Most glacier mass-balance data are collected through the use of stakes, pits, cores, or probing to or from a reference horizon. This is normally a summer horizon—either a winter snow/ice interface in the ablation area as measured in spring, or a snow/firn interface in the accumulation area as measured in spring or summer. This interface, here termed summer surface, may form at different times of the year in different parts of the world or even on the same glacier. A summer surface is normally recognized as a discontinuity between units of differing density or other physical property, or as a diagnostic, correlatable horizon such as a depth-hoar layer. On some glaciers it may not be possible to recognize summer surfaces; in this case conventional mass-balance measuring procedures cannot be used and the system presented here is not applicable.

In a pit or core in the accumulation area, the mass of ice material between two consecutive summer surfaces can be measured. This mass, in Mg/m² or meters of water equivalent, may be the balance (the difference between accumulation and ablation) at that point for the time interval between the formation of the two summer surfaces. However, in the percolation or soaked facies (Reference BensonBenson, 1962. p. 24–25) an appreciable part of the material deposited in this time interval may have been melted and subsequently redeposited (refrozen) in lower layers, below the lower of the two summer surfaces of interest. Detection of this problem is not easy, and analysis of the resulting balance may be even more difficult. We assume here that any appreciable mass redeposited below a summer surface of interest can be calculated from repeated depth–density profiles and added to the balance above the summer surface.

A more difficult problem, especially when relating mass-balance quantities to meteorologic and hydrologic quantities as in the Combined Heat, Ice, and Water Balances Program of the International Hydrological Decade stems from the fact that summer surfaces may form at different times in different places. This means that a simple integration over the glacier of mass-balance data related to summer surfaces produces a result that has no clear meaning with respect to time. Thus these data cannot be directly related to heat or mass-flux data obtained by other techniques.

This situation is somewhat analogous to the geologic problem of dealing with rock formations that were deposited at different times in different places, for example, a narrow zone of beach sand deposited as sea-water slowly encroached on a landmass over a span of geologic time. In geologic nomenclature one can speak of rock types, rock–stratigraphic units (such as a formation) which are correlatable units independent of time concepts; time–stratigraphic units which are the rocks deposited during a specific time interval; and time units (American Commission on Stratigraphic Nomenclature, 1961). Snow, firn and ice can be used either as rock-type or rock-stratigraphic terms. Confusion enters when a term such as firn is incorrectly used as a time–stratigraphic unit for a whole glacier.

The fact that a late snow layer and a summer surface may be time-transgressive (forming at different times in different places) can be illustrated by a typical sequence on a temperate Northern Hemisphere glacier (Table I).

Table I. Hypothetical Mass-Balance History for a Typical Glacier

Mass-balance terms based on observable summer surfaces were proposed by Reference MeierMeier (1962). The stratigraphic system (Anonymous, 1969; [IHD], 1970) is a modification in terminology of the basic mass-balance concepts. This stratigraphic system works well for individual points. However, in order to compare ice-balance data with hydrologic (water-balance) data, these point values must be integrated over a whole glacier or drainage basin. If, as is usually the case, the summer horizons are not formed synchronously over the whole area, this integration is an invalid measure of snow and ice storage. Therefore, a different system, the annual system (fixed-date system), has been conceived to relate glaciological data to hydrological data. Unfortunately, glaciological programs using only the annual system cannot take advantage of convenient reference horizons in the field, so the field work may be extremely difficult or exorbitantly expensive. These two systems are described in [IHD] (1970) and in Anonymous (1969); but no attempt was made to show how they might be combined.

We feel that to document properly the relation of glaciers to climate, and to provide a check on mass-balance measurements, it is necessary to combine these two systems into a unified whole. The vital key to a combination of these systems is identification of the material under consideration. This identification is, of course, useful additional information for any description of the meteorological–hydrological environment. We define four types of material which may be found on a glacier in one specific year—snow, old firn and ice, late snow and new firn—as follows: the highest (most recent) summer surface found in a pit dug in winter (or early spring) before the beginning of appreciable, continuous melting is termed ss ₀. The material above ss ₀ is termed snow and the material below it is old firn and ice. The highest (most recent) summer surface found in a pit in the upper regions of the glacier late in the same year after the beginning of snow accumulation following a period of melting in summer is termed The material above ss ₁ is termed late snow and the material below ss ₁ yet above ss ₀ is new firn.

