Trophic State Equations

Portions of the text below have been excerpted from the following NALMS publications:

Carlson, R.E. and J. Simpson. 1996. A Coordinator’s Guide to Volunteer Lake Monitoring Methods. North American Lake Management Society. 96 pp.

A Trophic State Index

A frequently used biomass-related trophic state indices is that of Carlson (1977). It is relatively simple to use, requires a minimum of data, and is generally easy to understand, both in theory and use. It is numerical, but the traditional nutrient-related trophic state categories fit into the scheme. It seems to be ideal for use in volunteer programs.

We define trophic state as the total weight of living biological material (biomass) in a waterbody at a specific location and time. Time and location-specific measurements can be aggregated to produce waterbody-level estimations of trophic state. Trophic state is understood to be the biological response to forcing factors such as nutrient additions (Naumann, 1919, 1929), but the effect of nutrients can be modified by factors such as season, grazing, mixing depth, etc.

In accordance with the definition of trophic state given above, the trophic state index (TSI) of Carlson (1977) uses algal biomass as the basis for trophic state classification. Three variables, chlorophyll pigments, Secchi depth, and total phosphorus, independently estimate algal biomass. Unlike Naumann’s typological classification of trophic state (Naumann, 1929), the index reflects a continuum of “states.” There are no lake “types.” The trophic continuum is divided into units based on a base-2 logarithmic transformation of Secchi depth, each 10-unit division of the index representing a halving or doubling of Secchi depth. Because total phosphorus often correlates with transparency, a doubling of the total phosphorus often corresponds to a halving of Secchi depth. Chlorophyll pigments double every 7 units rather than every 10 units (Carlson 1980).

The range of the index is from approximately zero to 100, although the index theoretically has no lower or upper bounds. The index has the advantage over the use of the raw variables in that it is easier to memorize units of 10 rather than the decimal fractions of raw phosphorus or chlorophyll values. An early version of the index was based on a scale of one to ten, but it became tempting to add 1, 2, or more numbers after the decimal. For this reason, the scale was multiplied by ten to discourage any illusory precision obtained by using more than whole numbers.

The logarithmic transformation of the data normalizes the skewed data distribution, allowing the use of parametric statistics (mean, standard deviation, parametric comparison tests). This facilitates not only comparison and data reduction, but communication as well, because the user does not need to resort to graphs with logarithmic axes.

The three index variables are interrelated by linear regression models, and should produce the same index value for a given combination of variable values. Any of the three variables can therefore theoretically be used to classify a waterbody. This is particularly useful in citizen lake monitoring programs, where Secchi depth is often the only variable that can be inexpensively measured. For the purpose of classification, priority is given to chlorophyll, because this variable is the most accurate of the three at predicting algal biomass. According to Carlson (1977), total phosphorus may be better than chlorophyll at predicting summer trophic state from winter samples, and transparency should only be used if there are no better methods available.

Calculating the TSI

The index is relatively simple to calculate and to use. Three equations are used: Secchi disk, TSI(SD); chlorophyll pigments, TSI(CHL); and total phosphorus, TSI(TP). The original Secchi depth equation in Carlson (1977), reproduced below looks forbidding, but illustrates how the index was constructed.

$TSI(SD)=10\left [ 6-\frac{ln\: SD}{ln\: 2} \right ]$

The basic Secchi disk index was constructed from doublings and halvings of Secchi disk transparency. The base index value is a Secchi disk of 1 meter, the logarithm of which is zero.

ln 1 = 0

6 – 0 = 6

6 = 60

Therefore, the TSI of a 1 meter Secchi depth is 60. If the Secchi depth were 2 meters,

ln 2 / ln 2 = 1

6 – 1 = 5

10 x 5 = 50

The indices for the chlorophyll and total phosphorus are derived in a similar manner, but, instead of a Secchi depth value in the numerator, the empirical relationship between chlorophyll or total phosphorus and Secchi depth is given instead. For example, the chlorophyll TSI is:

The above forms of the TSI equations may illustrate how the indices were derived, but they can be simplified for everyday use. The simplified equations are below:

TSI(SD) = 60 – 14.41 ln(SD)

TSI(CHL) = 9.81 ln(CHL) + 30.6

TSI(TP) = 14.42 ln(TP) + 4.15

Averaging TSI Values

There has been a tendency to average the three variables rather than to prioritize their use (Osgood 1982; Kratzer and Brezonik 1981). Perhaps this is just a natural tendency for humans to seek the central tendency, or it might reflect the concept that trophic state is defined by a number of variables. Whatever the reason, averaging makes no sense at all. The index is predicated on the idea that it is predicting algal biomass. Chlorophyll is a better predictor than either of the other two indices. There is no logic in combining a good predictor with two that are not (Carlson 1983).

