The Editor

Journal of Glaciology

Sir,

Reference MacAyeal, Firestone and WaddingtonMacAyeal and others (1991) have introduced a type of inverse method called control methods into glaciology. They suggested that control methods are the best way of deriving information about past surface temperatures from temperature-depth profiles in polar ice sheets. Although we believe that the method has promise, we have serious reservations about the way it is used in their paper. They asserted that (1) the uncertainty of an analysis by this method “can be established quantitatively” and (2) a temperature-depth profile calculated by their method fits the profile measured at Dye 3, Greenland (Reference Gundestrup and HansenGundestrup and Hansen, 1984) more closely than the one calculated by a simpler method by two of us (Dahl-Jcnsen and Johnsen, 1986). We believe that the authors have not demonstrated the truth of the first statement and that the improved fit is illusory. Further-more, their inferred surface-temperature history at Dye 3, which shows oscillations of up to 11 deg, peak-to-peak, during the last 10000 a, is not supported by any climatic data, from Greenland or elsewhere, known to us. Some further discussion of the method seems called for.

MacAyeal and others specified the inversion problem as: to find the surface-temperature history T_s(t) that will minimize the quantity

Here, t is time, t_f represents the present, z is depth, H is ice thickness and θ(Z) is the measured temperature profile. The quantity T(z,t_f) is the solution of the heat-transfer equation with surface-boundary condition T_s(t), and specified basal boundary condition (constant heat flux) and initial condition. The quantity η(t) is a “preconceived” surface-temperature history. In the present case, it is taken as a constant so that the effect of minimizing the second integral is to minimize the amplitude of the surface-temperature oscillations needed to fit the measured profile. The quantity E is a weighting factor; a zero value implies that the preconceived temperature history is ignored, whereas a sufficiently high value makes the computed surface-temperature history identical with η(t).

We have some concern about this formulation of the problem; the solution is forced to oscillate about the chosen η(t) and this may distort or obscure some of the paleoclimatic information in the data.

The inferred surface-temperature history depends sensitively on the value of E, as the authors’ figure 11 shows. As the e is increased, the amplitude of the inferred temperature oscillations diminishes. However, the authors gave no objective method of choosing the value. The difference between the observed and calculated temperature profiles cannot be used as a criterion because, as their figure 12 shows, a good fit can be obtained for a wide range of values of e. The value chosen for the Dye 3 analysis (2.5 x 10⁻⁹ m s⁻¹) appears to be the one that reduces the amplitude of the oscillations to what the authors considered reasonable, but which we would regard as unreasonably high. Moreover, they stated that the first temperature minimum after 10 000 year BP, a value of-24.35 °C at 7900 year BP and the subsequent maximum (−13.5 °C at 4125 BP) are reliable. However, the next minimum (−24.1 °C at 2475 year BP) is only “probably reliable” and the subsequent oscillations are “insignificant” even though they have amplitudes of several degrees. No reason for these assessments was given. The emphasis throughout the paper was on how closely a calculated temperature profile fits an observed one; the inferred temperature history was never compared with other paleoclimatic data and whether the observed history is even plausible was never discussed. The authors stated that the size of the oscillations also depends on the size of the steps in depth and time that are used in the numerical analysis. We do not understand how they can claim that the uncertainty of this analysis can be established quantitatively or even that their analysis yields any useful paleoclimatic information.

The analysis of Dahl-Jansen and Johnsen (1986) reproduced the observed temperature profile at Dye 3 to within 0.03 deg. This is also the precision of the calibration of the thermistors used to make the measurements (Reference Gundestrup and HansenGundestrup and Hansen, 1984). Further reduction of the discrepancy between calculated and observed temperatures, as achieved by Reference MacAyeal, Firestone and WaddingtonMacAyeal and others (1991), seems pointless. Their inversion method is highly unusual in that it takes no account of the uncertainties in the data. Indeed, their inferred surface-temperature history (their fig. 6) looks to us like an example of overfitting, that is, fitting noise as well as the signal. The possibility of small-scale convection in the borehole fluid, as discussed by Reference Gundestrup and HansenGundestrup and Hansen (1984), makes us doubt whether measuring temperatures with a precision of better than 0.01 deg, even if feasible, would reveal further details of the paleoclimate, as MacAyeal and others claim it would.

