Journal of Foraminiferal Research, v. 28, no. 3, p. 233-239, July 1998 4^^C? SHE ANALYSIS FOR BIOFACIES IDENTIFICATION MARTIN A. BUZAS' AND LEE-ANN C. HAYEK^ ABSTRACT Most quantitative methodologies for designating bio- facies do so by ordering, in some way, all the possible similarities between stations along some gradient. The procedure is considered successful when the ordering produces mappable units (biofacies or biotopes). In con- trast, SHE analysis for biofacies identification (SHEBI) serially accumulates contiguous stations and designates biofacies as groups of contiguous stations that exhibit the properties of known statistical distributions for biologi- cal communities. SHEBI identifies biological communi- ties and always insures mappability. The number of species, S, the information function, H, and the measure of evenness, E, are related by the de- composition equation H = InS -H InE. Each of these quantities is calculated as N, the number of individuals, is accumulated by adding successive samples (stations). In all cases, for a multispecies population (community) these variables form Hnear trends on a log scale. As N accumulates with each sample, S usually increases and the decomposition equation establishes constraints on H and InE. If H remains constant, then InE must decrease precisely as InS increases. If InE remains constant, then H must increase precisely as InS increases. If H increases and InE decreases as InS increases with accumulation, then the ratio InE/lnS may remain constant. Sometimes, InS and H increase while InE and InE/lnS decrease. De- partures from linear trends indicate a mixture of com- munities and, hence, a new biofacies. The total population from 35 stations ranging in depth from 20m to 2696m in traverse 6 of Parker (1954) in the northeastern Gulf of Mexico was analyzed using the SHEBI technique. Seven biofacies were recognized. These biofacies compare favorably with the six recog- nized by Parker. However, the SHEBI procedure rec- ognized three biofacies at depths shallower than 100m while Parker recognized one. At depths greater than 200m SHEBI recognized two biofacies while Parker rec- ognized three. SHEBI establishes boundaries by examining depar- tures from the linear trends expected for the statistical distribution describing a multispecies population or community. Thus, SHEBI provides us not only with a means of designating biofacies, but also with a quanti- tative definition of a community. INTRODUCTION Wherever modern benthic foraminifera are surveyed along a depth gradient, the species examined exhibit a zo- nation with depth. While the distribution of each species is ' Department of Paleobiology, NHB MRC-121, Smithsonian Insti- tution, Washington, D.C. 20560, U.S.A. ^ Mathematics &. Statistics, NHB MRC-I36, Smithsonian Institution, Washington, D.C. 20560, U.S.A. unique (Culver and Buzas, 1981), groups of species can be recognized as principally inhabiting particular areas such as inner shelf, outer shelf, and so on. These groups of species are often called biofacies (Gressly, 1838) even though the original term referred to the biological aspect of rocks or sediments. Initially, investigators examined their data and proposed depth zonations based on the distribution and abundance of the taxa contained therein (e.g., Natland, 1933; Phleger, 1960). When computers and multivariate methods became available, researchers took advantage of the many numerical techniques to quantify biofacies more objectively (e.g., Kaesler, 1966; Buzas, 1967, 1969; Culver, 1988). Today, the use of a myriad of numerical methodol- ogies for the designation of biofacies is commonplace. As might be expected, biofacies, regardless of how they are recognized, do not all contain the same number of spe- cies. Traverses with numerous stations distributed from shal- low to deeper water usually show an increase in species richness (S) per station to at least the outer shelf (Sen Gupta and Kilbourne, 1974). At greater depths, species richness per station exhibits one of three patterns: increase, decrease or relatively constant (Gibson and Buzas, 1973). The infor- mation function, H (Shannon, 1948), shows a similar pattern while, E, a measure of evenness derived from it (Buzas and Gibson, 1969), often shows no apparent trend (Gibson and Buzas, 1973; Sen