Although transparency and phosphorus may co-vary with trophic state, the changes in transparency are caused by changes in algal biomass and total phosphorus may or may not be strongly related to algal biomass. Neither transparency nor phosphorus are independent estimators of trophic state. Using transparency or phosphorus as an estimator of chlorophyll is very different from assuming equal and independent status of the variables. Carlson (1983) emphasized that the averaging of chlorophyll with the predicted chlorophyll based on Secchi depth is equivalent to assuming that temperature is better estimated by averaging the reading from a thermometer with the number of cricket chirps per minute. Secchi depth should be used as a surrogate, not covariate, of chlorophyll.

Relating Trophic State to the State of the Waterbody

Any trophic state index gains value when it can be correlated with specific events within a waterbody. Below is a table of attributes that could be expected in a north temperate lake at various TSI values. Some characteristics, such as hypolimnetic oxygen or fisheries may be expected to vary with latitude and altitude and the table may not place these changes in the proper TSI category. We have used the classic terms of oligotrophy, mesotrophy, and eutrophy in their original context of the amount of algae in the water, not hypolimnetic oxygen concentration, so it is quite possible for an oligotrophic lake to have no hypolimnetic oxygen.

 A list of possible changes that might be expected in a north temperate lake as the amount of algae changes along the trophic state gradient. TSI Chl(µg/L) SD(m) TP (µg/L) Attributes Water Supply Fisheries & Recreation < 30 < 0.95 > 8 < 6 Oligotrophy: Clear water, oxygen throughout the year in the hypolimnion. Water may be suitable for an unfiltered water supply. Salmonid fisheries dominate. 30 – 40 0.95 – 2.6 8 – 4 6 – 12 Hypolimnia of shallower lakes may become anoxic. Salmonid fisheries in deep lakes only. 40 – 50 2.6 – 7.3 4 – 2 12 – 24 Mesotrophy: Water moderately clear; increasing probability of hypolimnetic anoxia during summer. Iron, manganese, taste, and odor problems worsen. Raw water turbidity requires filtration. Hypolimnetic anoxia results in loss of salmonids. Walleye may predominate. 50 – 60 7.3 – 20 2 – 1 24 – 48 Eutrophy: Anoxic hypolimnia, macrophyte problems possible. Warm-water fisheries only. Bass may dominate. 60 – 70 20 – 56 0.5 – 1 48 – 96 Blue-green algae dominate, algal scums and macrophyte problems. Episodes of severe taste and odor possible. Nuisance macrophytes, algal scums, and low transparency may discourage swimming and boating. 70 – 80 56 – 155 0.25 – 0.5 96 – 192 Hypereutrophy: (light limited productivity). Dense algae and macrophytes. > 80 > 155 < 0.25 192 – 384 Algal scums, few macrophytes Rough fish dominate; summer fish kills possible.

An unfortunate misconception concerning trophic state is that the term is synonymous with the concept of water quality. Although the concepts are related, they should not be used interchangeably. Trophic state is an absolute scale that describes the biological condition of a waterbody. The trophic scale is a division of that variable(s) used in the definition of trophic state and is not subject to change because of the attitude or biases of the observer. An oligotrophic or a eutrophic lake has attributes of production that remain constant no matter what the use of the water or where the lake is located. For the trophic state terms to have meaning at all, they must be applicable in any situation in any location.

Water quality, on the other hand, is a term used to describe the condition of a water body in relation to human needs or values. Quality is not an absolute; the terms “good” or “poor” water quality only have meaning relative to the use of the water and the attitude of the user. An oligotrophic lake might have good water quality for swimming but be considered poor water quality for bass fishing. Confusion can ensue when the term trophic state is used to infer quality.

Suppose, for example, that a manager were to establish fishing goals based on trophic state. Generally fish yield increases as the production of the lake increases, but there may be changes in the dominant fish species as a lake eutrophies (Oglesby, et al. 1987). In northern lakes, salmonids might dominate in clear lakes having oxygenated hypolimnia. When production increases to the point where the hypolimnion becomes anoxic, then salmonids may disappear, to be replaced by percids, then centrarchids, and finally rough fish such as carp or bullheads. If a fisheries manager wished to manage all lakes based on fish production, then the greener the lake the better. However, what is meant by good water quality would be different for a person wanting to catch lake trout than a person wanting only bass. In fisheries management, the relationship between fish production and fish community structure and trophic state do not change. What changes is the perception of what is good or bad water quality. In this case, the meaning of quality water heavily depends on the goals and expectations of the fishery and the fishermen.

