a linear combination:

$\log q = e_1 \log q_1 + e_2 \log q_2 + \dots + e_L \log q_L$

As a consequence, when considering levers that act on the factors $q_i$, quantifying these actions in centibels can make it very straightforward to work out their impact, as illustrated by the following example on cargo ship emissions:

Example: reducing emissions of cargo ships by slow-steaming

Assume that we are operating a fleet of cargo ships; by adjusting the number and speed of cargo ships, we want to minimize GHG emissions while achieving a certain transportation throughput.

GHG emissions and transportation throughput are modeled by power laws:

$\text{transportation throughput} = A \times \text{fleet size} \times \text{ship speed}$
$\text{GHG emissions} = B \times \text{fleet size} \times (\text{ship speed})^3$

We wonder if, by reducing speed while increasing fleet size, we can reduce GHG emissions, while preserving transportation throughput.

Because we are dealing with power laws, framing the problem in centibels makes it elementary to work out from the elasticities:

Action Impact on throughput Impact on GHG emissions

-1 cB speed

-1 cB

-3 cB

+1 cB fleet size

+1 cB

+1 cB

Both actions

+0 cB

-2 cB

Therefore, the answer is yes, a reduction in ship speed compensated by an increase in fleet size can preserve transportation throughput, while reducing GHG emissions. Also observe that:

1. The answer is quantitative, not just qualitative.

2. There is no sign of logarithms or exponentiation in the above table; it is potentially very accessible to a decision maker with little scientific background.

## Carbon budgets and emission pathways

It has been estimated by the IPCC that remaining below +2°C of global warming corresponds to the constraint that future CO₂ emissions should not exceed a certain "carbon budget" (estimated to around 650 GtonCO₂ at the time of writing, but the exact number is irrelevant to our analysis). In this light, it makes sense for various actors to plan their future emissions such that they don’t accumulate beyond a certain threshold: we call this the Carbon Budget Problem.

The Carbon Budget Problem: planning future emissions such that they don’t exceed a certain threshold.

Note that we are not talking about annual emissions here: the limit is measured in 650 GtonCO₂, not in 650 GtonCO₂/year. In particular, this implies that yearly emissions must asymptotically near zero.

### Exponential-decay pathways

Exponential-decay pathways are frequently used to communicate about reducing GHG emissions; they arise from admonitions such as: "to avoid depleting our carbon budget, we should reduce emissions by 5% each year compared to the previous year." In other words, exponential-decay pathways reduce emissions by a constant CAGR .

Formally, an exponential-decay pathway starting at time $t_0$ plans a reduction of the annual emissions $E(t)$ given by the formula:

$E(t) = E(t_0) e^{- \frac{t - t_0}{T}} \quad \text{for } t \geq t_0$

in which $T$ is a duration constant (in years) that determines the "pace" of reduction (smaller is faster).

The cumulated emissions after $t_0$ are given by:

$\int_{t_0}^{+\infty} E(t) \,dt = T \cdot E(t_0)$

Therefore, if $B_0$ is the remaining carbon budget at $t_0$, this yields the constraint on $T$

$T \leq T_{\text{max}} := \frac{B_0}{E(t_0)}$

In other words, $T$ must be no more than the time in which the Carbon Budget would be depleted if emissions levels were kept constant over time rather than reduced.

### Centibels-based characterization: constant centibel-speed

In centibels, exponential-decay pathways take a very simple form: measured in centibels, the reduction in annual emissions levels is proportional to elapsed time, i.e emissions are reduced at constant centibel-speed (in cB/year).

This is illustrated by the following chart: Indeed, denoting $c_b(t)$ the centibels-variation of emissions levels since $t_0$, we have:

$c_b(t) = 100 \cdot \log \frac {E(t)}{E(t_0)} = 100 \cdot \log e^{- \frac {t}{T}} = \frac{100}{\ln 10} \cdot \frac{-t}{T}$

The "centibel-speed" of reduction $\dot{c}_b$ is therefore given by:

$\dot{c}_b = \frac{-100}{T \ln 10}$

### Delaying reduction, and the "rendez-vous point rule"

It is frequently stressed that the more we delay in starting to reduce GHG emissions, the faster we will need to reduce once we’ve started. Centibels allow for a simple mental model to turn this intuition into a quantitative guideline.