Gupta and Kilbourne, 1974; Gibson and Hill, 1992). While biodiversity patterns are used to char- acterize areas such as nearshore vs offshore and so on, they are not used to define biofacies. In this paper we show how patterns of biodiversity can be used to recognize biofacies through analysis of S, H, and E. This is now possible because a decomposition equation for the relationship of S, H, and E has been derived (Buzas and Hayek, 1996; Hayek and Buzas, 1997). A new meth- odology is involved. The new approach differs from the usual procedure used by researchers in an important aspect. Instead of examining the data station by station or taking the mean of several contiguous stations, we accumulate the data so that S, H, and E are examined as a function of N (the number of individuals). That is, we compute the quan- tities S, H and E, related through the decomposition equa- tion, on each set of contiguous stations as the samples (sta- tions) are accumulated. The expected behavior of S, H, and E with the accumulation of N for statistical distributions of communities is known uniquely. Thus, departures from known expected patterns define boundaries. We examine these boundaries for a traverse in the Gulf of Mexico (Par- ker, 1954). THE SHE RELATIONSHIP The famiUar Shannon information function (Shannon, 1948) is H = -E p,ln p, (1) 233 234 BUZAS AND HAYEK where H is the information function, p? is the proportion of the ith species and S the number of species. The maximum value of H = InS or e" = S where e is the base of the natural logarithms. This maximum value occurs when all the Pi's are equal in which case e^/S = 1. When the values of the Pi's are not equal this ratio will be less then one and will indicate the degree of evenness of the p?'s. Consequent- ly, Buzas and Gibson (1969) defined a measure of evenness as E = eH/S (2) where E is the measure of evermess or equitability. Hill (1973) showed that this measure of evenness, E, complies with his mathematical continuum of diversity measures. Now (2) can also be written as e" = SE. Taking the natural log (In) of both sides we obtain H = InS + InE (3) This decomposition formula (3) uses H not in the usual way as a diversity measure, but as a vehicle to partition H into components of species richness, S, and evenness, E (Buzas and Hayek, 1996; Hayek and Buzas, 1997). Equation (3) is incisive because, for the first time, researchers have a simple way to examine evermess separately from richness within a single multispecies system. First, consider that it is always true that 0 < E < 1. Therefore, the value of InE is always negative and (3) in- dicates the H is made up of InS minus the amount of even- ness exhibited by the population. Because the maximum value of H = InS, (3) could also be written as H = H,?^, + InE. When a natural population is repeatedly sampled without replacement, the number of individuals observed increases along with S until all the species inhabiting the area are accounted for or, more hkely, the researchers' resources are exhausted. The number of individuals, N, is, of course, cor- related with the area, A, sampled. Consequently, S is a func- tion of, or can be related to N and A; that is, the statements S = f(N) and S = f(A) are both true. Both of these rela- tionships can be fit with power curves which can be written as S = aN* or S = cA'', where a, b, c and d are constants. Often these equations are expressed in a linear form as InS = In a + b In N (4) or hiS = In c -I- d In A (e.g. Hanskj and Gyllenberg, 1997; Hayek and Buzas, 1997). Power curves when plotted on an arithmetic scale form a "hollow curve". That is, the curve rises sharply at the outset and then levels off, sometimes asymptotically, at some value of N or A. When plotted on a log-log scale, such curves form a straight line. While the relationship be- tween S and N or A has been long established (e.g. Preston, 1948, 1962; Mac Arthur and Wilson, 1967), the conse- quences for H and InE through consideration of (3) have not been fully explored. If we form a total for the number of species in two sam- ples, say SI and Sj, we expect, in their accumulation, that usually S, < S?