These terms define units which can be correlated over large areas of a glacier in any one year; however, they may be formed in different times at different places. Thus they correspond to “formations” (rock–stratigraphic units) in the geological sense. A year later, a different set of formations is described, each unit dropping back one step:

Another important requirement of combining the two systems is making measurements of stratigraphic units at the beginning and end of a fixed-date hydrologic year.

The variation in these units of the mass balance may be illustrated by graphs showing the changing balance with time, b[t], at specific points on a glacier, expressed as mass per unit area (Mg/m²) or simply in water equivalent (m) (Fig. 1).Footnote * The balance quantities are designated by the letter b with qualifying symbols, as follows: the subscript ₀ refers to the initial measurements made at or near the beginning of the year to relate fixed-date system measurements to stratigraphic units; the subscript ₁ refers to the final measurements made at or near the end of the year to relate the two systems; the subscript _a refers to certain measurements made (or calculated) exactly at the end of the hydrologic year, and the subscript _n refers to measurements related to the minimum firn and ice or the minimum total mass near (but not necessarily at) the end of a hydrologic year; the letter _x identifies balance quantities at the time of the maximum total balance in the hydrologic year. Letters in parentheses following the b indicate the material being measured: snow (s), old firn and ice (i), late snow (ls), and new firn (f). If superimposed ice accumulates or melt water is redeposited in lower layers it is added to either the snow (s) or new firn (f), to whichever the superimposed ice is related. A lack of parentheses following the b indicates that the total mass (undifferentiated) is considered. The hydrologic year, usually defined as 1 October through 30 September, runs from t ₀ to t ₁. Arrows pointing up indicate an addition of mass as time proceeds; the corresponding balance quantities are taken as positive.

Fig. 1. Balance quantities as measured at specific points on a glacier using a stratigraphic system referenced to summer surfaces. Shown here are typical balance curves b[t] for points in the accumulation area (top diagram) and in the ablation area (bottom diagram). The vertical scale is meters of water equivalent, with an arbitrary zero. The zero may be placed in any other position depending upon the needs of the data analysis. This and succeeding illustrations use the following sign convention: balance terms which increase with time, or represent an increment of mass added to the system, are considered positive and are shown with arrows pointing up. Note the formation of summer surfaces, and how they form at different times at the two points. Measurements at the beginning (t ₀) and end (t ₁) of the hydrologic year, in spring, and at a time after the hydrologic year (perhaps next spring) define these quantities b ₀(s) : the initial snow balance; b ₀, the initial ice balance; b _m(s), the measured winter snow balance; b ₁(ls), the final late snow balance; b ₁(i), the final ice balance; b _n(f), the net firnification; and b _n(i) the net ice balance. Also shown is b _w(s), the maximum winter snow balance which usually is not measured, being dependent on knowledge of b[t] at each point. Note that the maximum winter snow balance occurs at different times at different points.

Measurements made at specific points define the following quantities:

1. 1. b ₀(s), the initial snow balance, is the snow at the beginning of the hydrologic year. Often b ₀(s) = 0 in the ablation area, and sometimes even in the accumulation area. It is measured by probing or in pits by field work at the beginning of the hydrologic year or by computation using other field results and interpolation, perhaps using weather records.

2. 2. b ₀(i), the initial ice balance, is the old firn and ice loss after the start of the hydrologic year and before ablation ceases in winter and is usually measured by observing stakes in early winter or the next summer before snow has disappeared from the ice. Always b ₀(i) = 0 in the accumulation area and sometimes in the ablation area also.