Let us call Pivot Year the time $t_P$ at which our Carbon Budget would be exhausted if emissions levels remained constant. It can be proved that, whatever the time $t_0 \leq t_P$ at which we start reducing emissions by an exponential-decay pathway, and assuming emissions levels remain constant before $t_0$, we have

$\frac{E(t_P)}{E(t_0)} = \frac{1}{e}$

when the pace of reduction is chosen to be the minimum required to avoid overshooting the Carbon Budget.

In centibels, this constraint becomes:

$c_b(t_P) = 100 \cdot \log \frac{1}{e} = \frac{-100}{\ln 10} \approx -43.4 \text{ cB}$

This invariant provides a very simply guideline for adjusting the "speed" of emissions reductions: "no matter when we start reducing, by the Pivot Year, we must have achieved a -43.4 cB reduction." In particular:

• If we started reducing 10 years before the Pivot Year, we would need to reduce at a pace of -4.34 cB/year

• If we started reducing 15 years before the Pivot Year, we would need to reduce at a pace of -2.89 cB/year

• If we started reducing 5 years before the Pivot Year, we would need to reduce at a pace of -8.68 cB/year

Intuitive interpretation: there is a "reduction mileage" of -43.4 cB to be walked before the Pivot Year, from which the required speed of reduction can be inferred.

This is illustrated in the following chart: Caution: we emphasize that this guideline only works if the reduction pathway is indeed exponential, which implies in particular a sharp decline in early years. The "-43.4 cB at Pivot Year" target is not in general sufficient to solve the Carbon Budget Problem. This guideline should be considered a mnemonic, not an objective.

### Generalization to other reduction pathways

"Rendez-vous point" rules as described in the previous section are not unique to exponential-decay pathway: in fact, every pathway for which delay gets compensated by a uniform increase in "playback speed" will have an invariant of the form

$\forall t_0 \lt t_p, \frac{E(t_P)}{E(t_0)} = C$

in which $C$ is a constant determined by the shape of the pathway.

For example, rather than constant centibel-speed pathways (for which $C = 1/e$), we could imagine constant centibel-acceleration pathways (which are shaped as the decreasing half of a bell curve), for which $C = e^{-\frac{\pi}{4}}$.

## Centibels as a proxy for reduction effort

We now turn our attention to the problem of estimating the effort or cost of reducing GHG emissions. This is important in particular for planning emissions targets and the pathways to achieve them.

First, let us note that the amount of avoided emissions is not proportional to the cost of avoiding them. For instance, when reducing a person’s carbon footprint from 8 tonCO₂e/year to 2 tonCO₂e/year, transitioning from 8 to 6 tonCO₂e/year can be expected to be much easier than transitioning from 4 to 2 tonCO₂e/year, even though both transitions are 2 tonCO₂e/year reductions.

Centibels naturally account for this "law of diminishing returns": in the above example, the first transition is a -12 cB variation, whereas the second is a -30 cB variation.

We’ll now provide theoretical support for how centibels can be a better proxy for reduction cost than avoided emissions. More precisely, we’ll show that under relatively weak assumptions on how GHG emissions decrease with the investment made for reducing them, predicting the required investment is always more accurate when extrapolating from the marginal cost-per-centibel-reduction than from the marginal cost-per-percent-reduction.

We model emissions reductions by a function $x \mapsto G(x)$, in which:

• $G(x)$ is the GHG-intensity of the studied process, in tonCO₂e/FU (Functional Unit)

• $x \geq 0$ is the effort / cost invested for reducing $G$.

Assumptions about $G$: obviously, we expect the function $G$ to be decreasing. What’s more, as stated above, our "law of diminishing returns" is equivalent to $G$ being convex. Finally assuming that the studied process cannot result in negative GHG emissions, we expect $G$ to be positive.

On their own, these assumptions don’t imply that centibels are a better proxy for cost than avoided emissions.

However, it’s typically safe to make a much stronger assumption on $G$:

Assumption A1: denoting $E := \frac{1}{G}$ the GHG-efficiency of the studied process, we assume that gaining $+p\%$ on $E$ costs more and more as $E$ increases, i.e that the marginal efficiency returns are decreasing.