Multiple use situations can cause numerous conflicts because of differing perceptions of water quality by different users. Fishermen may want the optimal water quality for their particular species of game fish, boaters will want to minimize weeds, swimmers will want to see their feet. Other users, such as drinking water utilities, may want the clearest water possible, but ignore weeds completely. Vant and Davies-Colley (1988), for example, found that lakes in New Zealand ceased to be acceptable for swimming at Secchi depths less than one meter, but Secchi depth apparently did not affect fishing, passive recreation (relaxation/observation/picnics/camping), sailing, or power boating. For each use, the trophic spectrum is being referred to, but the needs of the users, and thus the perception of quality at any given trophic state, vary considerably.

Attitude about water quality is also affected by the general background of the user. General background means the attitude of the user that is related to his or her upbringing, geographical location, and virtually all attitudes that the user brings to lake evaluation other than that of a user. In a study of lay attitudes about water quality, Smeltzer and Heiskary (1990) queried volunteers as to whether their lakes were beautiful or if enjoyment was slightly impaired, substantially reduced, or nearly impossible. They found that the volunteer responses varied geographically. In Vermont and in the northeastern portion of Minnesota, volunteers were more sensitive to changes in trophic state. In the agricultural region of southwest Minnesota, lakes that were considered to have minor problems would have been considered impaired in the other regions. The lesson here is that what is judged to be good or poor water quality is affected by regional attitudes.

Nitrogen

Other indices have been constructed to be used with the basic three. Since nitrogen limitation still classifies a lake along Naumann’s nutrient axis, the effect of nitrogen limitation can be estimated by having a companion index to the Total Phosphorus TSI. Such an index was constructed by Kratzer and Brezonik (1981) using data from the National Eutrophication Survey on Florida lakes. This index is calculated using the formula:

TSI(TN) = 54.45 + 14.43 ln(TN)

(Nitrogen values must be in units of mg/L.)

The index of Kratzer and Brezonik were designed to be used in nitrogen-limiting conditions, but in reality, is relatively insensitive to the nitrogen : phosphorus ratio, while the phosphorus TSI of Carlson deviates at low nitrogen phosphorus ratios. This suggests that a nitrogen index value might be a more universally applicable nutrient index than a phosphorus index, but it also means that a correspondence of the nitrogen index with the chlorophyll index cannot be used to indicate nitrogen limitation. If, however, nitrogen and phosphorus indices were plotted at the same time, then a deviation of only the phosphorus index might indicate nitrogen limitation, while deviations of both nitrogen and phosphorus indices might indicate situations where nitrogen or phosphorus are not limiting.

Macrophytes

The TSI in its present form is based solely on algal biomass. It is therefore blind to macrophyte biomass and may, therefore, underestimate the trophic state of macrophyte-dominated lakes. This is a serious drawback that needs to be addressed. The solution could be very simple. Canfield et al. (1983) proposed a method to measure the total phosphorus content of lakes. The total macrophyte biomass in the lake is estimated by the equation:

TSMB = SA x C x B

where TSMB = total submersed macrophyte biomass, SA = lake surface area, C = percent cover of submersed aquatic macrophytes, and B = average biomass collected with a sampler.

Canfield et al. (1983) estimated the total phosphorus in plant biomass based on the phosphorus in each species and the relative abundance of each species. The total phosphorus content of the lake was obtained by adding the amount of phosphorus in the macrophytes to the amount estimated to be in the water column. There seems to be no reason why he same approach could not be used to measure total plant biomass or chlorophyll. If it were used, trophic state could include both macrophytes and algae, and have internally consistent units.

Using the Indices Beyond Classification

A major strength of TSI is that the interrelationships between variables can be used to identify certain conditions in the lake or reservoir that are related to the factors that limit algal biomass or affect the measured variables. When more than one of the three variables are measured, it is possible that different index values will be obtained. Because the relationships between the variables were originally derived from regression relationships and the correlations were not perfect, some variability between the index values is to be expected. However, in some situations the variation is not random and factors interfering with the empirical relationship can be identified. These deviations of the total phosphorus or the Secchi depth index from the chlorophyll index can be used to identify errors in collection or analysis or real deviations from the “standard” expected values (Carlson 1981). Some possible interpretations of deviations of the index values are given in the table below (updated from Carlson 1983).