3. 3. b _m(s), the measured winter snow balance, is the snow above the summer surface ss ₀ as measured directly by field work in late spring as near as possible to the time of greatest glacier mass (not necessarily the time of greatest mass at any one point). Normally b _m(s) is measured over the whole glacier at about the same time.

4. 4. b ₁(ls), the final late snow balance, is the late snow at the end of the hydrologic year, the same as b ₀(s) for the year following.

5. 5. b ₁(i), the final ice balance, is the old firn and ice loss after the end of the hydrologic year before ablation ceases for the next winter, the same as b ₀(i) for the year following.

6. 6. b _n(f), the net firnification, is the amount of new firn formed at about the end of the hydrologic year (either just before or just after). It is therefore the mass between the successive surfaces ss ₀ and ss ₁, and is usually measured in pits well after its time of formation.

7. 7. b _n(i), the net ice balance, is the corresponding change in mass between ss ₀ and ss ₁ in the ablation area where this change is negative; thus it records the loss of ice and old firn from the end of one melt season to the end of the next.

8. 8. b _w(s), the maximum snow balance, is the hypothetical maximum mass of the snow during the hydrologic year. This value will occur at a different time at each place and thus will probably not be measured directly.

Stratigraphic balance quantities at a point have been defined previously (Reference MeierMeier, 1962; Anonymous, 1969; [IHD], 1970) and their measurement is straightforward. The quantities and symbols presented here are consistent with the last two references. Difficulty appears only when one attempts to integrate these point values over a whole glacier. For instance, if values of maximum winter snow balance, b _w(s), are determined at numerous points on a glacier, a map of these values represents a time-transgressive maximum which is greater than the total amount of snow on the glacier at any moment. This peculiar “snow flood crest” map has no real meaning for most hydrologic balance computations. Similar problems exist with b ₀(i), b ₁(i), b _n(i), and b _n(f), but not with b ₀(s), b ₁(ls), and b _m(s).

We may now consider mass-balance data in terms of diagrams showing the area-average balance curve (Figs. 2–4). Alternatively, one could plot the total balance curve B[t] using a different vertical scale of m³ instead of meters of water equivalent. In the material to follow, a bar over a symbol indicates an area average, as in [IHD] (1970).

Fig. 2. Area-averaged balance quantities shown without relation to summer surfaces or type of material measured. The vertical scale is meters of water equivalent averaged over the entire area of the glacier, with an arbitrary zero (the scale could just as well be in kilograms of mass or cubic meters of water equivalent). Sign convention as in Figure 1. Figure 2a shows area-averaged terms equivalent to the stratigraphic system terms of [IHD] (1970), including b _w, the winter balance, and , the total mass net balance. The period between two balance minima in successive years is the balance year. Figure 2b shows the area-averaged term , the annual balance, and , the maximum balance, which are related to the annual (fixed date) system if [IHD] (1970). Also shown are the terms and , the initial and final balances, which relate the two systems. Field determination of all these quantities except is difficult because knowledge of the area-averaged balance curve is required to know magnitude and timing of these balance quantities. However, the curve is directly related to the changing mass storage, equal approximately to precipitation minus run-off in many environments.

Fig. 3. Area-averaged balance quantities that can be conveniently measured in the field. The vertical scale is meters of water equivalent averaged over the entire area of the glacier, with arbitrary zero. Sign convention is as in Figure 1. Figure 3a shows terms in the stratigraphic system including , the initial ice balance; , the measured winter snow balance; , the net ice balance; , the net firnification; and , the final ice balance. Nearly all the firn and ice ablation at this time (after t ₁ ^′) will occur on ss ₀ rather than ss ₁. However, the final ice-balance decrease, , is shown on ss ₁ and not ss ₀ because, if no snow ablation occurs after t ₁ ^′, will automatically become for the succeeding year. The terms and , the initial snow balance and final late snow balance, relate stratigraphic and annual (fixed-date) systems. Figure 3b shows in the annual system, including in addition to those above, the annual ice balance and the annual firnification; the latter is defined only if new firn formation is complete by the end of the hydrologic year. As in Figure 1, these terms can all be measured in a continuing program involving only two trips per year: one at the beginning of the hydrologic year and one in the spring. Two additional terms (open arrows) can be calculated easily from observed quantities: the firn and ice net balance , and the firn and ice annual balance . Figure 3c shows the situation if residual snow, not yet completely covered by new snow, remains at the end of the hydrologic year. In this case [and ] cannot be defined.