+2) where 8,1+2) is the total in both. Consid- eration of (3) indicates that if InS increases during accu- mulation, then to satisfy equation (3) either H or InE or both must also change. Let us examine the possibilities. If two samples are accumulated so that S, < 5,1+,), then InS, < lnS(|+2)- In this situation if either H or InE remains constant, then equation (3) requires the non-constant mem- ber to vary exactly as InS. For example, if H, = H(,+2) and H, therefore, remains constant with accumulation, then to satisfy (3) InE must become more negative by exactly the same amount that InS becomes more positive. If a number of samples are accumulated and H remains constant, then InE must become increasingly negative as InS becomes in- creasingly positive. At each step in the accumulation pro- cedure equation (3) must be satisfied. Much of the increase in S is by the addition of rare spe- cies. In a natural population, more species will be repre- sented by one individual, then by two, three, and so forth. The inclusion of these rare species is necessary for a mean- ingful analysis and does not present any problem, as it would if the analysis depended on particular rare species. If a SHE analysis were conducted again over the same area, we would probably not encounter the same rare species rep- resented by, say, one individual, but would instead encoun- ter others represented by one individual. In like fashion, as N accumulates some individuals originally represented by one individual move into a category represented by two, and so on. However, in the process of accumulating N, pre- viously undiscovered species appear which are represented by one individual. One way a constant H over accumulated samples can oc- cur is with an increase in the proportion of the abundant species and/or a decrease in the proportions of rare species, either of which will result in a lower value of E (more dom- inance). A special case conforming to this situation is Fish- er's log series (Fisher, 1943). Bulmer (1974) showed that the expected value of H for the log series is H = Ina -I- 0.58 where a is Fisher's measure of diversity and 0.58 is Euler's constant. Now Fisher's a, the parameter of the log series, is also a constant for any particular log series and, consequently, for any log series H is a constant. Hayek and Buzas (1997) found a constant value of H, but not with the value expected for a log series, for a hectare of trees in BoUvia. Another possibility as samples are accumulated with a concomitant increase in S is that InE can remain constant. In this circumstance (3) tells us that H must increase pre- cisely as InS. A way of keeping E constant is for a decrease to occur in the proportions of the more abundant species as samples accumulate. A special case is Mac Arthur's (1957) broken stick model. May (1975) indicated that for the bro- ken stick model the expected value of H is H = InS -0.42. Note the resemblance of this equation to (3), indicating that InE = -0.42, which gives an E value of 0.66. Buzas and Hayek (1996) found a constant InE for a one hectare plot of trees in Guyana, but the value of InE was not the expected one for a broken stick model. Earher we indicated that the maximum possible value of H is H = InS. When this is true, then H/lnS = 1. When this is not the case the value of the ratio will be less than one, and defines another measure of evenness, J, introduced by Pielou (1966) as J = H/lnS. This can also be written as H = JlnS. May (1975) indicated that for a log normal dis- tribution (Preston, 1948) the expected value of H is H = (1 - ?y^)lnS where -y is a parameter (a constant) of the log SHE ANALYSIS 235 normal. A comparison of these results for H shows that J = (1 - y^); consequently for the log normal, J is a constant. With a little ajgebraic manipulation, in terms of (3), H = (1 + lnE/lnS)LiS. Consequently, this shows that for a log normal, the ratio InE/lnS is a constant. A way to achieve a constant InE/lnS in a multispecies population as species ac- cumulate is to have the proportions of the more abundant species get smaller. However, this reduction in species pro- portions will be less than when InE remains constant. Thus, the accumulation when InE/lnS stays constant will show a shght decrease in the evenness as measured by E. A constant InE/lnS was found for a one hectare plot of trees in Front Royal, Virginia (Hayek and Buzas, unpublished). In the sit- uation for which InE or E remains constant (as indicated