This assumption can be formulated in the following equivalent ways:

1. As we progress in efficiency, gaining $+p\%$ in efficiency costs more and more.

2. As we progress in intensity, reducing intensity by $-p\%$ costs more and more.

3. Sustaining a constant CAGR in efficiency costs more each year.

4. The function $x \mapsto \log \frac{E(x)}{E(0)}$ is concave.

5. The function $x \mapsto \log \frac{G(x)}{G(0)}$ is convex. (This is sometimes phrased as $G$ being log-convex.)

Under this assumption, it can be proved that it is always strictly more accurate to (locally ) estimate the reduction cost by a constant effort-per-centibels ratio $\left(\frac{d x}{d c_b}\right)_{x=0}$ than by a constant effort-per-avoided-emissions ratio $\left(\frac{d x}{d p_{\%}}\right)_{x=0}$. More precisely, the actual cost will be underestimated by both approximations, but less so by the centibel-based approximation. This is illustrated by the following sketch: The above conclusion formalizes the reason why exponential-decay pathways are so popular: when the reduction effort is assumed to be proportional to the centibels variation, exponential-decay pathways are those that spread the effort evenly over the years.

The main objection we see to the realism of assumption A1 lies in threshold effects: in some situations, initial investments will not significantly reduce GHG emissions, until a critical point where they go down a cliff, such that $G(x)$ has a stair-shaped curve. Arguably, this does not recommend for or against centibels as a proxy for cost: it rather means that long-term cost cannot be extrapolated from marginal costs in such situations.

There is cause to believe that hypothesis A1 is largely applicable in industrial settings. As historical evidence, consider the evolution of energy efficiency of computing hardware. For 50 years, the number of computer operations per dissipated energy has doubled every 1.6 years, an exponential trend identified as Koomey’s Law, and understandably heralded as one of the most impressive trajectories in energy efficiency across all technology domains. From this exponential trend over time, it is safe to assume  that this energy efficiency problem follows hypothesis A1, and most other industrial technologies can be expected to have faster-declining ROIs in energy efficiency.

## Applicability and limitations

### Broad-scope GHG-accounting models are usually not multiplicative

In general, we recommend against using centibels for describing "broad-scope" situations encompassing many diverse activities, such as describing the entire GHG footprint of a company or society.

Indeed, we find that for such situations, accurate models are usually additive, not multiplicative (typically, GHG emissions are expressed as a long linear combinations of activity levels weighted by emission factors).

It could be objected that the Kaya equation is a counter-example to the above recommendation, since it provides a multiplicative decomposition of the GHG emissions of an economy. However, it must be noted that the factors in the Kaya equation are merely global statistical aggregates, and do not map directly to physical mechanisms or concrete action levers: for example, although the Kaya Equation has a GHG/energy factor, a large fraction of the world’s GHG emissions arise from non-energetic activities (deforestation, agricultural methane emissions, lime calcination).

Therefore, our general recommendation is to restrict the use of centibels to situations narrow enough that emissions can be faithfully described by a homogeneous model.

How can we reconcile these 2 views, the global use of tonCO₂e versus the local use of centibels? We imagine 2 approaches:

• Top-down: objectives are assigned to each narrow scope, at which point centibels are used to quantify the the implications of those objectives.

• Bottom-up: centibels are used to study the constraints, costs and opportunities of each narrow scopes (in tonCO₂); these possibilities are then aggregated into a global carbon strategy.

Of course, a more realistic method might consist of moving back and forth between both approaches.

### Limitations and corrections of multiplicative models

As we have seen, centibels are suitable for multiplicative emissions models. It may happen, however, that we want to refine a multiplicative model by making an additive correction to it. For example, we might at first model the emissions of car-driving as proportional to driven distance, and then add a term accounting for the manufacturing of the car.

As soon as such a correction is made, the model is no longer multiplicative, and the use of centibels becomes questionable. This raises the question: are centibels too fragile to be relied upon?

Experience shows a very strong appeal for multiplicative models, even when they’re grossly inaccurate. For example, researches will report emissions factors for photovoltaic electricity in gCO₂/kWh, despite the fact that the emissions of a photovoltaic panel are virtually unrelated to how much electrical energy we get from it; adopting such a model artificially adds significant uncertainty to estimating the lifecycle emissions of photovoltaic electricity, yet this disadvantage is considered a reasonable price to pay for the usability of a multiplicative model.

We do not systematically recommend for or against the use of simplistic multiplicative models for GHG emissions: this is a decision that has to be made for each application, as a tradeoff between accuracy and usability. We do recommend, however, that when accuracy is sacrificed the most be made of usability; in the case of multiplicative model, that may involve the use of logarithmic representations.