The large oscillations in the inferred surface temperature may arise partly because the heat-transfer equation is difficult to solve in the space and time domain that the authors used (constant depth intervals of 50 m and time step 25 a). A transformation of the time variable might be an improvement. Reduction of the depth interval in the upper part of the profile might also help, although this change is constrained by the fact that the temperature was measured only every 25 m.

The authors tested their method with synthetic data. They calculated a temperature–depth profile by solving the heat-transfer equation with a surface-boundary condition consisting of two cycles of a sinusoidal temperature oscillation of period 2500 a and amplitude 5 deg. They then saw how well their method recovers this oscillation from the profile. Their figure 2 shows discrepancies of up to 2 deg, which is 40% of the amplitude. The authors never mentioned this large discrepancy, let alone discussed possible reasons for it. Because their analysis of the Dye 3 data covered the past 10000 a, running this test for four cycles, rather than merely two, might have been enlightening because the error increases with time before the present.

There is an extensive literature about deriving paleoclimatic information from temperature-depth profiles in rock. Since the classic paper by Reference BirchBirch (1948), a variety of forward and both Bayesian and non-Bayesian inverse methods has been used. Reference WangWang (1992) has recently summarized these. This is a simpler problem than the interpretation of ice-sheet temperatures; conduction is the only means of heat transfer and so there is no need for assumptions about past changes in precipitation rate and how the vertical velocity component varies with depth. If the authors wish to develop their method, applying it to some of these data, and comparing their results with those obtained by other methods, would be a useful first step.

In their final paragraph, the authors suggested that temperature-depth profiles can provide a check of paleotemperature records derived from oxygen-isotope ratios measured in ice cores, as proposed by Reference RobinRobin (1976). They then tentatively identified the cold period from 10000 to 7500 year BP in their inferred surface-temperature history with the Younger Dryas event. They pointed out that this differs from the record of this event in the oxygen-isotope profile at Dye 3, namely a cold period lasting less than 1000 a immediately preceding the end of the glaciation at about 11 000 year BP. We would ascribe the discrepancy to the defects in their analysis discussed above, combined with the use in their calculations of an accumulation rate of only 75% of the present value. The amplitude, timing and duration of the cold event depend sensitively on the accumulation rate, as their figure 10 shows. The authors, on the other hand, took the discrepancy as support for the suggestion of Reference FairbanksFairbanks (1989) that the low (i.e. highly negative) values of δ¹⁸O in the Younger Dryas section of the Dye 3 record result, not so much from low temperatures in Greenland as from the presence of a surface layer of glacial meltwater in the North Atlantic, in the source region of Greenland precipitation, at that time. The low values of δ¹⁸0 are, however, accompanied by the high concentrations of wind-blown dust and also chloride and sulphate, which come mainly from the ocean, characteristic of a glacial period (Reference Hammer, Clausen, Dansgaard, Neftel, Kristinsdottir, Johnson, Langway, Oeschger and DansgaardHammer and others, 1985; Reference Herron, Langway, Oeschger and DansgaardHerron and Langway, 1985; personal communication from M.M. Herron to W.S.B. Paterson); a surface-meltwater layer cannot account for these features. Furthermore, deuterium-excess data (Reference Dansgaard, White and JohnsenDansgaard and others, 1989) excluded Fairbanks’ explanation because evaporation from mid- to high-latitude source regions would result in much lower excess values (Reference Johnsen, Dansgaard and WhiteJohnsen and others, 1989) than observed. Again, Reference Lehman and KeigwinLehman and Keigwin (1992) have recently shown that the meltwater peaks coincide with the high rather than the low 5¹⁸0 parts of the Dye 3 record (Reference Lehman and KeigwinLehman and Keigwin, 1992, fig. 3c and d). For these reasons, we prefer the straightforward explanation that the low values of δ¹⁸O in the Younger Dryas sections of the Dye 3 (and Camp Century) cores do indeed reflect low temperatures in Greenland at the time.

The accuracy of references in the text and in this list is the responsibility of the authors, to whom queries should be addressed.