in the previous paragraph), the ratio InE/lnS or J will show a shght increase. We have also encountered data sets where none of the above quantities are constant. Sometimes as S increases with accumulation, H increases slightly, and InE and InE/ InS decrease. One way this can be accomplished in nature is having the abundant species remain relatively constant while adding species represented by one individual (single- tons). As mentioned previously, we have observed instances where H or InE remain constant, but the values are not those expected for a particular distribution. This is because in eco- logical studies only three statistical distributions are most often used in data fitting. Hayek and Buzas (1997) discussed the interrelationships of these three (broken stick, log series, log normal). For more complete identification further math- ematical derivation is required to identify entropy (H) uniquely for additional members of this family of distribu- tions. In all of the situations described above, we have been referring to observations obtained from a relatively uniform habitat that is inhabited by a multispecies population, name- ly, a biofacies. As we accumulate N and S, the value of E or InE remains constant or decreases. However, as we ac- cumulate across biofacies this pattern will show disruption as we pass from one biofacies to another It is this disruption in pattern that we look for when establishing biofacies through SHEBI. NORTHEASTERN GULF OF MEXICO In her classic paper, Parker (1954) enumerated the fora- minifera from about 200 samples from 11 traverses in the northeastern Gulf of Mexico. We arbitrarily selected traverse 6 for SHE analyses. Although Parker (1954) enumerated live as well as total (live + dead) populations, we selected the total populations because of the much larger numbers observed. Strictly speaking, a biofacies can be considered as the biological component of a formation and is a geo- logical term. The living population represents a community in the ecological sense. However, it can also be argued that the total population gives a better representation of a fauna over ecological time and often resembles the live population anyway, taphonomic effects not withstanding (Buzas, 1965; Scott and Medioli, 1980; Loubere and Gary, 1990; Loubere et al, 1993). Consequently, we will refer to our analysis of the total population as an analysis of biofacies. For our pur- A 24m 55m 146m 24m 55m 24m 55m 146m FIGURE I. Plot,s of SHE analysis of Parker's (1954) traverse 6. Dashed lines show depths for particular values of accumulated InN. (A) MS vs InN, (B) H vs InN. (C) InE vs InN. poses the terms community and biofacies are regarded as synonymous (Schopf, 1996). In traverse 6, Parker (1954) occupied 35 stations ranging in depth from 20 to 2697 m. She presented her data as the percent of each species observed at each station. In all, 201 taxa were encountered in traverse 6. For the procedure fol- lowed here we converted the percent values back to the number of individuals observed. At the first station, 20m, the values are hiS = 2.89, H = 2.14, and InE = -0.75 (E = 0.47). After these values are calculated for the initial sam- ple in the traverse, a stepwise accumulation procedure is used for the remaining samples. That is, the number of in- dividuals for each unique species from the first and second station are sunmied and InS, H, and InE are recalculated. The number of individuals in the first, second, and third are then summed and InS, H, and InE are recalculated, and so on. Although tedious, the accumulation procedure is straightforward and can be carried out easily on any spread sheet package. We used QUATTRO (1994) for the calcu- lations. The results are shown in Fig. lA, IB, IC. Notice 236 BUZAS AND HAYEK TABLE 1. SHEBI analysis for Parker (1954) traverse 6. Depth is in meters, N is the number of individuals in the sample, lnN?,? is the In of cumulative N, Sobs is ihe number of species in the sample. S+ is the number of species not observed in the previous sample, S^?,? the cumulative number of species, lnS?i,? the In of S(??,, H is the Shannon information function for cumulative samples, InE the In of E, the Buzas and Gibson measure of evenness, and (I + I lnE/lnS?,?) is equal to J, Pielou's measure of evenness. (1 + Inlj Biofacies Depth