### Centibels versus logarithmic scales

Most centibels-based data visualizations are graphically equivalent to displaying one of the axes in a logarithmic scale. Indeed, behind both centibels and logarithmic scales, there lies a logarithmic transform.

However, logarithmic scales can easily confuse an audience by the unusual fact that displayed quantities cannot be mapped to lengths on the chart, leaving the presenter with the challenge of explaining and justifying logarithmic scales to the audience.

Centibels are less subject to this "leaky abstraction" problem: once the audience has accepted that centibels quantify change, they can be plotted in a way that is consistent with visual intuition.

### Are centibels accessible to everyone?

A typical objection to logarithmic representations goes as follows: "logarithms are too mathematically advanced to be understood outside of a few technical niches, so it’s useless to communicate using centibels". And indeed, we do not plan on a widespread mastery of logarithms in time to tackle climate change.

However, we argue that using centibels does not require learning the mathematical theory of logarithms, and is in fact much more accessible.

In particular, presenting centibels as a unit of measure, to be manipulated like dollars, miles or kilograms, can provide a significant foothold to intuition. Forget about logarithms: just express the objectives and opportunities in centibels, and leave the rest to intuition.

Admittedly, leaving aside the question of intellectual accessibility, percentages have over centibels the advantage of familiarity. That is true, but this familiarity is double-edged: when applied to multiplicative models, percentages commonly lead to reasoning errors.

In a similar line, consider hindu-arabic numerals (this handy notation for numbers, enabling us to write '3426' rather than 'three thousand two hundred forty-six' or 'MMMCCXLVI'). Almost everyone learns to use hindu-arabic numerals as soon as elementary school , yet almost no one learns the mathematical theory underlying them.

In order to become proficient with centibels, being able to make quick and approximate conversions between centibels and classical representations, without using a calculator, is quite useful. We give a few guidelines to achieve that.

### Guideline 1: good-to-know centibel conversions

The following table gives a few useful conversions to remember:

% × cB

+0 %

× 1

+0 cB

-50 %

× 1/2

-30.1 cB

-66.7 %

× 1/3

-48 cB

-90 %

× 1/10

-100 cB

### Guideline 2: via algebraic rules

Recalling that centibels turn multiplications into additions, other conversions can be readily derived, for example:

• -95 % = × 5/100 = × 1/20 = × 1/2 × 1/10 = -30 cB -100 cB = -130 cB

• × 1/5 = × 2/10 = × 2 × 1/10 = +30 cB -100 cB = -70 cB

• -25 % = × 3/4 = × 3 × 1/2 × 1/2 = +48 cB -30 cB -30 cB = -12 cB

### Guideline 3: small variations

In this section, c_b and p_% respectively denote the centibels variation and the relative percentage for a given variation, as defined in the introduction.

For small variations (e.g between -5% and +5%), c_b can be approximated to being proportional to p_%, the coefficient being $\left(\frac{d p_{\%}}{d c_b}\right)_{c_b = 0} = \ln (10) \approx \frac{7}{3} \approx 2.30 \%.\text{cB}^{-1}$, i.e:

$\text{For small } p_{\%} \text{,} \quad p_{\%} \approx \ln(10) \times c_b \approx 2.30 \times c_b \approx \frac{7}{3} \times c_b$

For example, $-2 \text{ cB} \approx -4.6\%$.

And reciprocally:

$\textrm{For small } c_b \text{,} \quad c_b \approx \frac{1}{\ln(10)} \times p_{\%} \approx 0.43 \times p_{\%} \approx \frac{3}{7} \times p_{\%}$

For example, $-2\% \approx -0.86 \text{ cB}$.

1. The source code and data for all graphics in this article may be found at github.com/vvvvalvalval/ecolog10
2. Compound Annual Growth Rate
3. Why are we restricting our estimations to extrapolations from local behaviour? We assume that future costs are difficult to foresee, such that only local variations of $G(x)$ are known: this is why the approximation ratios are the derivatives at $x=0$ $\left(\frac{d x}{d c_b}\right)_{x=0}$ and $\left(\frac{d x}{d p_{\%}}\right)_{x=0}$
4. Justification: an exponential time trend is is log-affine and thus log-convex as a function of time, and annual R&D investments in computing hardware can be expected to have been increasing over time
5. That is of course where elementary schools exist.