Fig. 4. All area-averaged quantities shown together on one diagram for reference purposes. No one program would expect to report all of these possible terms, but only those selected as most relevant to the particular environment and observation scheme. The vertical scale is meters if water equivalent averaged over the entire area of the glacier, with arbitrary zero. Conveniently measurable quantities (solid arrows) are those shown in Figure 3. Quantities determined through knowledge of the area-averaged balance curve (dashed arrows) are those shown in Figure 2. Calculated quantities are shown with open arrows.

Figure 2a is a plot of on which are shown the area-averaged terms (winter balance) and (here termed the total mass net balance) which are analogous to the stratigraphic system point terms b_w and b_n in [IHD] (1970). The time interval t ₀ ^′ to t ₁ ^′ is a balance year which is unique to each glacier and each year, and is not necessarily 365 days long. Figure 2b is a plot of showing the area-averaged annual balance , which is analogous to the fixed-date system point term b _a in [IHD] (1970). Also included in this figure is a new term , the maximum balance [of the hydrologic year], In order to relate to to , or vice versa, two additional terms are proposed: , the initial balance increment, and , the final balance increment. These curves show mass-balance quantities that can be directly related to other hydrologic balances.

Unfortunately the scheme shown in Figure 2a and b is not fully workable in practice. This is because the terms , , and have no relation to the corresponding values measured relative to summer surfaces at the individual points. This scheme is workable only if point data are integrated in the following way:

Suppose that curves such as those shown in Figure 1 have been obtained at a large number of stakes on a glacier in a given year. The balance curve for the whole glacier, B[t], might then be obtained by summing the individual balance curves (b ₁[t], b ₂[t] …), multiplying each by a weighting factor (a ₁, a ₂ …) according to its proportion of the total area (A):

where is the area-average balance curve for the whole glacier. Note especially that the balance curves are combined to obtain , not the individual point measurements such as b ₀(i) which might have been made at different times in different places.

This computation procedure requires more measurements than can normally be made. Needed is a scheme in which the point data, taken only a limited number of times during a year and usually referenced to summer surfaces, can be combined directly. In order to do this, the summer surfaces must be included in the area-average diagrams. This is done by dividing the balance curve into its four components: old firn and ice, snow, new firn and late snow(Fig. 3). The largest mass, old firn and ice, is plotted at the bottom—it can only decrease or remain constant during any one year. Above it is plotted snow, which increases during the first part of the year; during the last part of the year it decreases due to ablation and is converted to new firn. Toward the end of the year, late snow is deposited on top of a melt surface, causing the snow below the surface to be converted to firn. The amount of new firn after all of the snow is eliminated and/or converted remains relatively constant. The interface between snow plus new firn and old firn and ice is summer surface ss ₀. The interface between new firn and late snow is summer surface ss ₁. The interface between snow and new firn is shown as a jagged line; it has little physical significance.

The point data taken at specific times during the field season (or determined after the fact) can now be averaged over the glacier and shown on the area-average balance diagram (Fig. 3a–c). Now several important balance terms are determined. One is the annual balance . Another important balance quantity is the difference between old firn and ice melt, , the net ice balance, and snow which lasts through the melt season and is preserved as new firn, , the net firnification. This difference is here called the firn and ice net balance, . This is in fact the quantity most often reported by glaciologists as “net balance” but is not the area-average net balance (Fig. 2) indicated in [IHD] (1970).

On some glaciers, all of the snow is normally converted to firn before the end of the hydrologic year, and it is convenient to measure firn and ice balances at this time. We therefore define three additional terms: the annual firnification , the annual ice balance , and the firn and ice annual balance (Fig. 3b). These units are analogous to , and , respectively.

