1. Statistics and the Interpretation of Glacier Variability

Climate is defined as the statistics of weather, averaged over some period of interest. The World Meteorological Organization takes 30 years as the time interval over which those statistics should be determined, although other intervals are equally valid depending on the purpose. The statistics of weather includes the average and the standard deviation, as well as higher-order statistical moments. By definition, a constant climate means constant (or stationary) statistics. Variability, as manifest in the standard deviation, is therefore inherent in a constant climate. What does this mean for how glaciers behave in such a climate? Of particular importance are the year-to-year stochastic fluctuations in accumulation and ablation. Glaciers are dynamical systems with a finite memory, and a fundamental property of such systems is that they will integrate such stochastic fluctuations to produce persistent fluctuations on longer timescales (e.g. Reference HasselmannHasselmann, 1976; Reference RoeRoe, 2009).

Reference OerlemansOerlemans (2000) and Reference Reichert, Bengtsson and OerlemansReichert and others (2002) modelled two well-studied glaciers in Scandinavia and the European Alps and concluded that Little Ice Age-scale fluctuations will occur every so often, even in a constant climate. Reference Roe and O’NealRoe and O’Neal (2009) showed that, for the setting of Mount Baker in the Cascade Range of Washington State, USA, glaciers will undergo kilometre-scale, century-scale fluctuations even in a constant climate. Sorting out real climate change from the variability intrinsic to a constant climate is crucial for correctly interpreting the climatic cause of past glacier variations and for the detection and attribution of modern climate change from the modern glacier record.

Reference Burke and RoeBurke and Roe (2009) and Reference RoeRoe (2009) gave a spectral interpretation of this argument, which we review briefly here. The true, physical measure of climatic persistence is whether climate variables are autocorrelated. In other words, does one year’s climate bear any relationship to that of previous years? Consider a climate that has year-to-year variability (drawn randomly from the probability distribution of that climate), but no memory. The time series of such a climate is characterized by a ‘white noise’ power spectrum; that is to say, it has equal power at all frequencies. By construction, a climate that has no persistence nonetheless has power at all frequencies. The reason is that the phase of individual frequencies in the spectrum is random. On average they destructively interfere, leaving no persistence in the time series constructed from that spectrum.

A glacier can be thought of as acting as a low-pass spectral filter: the glacier’s response to this white-noise climate is characterized by a ‘red noise’ power spectrum. Analogous to red light, higher frequencies are damped compared to lower frequencies. Because of this damping, different frequencies will no longer cancel out and the time series of glacier length variations exhibits long-term fluctuations with a timescale related to the spectral filter of the glacier dynamics.

In the above example, the climate was chosen to be white noise (i.e. with no persistence). Weak interannual persistence in sea-surface temperatures does exist because the ocean mixed layer has some thermal intertia (e.g. Reference Deser, Alexander and TimlinDeser and others, 2003). Such weakly red persistence is captured in the 30 year statistics, and so should properly be included in the definition of the climate statistics. The response time of the mixed layer is ∼ 1 year or less, except near sites of deep ocean convection or in the vicinity of the sea-ice margin (e.g. Reference Stouffer, Hegerl and TettStouffer and others, 2000). In other words, it is much less than typical glacier response times and so the above argument is unaffected: the persistence of glacier fluctuations is due to the memory intrinsic to the glacier and not any persistence intrinsic to the climate. Reference Burke and RoeBurke and Roe (2009) analysed the persistence of relevant climate fields and mass-balance records for Europe and, after linearly detrending to account for anthropogenic trends, found no evidence in the instrumental or glacier mass-balance record for decadal-scale persistence.

Lastly, even where such persistence in climate does exist, it is typically only a small fraction of the overall variance. It is always possible to split the time series of climate forcing into a piece that is due to persistence (i.e. what can be related to previous years) and a piece that is a residual without persistence. Reference Burke and RoeBurke and Roe (2009) calculated the relative importance of these two pieces in driving the variance of glacier fluctuations in Europe, and concluded that the residual piece is of greatest importance.