N InN S.,h. s + s InS H InE E InS, ..1 1 20 250 5.52 18 18 18 2.89 2.14 -0.75 0.47 0.74 22 1798 7.62 46 29 47 3.85 3.00 -0.84 0.43 0,78 24 6700 9.08 39 2 49 3.89 2.92 -0.98 0.38 0.75 24 848 9.17 38 1 50 3.91 2.92 -0.99 0.37 0.75 2 29 375 5.93 31 31 31 3.43 2.87 -0.56 0.57 0.84 27 401 6.65 35 13 44 3.78 3.00 -0.79 0.45 0.79 24 1092 7.53 47 11 55 4,01 3.15 -0.86 0.42 0.78 39 1096 7.99 67 15 70 4.25 3.27 -0.98 0.38 0.77 43 273 8.08 32 2 72 4.28 3.28 -1.00 0.37 0.77 49 1693 8.50 44 1 73 4.29 3.16 -1.13 0.32 0.74 55 4191 9.12 36 2 75 4.32 2.88 -1.44 0.24 0.67 3 64 3307 8.10 58 58 58 4.06 3.16 -0.90 0.41 0.78 91 4291 8.94 70 21 79 4.37 3.37 -1.00 0.37 0.77 4 100 28955 10.27 79 79 79 4.37 3.59 -0.78 0.46 0.82 106 24349 10.88 73 12 91 4.51 3.63 -0.88 0.42 0.80 113 27972 11.30 71 8 99 4.60 3.66 -0.94 0.39 0.80 119 12U2 11.44 65 4 103 4.63 3.68 -0.96 0.38 0.79 128 18625 11.63 68 1 104 4.64 3.68 -0.96 0.38 0.79 139 23529 11.82 76 6 110 4.70 3.74 -0.96 0.38 0.80 146 27473 12.00 79 5 115 4.74 3.78 -0.97 0.38 0.80 5 155 9542 9.16 72 72 72 4.28 3.64 -0.64 0.53 0.85 165 8991 9.83 68 17 89 4.49 3.67 -0.81 0.44 0.82 183 13772 10,38 68 5 94 4.54 3.59 -0.96 0.38 0.79 6 223 2083 7.64 65 65 65 4.17 3.41 -0.76 0.47 0.82 446 7816 9.20 74 30 95 4.55 3.55 -1.00 0.37 0.78 555 3106 9.47 66 10 105 4.65 3.61 -1.05 0.35 0.77 631 2207 9.63 73 7 112 4.72 3.67 -1.05 0.35 0.78 650 1624 9.73 63 3 115 4.75 3.68 -1.06 0.35 0.78 677 1600 9.82 66 3 118 4.77 3.70 1.06 0.34 0.78 823 2502 9.95 67 6 124 4.82 3.74 -1.08 0.34 0.78 860 2104 10.04 62 1 125 4.83 3.75 -1.08 0.34 0.78 960 3603 10.19 70 3 128 4.85 3.73 -1.12 0.33 0.77 1144 5100 10.36 64 3 131 4.88 3.72 -1.15 0.32 0.76 7 1573 1301 7.17 56 56 56 4.02 3.15 -0.87 0.42 0.78 2697 595 7.55 48 20 76 4.33 3.40 -0.93 0.39 0.78 that InE shows considerably more variation than InS or H even though they are all related by (3). From our outline of the SHE relationship given above, we recall that for a mul- tispecies population, the value of InE should either be con- stant or decrease. This is clearly not the case. An increase in the value of InE indicates the addition of new species with enough relative abundance to increase the evenness. Consequently, the graph confirms what we already knew, namely, that more than one foraminiferal biofacies exists between the depths of 20 to 2697 ra. Fig. lA, IB, and IC show that for InS, H, and InE changes in slope occur at hiN = 9,17 (24m), InN = 9.83 (55m), and InN = 12.15 (146m), In addition, InE (Fig. IC) indicates the presence of at least three other places of demarcation. In general, InE is the most sensitive of the trio and we will use it extensively in what follows. However, the accumulations after many thousands of individuals may distort the appearance of the later bio- facies because of the influence in the accumulation proce- dure of species encountered earlier in the traverse which are no longer present at later depths. Consequently, we will break up the traverse and examine it piece by piece. Looking at Fig. IC notice that the first four stations have decreasing values of InE. We designate these four stations ranging in depth from 20 to 24 m as the first biofacies (first group in Table 1). The data from these stations are now eliminated from consideration. The fifth station (29 m) now becomes the first for the next SHEBI procedure. The results are shown in Fig. 2A which indicates that the first seven stations show a steadily decreasing trend for InE. These stations ranging in depth from 29 to 55 m are designated as biofacies 2. These stations are now deleted from the data set as we proceed with further calculations. The remaining stations beginning with Parker's 12th station (station 62 at 64 m, Parker, 1954) are now used to continue the SHEBI. Fig. 2B illustrates that two stations at 64 and 91 m display a set of decreasing values of InE, These are designated as biofacies 3. Deleting these stations, we re-accumulate and produce Fig, 2C. Here three stations show a decreasing trend fol- lowed by four more with a constant InE, and then a sharp decline after 146m, These seven stations ranging in depth from 100 to 146 m are designated as biofacies 4. Deleting these, we produce Fig. 2D which shows three stations before a sharp break in the slope. These stations, ranging in depth from 155 to 183 ra, are designated as