 Relationship Between TSI Variables Conditions TSI(Chl) = TSI(TP) = TSI(SD) Algae dominate light attenuation; TN/TP ~ 33:1 TSI(Chl) > TSI(SD) Large particulates, such as Aphanizomenon flakes, dominate TSI(TP) = TSI(SD) > TSI(CHL) Non-algal particulates or color dominate light attenuation TSI(SD) = TSI(CHL) > TSI(TP) Phosphorus limits algal biomass (TN/TP > 33:1) TSI(TP) >TSI(CHL) = TSI(SD) Algae dominate light attenuation but some factor such as nitrogen limitation, zooplankton grazing or toxics limit algal biomass.

The simplest way to use the index for comparison of variables is to plot the seasonal trends of each of the individual indices. If every TSI value for each variable is similar and tracks each other, then you know that the lake is probably phosphorus limited (TN/TP = 33; Carlson 1992) and that most of the attenuation of light is by algae.

In some lakes, the indices do not correspond throughout the season. In these cases, something very basic must be affecting the relationships between the variables. The problem may be as simple as the data were calculated incorrectly or that a measurement was done in a manner that produced different values. For example, if an extractant other than acetone is used for chlorophyll analysis, a greater amount of chlorophyll might be extracted from each cell, affecting the chlorophyll relationship with the other variables. If a volunteer incorrectly measures Secchi depth, a systematic deviation might also occur.

After methodological errors can be ruled out, remaining systematic seasonal deviations may be caused by interfering factors or non-measured limiting factors. Chlorophyll and Secchi depth indices might rise above the phosphorus index, suggesting that the algae are becoming increasingly phosphorus limited. In other lakes or during the season, the chlorophyll and transparency indices may be close together, but both will fall below the phosphorus curve. This might suggest that the algae are nitrogen-limited or at least limited by some other factor than phosphorus. Intense zooplankton grazing, for example, may cause the chlorophyll and Secchi depth indices to fall below the phosphorus index as the zooplankton remove algal cells from the water or Secchi depth may fall below chlorophyll if the grazers selectively eliminate the smaller cells.

In turbid lakes, it is common to see a close relationship between the total phosphorus TSI and the Secchi depth TSI, while the chlorophyll index falls 10 or 20 units below the others. Clay particles contain phosphorus, and therefore lakes with heavy clay turbidity will have the phosphorus correlated with the clay turbidity, while the algae are neither able to utilize all the phosphorus nor contribute significantly to the light attenuation. This relationship of the variables does not necessarily mean that the algae is limited by light, only that not all the measured phosphorus is being utilized by the algae.

A Multivariate Comparison

A different way of looking at deviations is reported in Carlson (1992). If both of the deviations, TSI(CHL) – TSI(TP) and TSI(CHL) – TSI(SD), are simultaneously plotted on a single graph, it is possible to identify some of these systematic deviations. The possibilities are illustrated below.

A representation of possible explanations of deviations of the Trophic State Index equations.

Points lying to the right of the Y-axis indicate situations where the transparency is greater than expected from the chlorophyll index. These deviations may occur if large particulates, such as blue-green algae (Cyanobacteria), dominate, and transparency is less affected by the particulates. Deviations to the right may also occur if zooplankton grazing removes smaller particles and leaves only large forms. Points to the left of the Y-axis would be related to situations where transparency is dominated by non-algal factors such as color or turbidity or where very small particles predominate.If TSI (CHL) – TSI (TP) is plotted on the vertical axis, then points below the X-axis would be associated situations where chlorophyll is under-predicted by total phosphorus, i.e., situations where phosphorus may not be limiting chlorophyll. Carlson (1992) reported that this zero line is related to total nitrogen to total phosphorus (TN/TP) ratios greater than 33:1. Phosphorus is usually thought to become limiting at a TN/TP ratio of 10:1, therefore slight deviations below the zero line would not truly indicate nitrogen limitation. A better interpretation would be that the greater the negative deviation, the greater the probability of something other than phosphorus limits algal growth. A combined phosphorus and nitrogen TSI deviation could also be used for this axis to eliminate the effects of nitrogen as well as phosphorus limitation. As points go above the zero axis, it would suggest increasing possibility of phosphorus limitation.

Points lying on the diagonal to the left of the origin indicate situations where phosphorus and transparency are correlated, but chlorophyll is not. Points on or near this line would be found in turbid situations where phosphorus is bound to clay particles and therefore turbidity and phosphorus are related, but chlorophyll is not.

This form of graph collapses the deviations of the Secchi depth TSI onto the graph of the other deviations, allowing simultaneous viewing of the deviations of all three indices. The spatial location of the data for a single lake or for a number of lakes can therefore be used to infer possible relationships between the three variables. This use of the index is still being developed but holds considerable promise in the interpretation of data.