The fundamental and important point is that glaciers can undergo large and persistent fluctuations in a constant climate that has little or no persistence. This fact is often overlooked in the climatic interpretation of past glacier fluctuations. A central goal of palaeoglaciology (and palaeoclimate in general) is to identify glacier fluctuations that are either unusually large or unusually persistent and which are therefore ‘interesting’ to explain. They are interesting because we can then conclude that some definitive change in the climate dynamics or climate forcing has occurred and try to identify the cause. The importance of framing the problem in this way is that the alternative is unsatisfying and violates basic tenets of statistical analysis: if climate is defined as the statistics of weather averaged over some period of choice, and if it is also established that a particular glacier fluctuation is quite likely to occur given those statistics, then it makes no sense to conclude that the glacier fluctuation reflects a climate change.

The interpretation of the climatic cause of glacier fluctuations can therefore be distilled into a classic statistical exercise of correctly identifying changes due to a signal (i.e. the glacier response to a climate change) versus changes due to noise (i.e. the glacier response to interannual variability). What factors control the relative magnitude of this signal and noise? Under the assumption that a glacier is a simple dynamical system relaxing back to equilibrium with a single dynamical response time (e.g. Reference Jóhannesson, Raymond, Waddington and OerlemansJóhannesson and others, 1989a,b), some useful formulae can be derived (e.g. Reference Huybers and RoeHuybers and Roe, 2009; Reference Roe and O’NealRoe and O’Neal, 2009). This study extends these prior analyses to ask two more questions:

1. 1. What factors govern the likelihood of a glacier excursion of a given magnitude in a given interval of time?

2. 2. How can the statistical significance of a trend in observed or reconstructed glacier length be evaluated?

In both cases, formulae can be derived from the linear equations in which the dependencies on glacier geometry and climate setting can be clearly understood. The value of these formulae is that the parameters involved can be calculated from a glacier’s geometry, and so they give guidance as to which glaciers are likely to be best for detecting past climate change. A second and key part of the present study is to establish whether the formulae successfully predict the behaviour of a flowline glacier model, which obeys a nonlinear rheology.

We find that maximum glacier excursions are governed by high-frequency behaviour of the glacier, and the linear formulae hold provided the short-term lag correlations are used to calculate the response time. The probabilities of a given excursion are, however, very sensitive to the magnitude of the natural variability. We also show that glacier trends can be evaluated using a standard Student’s t test, provided that the correct degrees of freedom are used. These degrees of freedom can be accurately calculated from the linear model equations. Finally, we conclude that the current retreat of glaciers around Mount Baker is consistent with, but not by itself independent proof of, regional warming.

2. Linear and Flowline Models

3. The Response to Climate Variability

In the following we characterize the nature of the glacier response to interannual climate variability in more detail. Equation (2) can be developed further to derive some useful properties of glacier variability, whose dependence on glacier geometry and climate parameters can be clearly understood. As noted in the introduction, a focus of this present study is to evaluate the degree to which these expressions also govern the behaviour of the flowline model.

Reference Roe and O’NealRoe and O’Neal (2009) demonstrated that, after linearly detrending, the observed interannual variability in the Pacific Northwest in both the annual-mean accumulation and the melt-season temperature is consistent with random fluctuations that are Gaussian (i.e. normally distributed) and white (i.e. uncorrelated in time). In other words, after accounting for the trends in climate (presumably anthropogenic), the remaining natural variability has no interannual persistence. The analysis of Reference Roe and O’NealRoe and O’Neal (2009) is sufficient to prove that, even if some interannual persistence in climate does in fact exist for the region, it accounts for a statistically insignificant fraction of the climate variability over the period of the instrumental record.

Two 10 000 year long realizations of white noise were generated to simulate interannual variability in P′ and T′, characteristic of the Pacific Northwest (with standard deviations of 1 m a⁻¹ and 0.8°C, respectively). This is long enough to acquire good statistics on the glacier response. The two climate time series are assumed to be uncorrelated, also consistent with observations for the region (Reference Roe and O’NealRoe and O’Neal, 2009). Both the flowline and the linear models are then integrated forwards in time, using the same realizations of this simulated climate variability.

Figure 4 shows a 500 year segment of the climate and the glacier response. Both the linear and flowline models undergo kilometre-scale, centennial-scale fluctuations in response to a climate that we reiterate has no persistence. The standard deviation of the linear model can be derived from model parameters (Reference Roe and O’NealRoe and O’Neal, 2009) and is 360 m. The standard deviation of the flowline model must be calculated from the numerical integration and is 324 m, a difference of 10%; this is approximately what Reference Roe and O’NealRoe and O’Neal (2009) found for a similar calculation. The smaller standard deviation of the flowline model is evident in Figure 4 as is the fact that the flowline model is noticeably smoother. Finally, it is also clear that the response of the flowline model lags behind the response of the linear model.