biofacies 5. Proceed- SHE ANALYSIS B 237 8.CW 8,50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 -0.60 -0.70 -0.90 -0.90 \^ Biotacies 5 12.00 12,50 0.70 1144m 0.80 . ^^^^ Biofacies 6 0.90 ^^^^ 1.00 ^"\^ 860 m 1 1.10 555 m 1 FIGURE 2. Plots of InE vs InN with serial deletions for the accumulation procedure. Da,shed lines indicate depths at which biofacies are identified. (A) first 4 stations from Fig. 1 are deleted. (B) first 7 stations from Fig. 2a are deleted. (C) first two stations from Fig. 2B are deleted. (D) First 7 stations from Fig. 2C are deleted. (E) First 3 stations from Fig. 2D are deleted. ing in like fashion, we produce Fig. 2E, which shows three more stations before a slight change in slope at InN = 9.20 (555m). UnJiice the previous breaks, this one is more subtle. Another possible break is at InN = 10.04 (860m). However, neither of these are like the sharp dehneations experienced with biofacies 1 through 5. Additionally, the samples are quite widely spaced over a considerable depth range. Con- sequently, we took the conservative viewpoint and chose neither. Instead, we chose the upturn in InE at the second last station (InN = 10,40, 1573m). Biofacies 6 then consti- tutes the 10 stations from 223m to lJ44m, Finally, the last two stations are designated as biofacies 7 (1573 and 2697 m). We have then by this simple procedure designated 7 biofacies for Parker's traverse 6, Values are shown in Table 1, When referring to Table 1, the reader should keep mind that the accumulation procedure begins each time with the first entry in each biofacies, A comparison of biofacies obtained from SHEBI and Par- ker's generaUzed biofacies based on 11 traverses is shown in Fig, 3, The results are surprisingly similar considering the difference in criteria for discrimination. Parker chose biofacies based on the changes in the relative abundance of individual taxa while the SHEBI procedure examined changes in the multispecies structure of the total population. In shallower water, SHEBI indicates the presence of three biofacies while Parker only designated one. However, Par- ker did recognize a subfacies at 30-50 m and indicated con- siderable variability of relative abundance in shallow water, Parker recognized a boundary between 555 and 631m and while we noted a possible change in slope (Fig, 2E) at this location, we did not designate a biofacies. DISCUSSION Statistical distributions of naturally occurring multispe- cies populations (communities) exhibit a non-increasing lin- ear trend for InE vs InN. The criteria for biofacies identifi- cation using SHEBI is to identify a group of contiguous stations that exJubit such a linear trend. Departures from a non-increasing linear trend define the boundaries of the bio- facies. This criteria provides us with a quantitative definition of what constitutes a community. Unlike most ordination procedures that compare all pos- sible pairs of samples and sometimes produce uninterpret- able results, the SHEBI analysis assures continuity and mappability because all the areas recognized by SHEBI are 238 BUZAS AJND HAYEK Deptli (m) SHEBI Parker 20 22 24 1 24 29 27 24 1 39 2 43 49 55 64 91 100 3 106 113 119 4 2 128 139 146 155 165 5 3 183 223 446 4 555 631 650 6 677 823 5 860 960 1144 1573 6 2697 7 FIGURE 3. Comparison of biofacies obtained by SHEBI and Parker (1954) for traverse 6. contiguous. This contiguity of the samples does not imply that we must observe a trend with depth, only that SHEBI will uncover a series of biofacies with depth, if it occurs. We believe this is a most desirable property for biofacies analysis. On the other hand if the same biofacies repeats itself along a traverse, the SHEBI procedure would recog- nize only another additional biofacies. However, in all bio- facies analyses, regardless of the methodology used, the re- searcher should always examine the data to determine the constituency. The SHEBI procedure could not unequivocally discrimi- nate among the 10 stations distributed from 223m to 1144m. Parker (1954) recognized a boundary between 555m and 631 m in her scheme for the entire northeastern Gulf (Fig. 3). However, unlike the shallower stations, the stations deep- er than 183m are distributed over a very much wider geo- graphic area. The lack of replication and the wide geograph- ic range of the stations