Recommendations

Trophic state determination is an important aspect of lake surveys. Trophic state is not the same thing as water quality, but trophic state certainly is one aspect of water quality.  Several recommendations can be made with regard to the use of trophic state classifications.

1. Use the simplest definition of trophic state: the concept does not have to be so complex that it is cannot be simply explained or easily measured.
2. The recommended definition is that of plant biomass: it is historically correct, simple to measure, and simple to understand and explain. It also can be predicted from nutrient models and can be used to predict other biological characteristics.
3. Remove the mystery from the term eutrophication. Rather than linking the process to nutrients, which can cause all sorts of interpretational problems, simply define it as a movement of the lake’s trophic state in the direction of more plant biomass. The definition is simple and far more functional than any other definition.
4. The trophic state index of Carlson (1977) is recommended as the simplest method of calculating and explaining  trophic state concepts.
5. If data for chlorophyll and phosphorus are available, use chlorophyll as the primary index for trophic state classification. Use the deviations of the Secchi depth and total phosphorus indices from the chlorophyll index to infer additional information about the functioning of the lake.
6. Use the index as a teaching tool. You can use it to discuss all the possible factors, not just nutrients, that could make a lake more eutrophic. For example, you can explain that the deposition of erosional materials will cause the lake to become shallower, and therefore enhance macrophyte growth, thus affecting the total amount of biomass. Discuss the ramifications of change in plant biomass, how it affects hypolimnetic oxygen and fish species and its possible effect on food chains and recreational potential.
7. Be careful about using quality terms when speaking of trophic state. Even your own perception of quality is affected by your background and education. Be sensitive to the fact that not all users will want the same type of water quality that you do. Not everyone considers the ideal lake to be clear. Always be sensitive to the background and needs of the users.

Literature Cited

Canfield, D.E. Jr., K.A. Langeland, M.J. Maceina, W.T. Haller, J.V. Shireman, and J.R.Jones. 1983. Trophic state classification of lakes with aquatic macrophytes. Can. J. Fish. Aquat. Sci. 40: 1713-1718.

Carlson, R.E. 1977. A trophic state index for lakes. Limnology and Oceanography. 22:361-369.

Carlson, R.E. 1980. More complications in the chlorophyll-Secchi disk relationship. Limnology and Oceanography. 25:378-382.

Carlson, R.E. 1981. Using trophic state indices to examine the dynamics of eutrophication. p. 218-221. In: Proceedings of the International Symposium on Inland Waters and Lake Restoration. U.S. Environmental Protection Agency. EPA 440/5-81-010.

Carlson, R.E. 1983. Discussion on “Using differences among Carlson’s trophic state index values in regional water quality assessment,” by Richard A. Osgood. Water Resources Bulletin. 19:307-309.

Carlson, R.E. 1992. Expanding the trophic state concept to identify non-nutrient limited lakes and reservoirs. pp. 59-71 [In] Proceedings of a National Conference on Enhancing the States’ Lake Management Programs. Monitoring and Lake Impact Assessment. Chicago.

Carlson, R.E. and J. Simpson. 1996. A Coordinator’s Guide to Volunteer Lake Monitoring Methods. North American Lake Management Society. 96 pp.

Kratzer, C.R. and P.L. Brezonik. 1981. A Carlson-type trophic state index for nitrogen in Florida lakes. Water. Res. Bull. 17: 713-715.

Naumann, E. 1919. Nagra synpunkter angaende limnoplanktons okologi med sarskild hansyn till fytoplankton. Sv. Bot. Tidskr. 13: 129-163.

Naumann, E. 1929. The scope and chief problems of regional limnology. Int. Revue ges. Hydrobiol. 21: 423-.

Oglesby, R.T., J.H. Leach, and J. Forney. 1987. Potential Stizostedion yield as a function of chlorophyll concentration with special reference to Lake Erie. Can. J. Fish. Aquat Sci. 44(Suppl. 2): 166-170.

Osgood, R. 1983. Using differences among Carlson’s trophic state index values in regional water quality assessment. Wat. Res. Bull. 18: 67-74.

Smeltzer, E. and S.A. Heiskary. 1990. Analysis and applications of lake user survey data. Lake and Reservoir Management.

Vant, W.N. and R.J. Davies-Colley. 1988. Water appearance and recreational use of 10 lakes of the North Island (New Zealand). Verh. Internat. Verein. Limnol. 23: 611-615.