Fig. 4. A 500 year segment of a 10 000 year simulation of the glacier response to interannual climate variability. The lower panels are white-noise realizations of interannual fluctuations in melt-season temperature and accumulation, for which a 30 year running mean is also shown. The upper panel shows the response of the two glacier models. Kilometre-scale, century-scale glacier fluctuations occur in this simulated climate that, by construction, has no persistence. Also shown by the thin black curve is a linear fit to the flowline model, using the best-fit τ_bf of 73 years.

For comparison, Reference OerlemansOerlemans (2001) estimates a 1σ of 660 m for typical glacier parameters, and Reference Reichert, Bengtsson and OerlemansReichert and others (2002) model values of 550 and 290 m for Nigardsbreen, Norway, and Rhonegletscher, Switzerland, respectively. The exact values do, and should, vary with glacier geometry and climate setting. For a particular glacier, the importance of this natural variability relative to past and current changes must be judged on a case-by-case basis.

A linear model such as that of Equation (2) must have a normally distributed response to normally distributed forcing. For the flowline model, one test of its linearity is to calculate the probability density function (PDF) from the histogram of its fluctuations. The PDFs for both models are shown in Figure 5. The smaller standard deviation of the flowline model relative to the linear model is clear in its narrower clustering around zero. Visually, it appears there is a hint of skewness to central and negative values, although the skewness is in fact very slightly positive (0.06). The flowline model PDF is not quite normally distributed, however. A standard Kolmogorov–Smirnov test (e.g. Reference Von Storch and Zwiersvon Storch and Zwiers, 1999) rejects the normal distribution at >95% confidence. The probable reason is that kurtosis of the flowline model is 3.2, implying it is slightly more outlier-prone than a normal distribution for which the kurtosis is 3.0.

Fig. 5. The PDFs of the linear and flowline models. The linear model follows a normal distribution; the flowline model has a slightly non-normal distribution.

Despite some small differences, the response of the linear and flowline models to equilibrium climate changes, climate trends and climate variability has differed by only a few percent. This generally solid agreement between the flowline and linear models in these preliminary tests is a firm basis for proceeding to explore the response to climate variability, using the analytical power of the linear model to understand the reasons for the glacier behaviour.

4. The Likelihood of a Glacier Excursion

One way of characterizing the expected natural variability of a glacier in a constant climate is to answer questions such as: What is the expected return time, on average, of a particular glacier advance? How long, on average, does the glacier persist above or below its equilibrium length? How likely is an excursion of a given size in a given period?

For linear models of the form Equation (2), answers can be derived using standard formulae for threshold crossings of stochastic processes (first laid out by Reference RiceRice, 1948). Reference VanmarckeVanmarcke (1983) and Reference Leadbetter, Lindgren and RootzenLeadbetter and others (1983) provide good summaries. In the Appendix it is shown that, as long as τ ≫ Δt, the average interval between up-crossings of a particular threshold L ₀ is given by

R(L ₀) is also the average return time of a glacier advance of size L ₀. σ_L is the standard deviation of natural fluctuations. Reference Roe and O’NealRoe and O’Neal (2009) show that Equation (2) yields

provided that P′ and T′ are neither autocorrelated nor correlated with each other (consistent with climate in the Pacific Northwest). A general expression for σ_L without these restrictions is also possible (Reference Huybers and RoeHuybers and Roe, 2009).

From Equation (6) the exponent in Equation (5) contains τ; the return time of a given advance is therefore a very sensitive function of the response time. A larger value of τ means that the glacier is slower to return to equilibrium and has a weaker restoring tendency. All else being equal, the shorter the value of τ, the longer the return time will be for a given excursion magnitude (as seen in Fig. 7). Secondly, the in the exponent in Equation (5) means the average return time lengthens extremely rapidly as the size of the advance increases. For τ = 12 years, for example, an advance of 1 km will happen on average every 250 years. For an advance of 1.5 km, the average return time balloons to 7500 years (Fig. 7).

Fig. 7. The average return time of a glacier advance (i.e. the interval between up-crossings of glacier length beyond a given threshold) calculated from Equation (5). The three curves are for the range of parameters appropriate for a typical glacier on Mount Baker. Note the logarithmic scale on the y-axis and the acute sensitivity of the average return time to changes in glacier properties.