In some situations, snow melt may continue after the end of a hydrologic year. One can measure the residual snow on the glacier surface on 1 October, but only part of this (that part which is buried under new snow) is properly called firn. Thus the quantities and cannot be defined (Fig. 3c). If it is considered necessary to compute a quantity like , one can redefine terms so that relates to the new firn and residual snow and not material which everywhere underlies late snow. The quantity always definable.

Other area-averaged terms shown on Figure 3, exactly equivalent to the corresponding values shown on Figure 1, are , the initial snow balance; , the initial ice balance; , the final late snow balance; , the final ice balance; and , the measured winter snow balance.

The annual balance, , is an important quantity because it represents the total change in storage (of snow, firn and ice) during a hydrologic year. Thus this value can be compared directly with the difference between precipitation as snow and melt-water run-off if net evaporation/condensation and net changes in liquid water storage are negligible. can be measured directly, or computed as is definable. The annual balance can also be calculated, more indirectly, from balance year (stratigraphic system) quantities as .

It is important to define the maximum winter snow balance on the glacier during the year, because it is often impossible to measure the actual winter snow accumulation. Our term , a measured but lower value at about the right time of year, can often be used as a basis for computing if sufficient supplementary meteorological data are available. The same statement can be made about (Fig. 2a) and (Fig. 2b). Although and occur at the same instant in time, can occur at a later date. Note that . Normally, only one or two of these four balance quantities would be reported. It must be stressed that none of these can be calculated by averaging point values of b _w (winter balance) because the balance reaches a maximum at different times at the different points.

The terms defined and shown on Figures 2 and 3 can be combined on one diagram for reference and comparison (Fig. 4). The apparent complexity of this diagram is somewhat misleading; only half to two-thirds of these terms need be reported in any given study.

These terms define glacier balance with sufficient generality to report any condition of varying balance conditions at different points. At many glaciers, some of the correction terms, such as and , will be zero or small, and if so can be neglected to simplify the calculations.

Many of the terms taken at the end of one year automatically become the important initial terms for the next year. The symbols change, as follows:

The combined terms allow a description of the hydrology of a glacier involving process, material and time. Thus we can list initial, final and annual changes in snow cover and ice storage, and relate these to the hydrologically relevant quantities of precipitation and run-off.

This scheme appears, unfortunately, to be rather complicated. However, it cannot be simplified without omitting important features of glacier hydrology. One must recognize that some of these terms may be appreciable in magnitude at one glacier while not even definable or measurable at another. Lack of recognition of these terms has made analysis of some previous mass-balance data difficult, if not impossible. If ice-balance data are not analyzed together with heat or water-balance data, a reporting scheme which appears to be simple and concise can be devised. But such a system may be either useless or invalid for comparison with any kind of meteorological or other hydrological data.

Two examples of the use of the combined scheme to report data from glaciers are given in Table II. These early results from an International Hydrological Decade study illustrate a typical measurement and computation program (on other glaciers it may be necessary or desirable to measure values listed as computed or vice versa). Some of the values shown are redundant and would not be given in a routine report; they are shown here to illustrate all of the terms we have defined.

Table II. Mass-Balance Quantities for Gulkana Glacier, Alaska, and South Cascade Glacier, Washington, 1966

(Except for times, all quantities are reported in meters water equivalent averaged over the glacier. Data for Gulkana Glacier are based on a reconnaissance study and contain large uncertainties.)

Note that the terms are small or zero and are similar in magnitude for South Cascade Glacier. However, the term for Gulkana Glacier is appreciable, and the balances and are not even of the same sign. Obviously, on a glacier like Gulkana it is meaningless to express the “state of health” by a net- or annual-balance value unless it is absolutely clear exactly what is meant. This scheme, although appearing somewhat cumbersome, gives the author a code for expressing whichever units be prefers in exact, definable and comparable terms.