5. Trend Detection for Glaciers

How big a trend in glacier length is statistically significant? When does the trend exceed that expected from natural variability? We show here that two factors are of primary importance: (1) the magnitude of the trend relative to the magnitude of the natural variability, and (2) the amount of independent information in the observations. The latter factor depends on the degrees of freedom, which in turn depends on the length of observations and the glacier memory.

Let ρ be the correlation of the observations of glacier length and time at lag Δt, and let ν be the degrees of freedom in the dependent time series (i.e. glacier length in this case). A statistic (distinct from time t) can be calculated from the following combination of ρ and ν:

Basic textbooks on statistics (e.g. Reference Von Storch and Zwiersvon Storch and Zwiers, 1999) show that, in the absence of a real trend, the probability of finding a given value of will follow a Student’s distribution. Standard tables can be used to calculate how often would occur simply by chance. In general, the larger the absolute value of , the greater the confidence that the observed trend is significant.

Equation (11) can also be written as:

where b is the regression coefficient between time and glacier length and σ_t and σ _res are the standard deviations of time and of the residuals of glacier length after the time-correlated trend has been subtracted, respectively.

σ_t is the standard deviation of the independent variables (in this case, time) over a given interval of time (t _f − t _i), and is given by

Equation (12) therefore becomes

ΔL is equivalent to b(t _f − t _i) and is the change in glacier length that is attributable to the linear trend.

Equation (14) shows some basic and readily understood dependencies. The first factor on the right-hand side can be regarded as the signal-to-noise ratio; the greater the trend relative to the natural variability, the more significant the trend will be. Glaciers that exist in maritime climates are subject to a high degree of precipitation variability (e.g. Reference Huybers and RoeHuybers and Roe, 2009) and have a muted sensitivity to temperature. As such, a warming trend in melt-season temperature may be obscured by the natural variability. A continental glacier with less precipitation variability and a higher sensitivity to temperature may more directly reflect warming trends.

The second factor on the right-hand side of Equation (14) shows that the degrees of freedom (i.e. the number of independent pieces of information in the glacier record) is critical for assigning statistical confidence to an observed trend. If the glacier position was recorded annually, Δt = 1 year and there would be N = (t _f − t _i)/Δt observations.

A glacier has a dynamical response time, however, so it has memory of its previous positions. It is therefore autocorrelated and there are fewer than N degrees of freedom. Standard theory (e.g. Reference Von Storch and Zwiersvon Storch and Zwiers, 1999) yields the appropriate formula for the effective degree of freedom as:

where, as noted above, γ is the autocorrelation coefficient at a lag time Δt. Using the approximation that Δt ≪ τ and considering only first-order terms, Equation (3) yields:

For a 100 year glacier record and for τ _lin = 7 years, there are about seven effective degrees of freedom.

An important measure of whether the test can be used for trend detection is to determine whether random realizations of a dynamic glacier that is forced by a climate without a trend does, in fact, follow a Student’s distribution. In other words, is the glacier variability consistent with the assumptions of the test? To answer this, the correct number of degrees of freedom needs to be established.

Figure 11 shows the probability distribution of the statistic (Equation (11)) using 1000 randomly selected 100 year intervals from the flowline model output, assuming that ν = 100/(2τ _lin). Also shown are the theoretical Student’s distributions calculating ν using both τ = τ _lin = 7 years and τ = τ _bf = 73 years. The close agreement between the distribution derived from the model output assuming τ = 7 years demonstrates the model output has about 7 degrees of freedom per 100 years. We can also conclude that the significance of a trend can be evaluated using a standard test, using τ _lin to calculate the degrees of freedom.

Fig. 11. The distribution of the statistic calculated from 1000 randomly selected 100 year long periods from the flowline model output, using τ _lin to calculate the degrees of freedom (dof). It agrees very well with the theoretical distributions, meaning the model output has about 7 dof (100 a)⁻¹. For comparison, the theoretical distribution assuming τ _bf is also shown. The statistic from the model output for τ _bf is imaginary (Equation (14)). In general, though, smaller dofs from assuming larger τs would produce a narrower distribution of the statistic from the model output (Equation (14)), making it inconsistent with the theoretical distribution.

The statistic was also calculated for 50, 200 and 1000 year trends, and again using τ _lin for the degrees of freedom. For 50 year trends (just 3.2 effective degrees of freedom) the agreement with the theoretical distribution was slightly worse than that shown in Figure 11, but at no point does the error exceed 5%. For 200 year and 1000 year trends, the statistic approximated the theoretical distribution even more closely than that in Figure 11. We also note that annual observations of glacier length are not required to estimate trends; providing observations are frequent enough to sample the effective degrees of freedom, there can still be a correct assessment of significance.

The analysis highlights the importance of knowing the correct effective degrees of freedom for evaluating glacier trends in practice. If degrees of freedom were calculated using τ _bf = 73 years, it would formally mean less than one degree of freedom in a 100 year record; Figure 11 shows that significant trends with high statistics would go unrecognized. One conclusion is that the importance of having degrees of freedom in the observations means that the statistical significance of glacier trends can only be established on multi-decadal or longer timescales.

It is interesting that τ _lin is the correct timescale to use for evaluating trends; for the likelihood of large excursions, τ _bf works well. The reason seems to be that a trend, like an equilibrium step change, is a low-frequency behaviour of the glacier. It is therefore well described by the linear model, as seen in section 2.3. In contrast, maximum excursions depend on relatively abrupt changes that cause a threshold to be crossed and, for that reason, depend on the high-frequency behaviour of the glacier which is best characterized by τ _bf.

6. Summary and Discussion

Stochastic fluctuations are inherent in a constant climate. Distinguishing between climate records that simply reflect these fluctuations and those that reflect a true climate change is a central challenge in climate science. For the case of global mean temperature, for example, there is widespread agreement that the instrumental record shows a significant warming trend that exceeds the natural variability of the last 30 years. An identical issue arises in interpreting the climatic cause of past glacier fluctuations, which are almost always attributed to climate change. This study expands on earlier work and confirms that interannual variability alone can cause century-scale, kilometre-scale fluctuations in glacier length.

We have presented results for the geometry and setting of the glaciers on Mount Baker in the Pacific Northwest. These are relatively small and steep; to evaluate the impact of glacier geometry we repeated the analyses for a glacier with double the total area and half the basal slope and also for a glacier with quadruple the total area and one-quarter the basal slope (and double the width). For these cases, we obtained comparable agreement between the linear and flowline models to that for the Mount Baker glaciers.

We focused on glacier length because that is typically the clearest signature of past fluctuations. All of the metrics and formulae discussed in this study could instead be applied to glacier volume. The linear model under-predicts the response of glacier volume to climate variability because it does not account for thickness changes. For the spectrum of ice volume fluctuations (i.e. the equivalent of Fig. 6a), the linear and flowline models show very good agreement at high frequencies; however, the linear model under-predicts at low frequencies (by ∼30% for the parameters presented here). At high frequencies, mass-balance fluctuations are simply added to the existing volume and the dynamics have no time to respond. At low frequencies, the linear model does not allow for the thickness changes that amplify the volume fluctuations in the flowline model.

The autocorrelation timescale for volume fluctuations in the flowline model is much shorter than that for length fluctuations (15 years vs 73 years) and so is closer to the predictions of the linear model. Understanding volume fluctuations might find relevance in settings where glacier history is recorded in sediments of proglacial lakes or in trimlines on valley side-walls, or when the impact on sea level is of interest (e.g. Reference Raper and BraithwaiteRaper and Braithwaite, 2009). Another natural metric of glacier response to consider might be the mean thickness of the ice.

The important principle in this study is that stochastic interannual climate variability can cause large and persistent glacier fluctuations that should not be misinterpreted as being driven by a climate change. This principle is fundamental and does not depend in any way upon the details of the models used. These models are sufficient to gauge the magnitude of the effects, and sensitivity to different conditions has been reported here. Glaciers are complicated, and no model can capture all their facets. We note that our linear model does not incorporate the mass-balance (e.g. Reference Harrison, Elsberg, Echelmeyer and KrimmelHarrison and others, 2001) or thickness feedbacks (e.g. Reference Bahr, Meier and PeckhamBahr and others, 1997) that are sometimes included in linear glacier models, although such a modification is easily possible.

Other approaches to estimating the glacier response time have used volume–area scaling ratios that represent some nonlinearities and asymmetries (e.g. Reference Bahr, Pfeffer, Sassolas and MeierBahr and others, 1998; Reference Raper and BraithwaiteRaper and Braithwaite, 2009). For our geometry and setting, and also for the sensitivity experiments, we did not find it necessary to introduce these additional factors in order to successfully emulate a flowline glacier model for our purposes. Further work to establish when such factors or others, such as more complicated geometric settings, cause a breakdown of the relationships derived here will be useful.

One more assumption is that all of the glacier variability is driven by climate variability; in other words, that the glacier physics is fixed. This does not apply to surging or tidewater glaciers, where the physics of glacier flow is itself a function of the glacier state. Such factors will always be difficult to model, however, and it may be best to identify settings where those complications are minimized. Finally, it has also been assumed that climate variability itself does not change during the interval considered. On centennial timescales and longer, we expect greater variability than on interannual timescales (e.g. Reference Huybers and CurryHuybers and Curry, 2006). In general, this will tend to increase in the magnitude of longer-term glacier fluctuations.

The success of the linear model at emulating the flowline model at low frequencies means that, when climate change occurs on timescales longer than a couple of glacier response periods (meaning longer than about 15 years for Mount Baker), the glacier’s response to climate variability (such as the excursion probabilities) can be combined linearly with, and superimposed directly on, the glacier’s response to the climate change.

Extreme events and zero-crossings depend on short rapid advances and retreats, and so are governed by the high-frequency characteristics of the glacier. The flowline model is highly autocorrelated on short timescales (Fig. 6b), and hence the longer decorrelation timescale must be used in the formulae. We also note the extreme sensitivity of threshold-crossing statistics to σ_L . It may be very hard to determine the value of σ_L for a real glacier with the accuracy needed to formally establish whether a given glacier advance exceeds that expected from natural variability (cf. Reference OerlemansOerlemans, 2000; Reference Reichert, Bengtsson and OerlemansReichert and others, 2002).

On the other hand, trends, tests and equilibrium changes depend on the low-frequency characteristics of the glacier, for which it is acting as an essentially passive reservoir of ice. The shorter timescale of the linear model therefore provides excellent agreement; moreover, it can be efficiently estimated from the glacier geometry. Furthermore, the analyses are far less sensitive to parameter uncertainties. Lastly, although not formally as rigorous, by solving for the σ_L required for the observed trend to have a nominal confidence level of 95%, the expression for a statistic can be used to roughly gauge the significance of a trend and so circumvent the need for a comprehensive simulation of the natural variability. Nonparameteric tests that do not rely on the glacier adhering to a particular theoretical PDF might be applied for trend detection (Reference Morell, Fried, Schipp and KrämerMorell and Fried, 2008; Reference CotterCotter, 2009). Such tests are more flexible (although typically less powerful) than parametric tests, and care is needed to account for serial correlations.

Various methods have been used to estimate the glacier response time from observations (e.g. Reference HarperHarper, 1992; Reference OerlemansOerlemans, 2001, Reference Oerlemans2007; Reference Pelto and HedlundPelto and Hedlund, 2001; Reference Harrison, Raymond, Echelmeyer and KrimmelHarrison and others, 2003; Reference Klok and OerlemansKlok and Oerlemans, 2004). A useful exercise would be to repeat those various methods on the output from the flowline model, and see which best captures the correct effective degrees of freedom. A concern is that the results here suggest that short-term lag correlations, which are the most easily estimated from observations, may underestimate the actual degrees of freedom.

Finally, this study evaluated the observed retreat of glaciers around Mount Baker. We conclude there are about seven effective degrees of freedom in a 100 year long record, and that the retreat will have to be much larger than is observed to be considered independent evidence of regional warming. It can certainly be said that the retreat is consistent with the observed warming that is already established to be significant from the instrumental record. It should be made clear that the detection of a trend in glacier length is a different exercise from the detection of a trend in glacier mass balance, which is in many ways more closely related to the instrumental record of climate. Where available, local instrumental and local mass-balance records have more statistical power to resolve climate change than glacier length records. In the case of mass-balance records, changes in glacier area must be factored in (e.g. Reference OerlemansOerlemans, 2001).

Glaciers are consequential and captivating elements of the Earth system. Correctly understanding their dynamics and interpreting their history is a worthwhile challenge. Providing that care is taken to identify the correct timescale, the linear model and the formulae derived from its equations do an excellent job of characterizing some important properties of glacier behaviour. Such formulae can provide guidance as to which glaciers and settings are the most sensitive indicators of warming trends or precipitation trends, and which palaeoreconstructions are likely to be most indicative of past climate changes. Identifying such conditions is an important prerequisite for realizing the fullest potential of glacier records.