in this interval leaves the question of further subdivision unresolved. When enough samples are available within a biofacies, a SHE analysis will allow us to identify the properties or structure of the underlying statistical distribution. In other words, SHE analysis can be used for community structure identification (SHECSI). As an example of SHECSI, we will examine the three (2, 4, 6) biofacies with the most samples identified by SHEBI in traverse 6. Biofacies 2 (29 to 55 m, n = 7, Table I) is defined by a pattern of increasing InS and an accompanying decrease in values of both ln? and (1 -I- InE/lnS). The value of H increases and then decreases so that the final value at 75 species is nearly identical to the ini?al value at 31 spe- cies (Table 1). The regression coefficient b for the equation H = Ina + blnN is 0.05 indicating that the predicted values for H are nearly constant. Ideally, for a log series the ob- served values of H should remain constant. For a log series equation (3) requires that the values for the regression co- efficients for InS and InE should be nearly equal and of opposite sign. In our example, the former has a value of 0.29 and the latter -0.24, the difference of 0.05 is the re- gression coefficient for H. The predicted value of H for a log series is H = Ina + 0.58. With InN = 9.12 and S = 75, the value for a = 11.2 (see, Hayek and Buzas, 1997, for calculation procedure) which results in a predicted H = 2.99. The observed value at S = 75 is 2.88 (Table 1) and the average of all observed values in the biofacies is H = 3.09. These observations are consistent with a log series distribution for biofacies 2. Biofacies 4 with n = 7 is at depths of 100 to 146 m. The values of InS and H increase while the values of InE de- crease at the outset and then become constant (Table 1). The regression coefficient for H vs InN is 0.10 and for InE vs InN, -0. II. The values of (1 -I- InE/InS) are nearly constant throughout (Table 1). The regression coefficient for InE/hiS vs InN is -0.02. This constancy of InE/lnS is expected with a log normal distribution. The third example is biofacies 6 with n = 10 at depths of 223 to 1,144 m. As InS increases, InE decreases while H increases (Table 1). The regression coefficient for H vs InN is 0.13 and for InE vs InN, -0.14. The values for (1 + InE/ InS) shown in Table I are nearly constant and the regression coefficient for InE/lnS vs InN is -0.02. These results are also most consistent with a log normal distribution. The SHECSI analysis, then, allows the researcher to go beyond the designation of biofacies procedure of SHEBI and examine the underlying distribution. At the present time, however, the data are too scanty to speculate on an overall pattern or meaning. The main purpose of this paper is simply to outline the procedure for designating biofacies using the technique of SHEBI. SUMMARY 1. SHE analysis (Buzas and Hayek, 1996; Hayek and Buz- as, 1997) is a generaUzed methodological set of procedures for the study of multispecies population structure. SHE anal- ysis can be used for community structure identification (SHECSI) and biofacies identification (SHEBI). 2. A naturally occurring community consists of N individ- uals which are distributed among S species. The calculation of the informafion function, H, uses both the number of species and their relative abundances. H is decomposed by the equafion H = InS -I- InE where E = e^/S is a measure of evenness. SHE ANALYSIS 239 3. As samples are accumulated within a single community, N and S increase. The decomposition equation places con- straints on H and InE. Con:imunities have known statistical distributions that exhibit a non-increasing linear trend for InE vs ]nN. 4. The underlying premise when using the SHE approach for the identification of biofacies (SHEBI) is to identify de- marcations or departures from non-increasing linear trends. 5. Using SHEBI for the analysis of biofacies along some environmental gradient such as depth is a simple procedure. The procedure involves identifying a break and re-examin- ing the remainder of the data. We begin by subjecting the entire transect or traverse to a SHE analysis. A plot of JnE vs InN is examined for changes in a non-increasing linear trend. The first change is designated as a biofacies bound- ary. The station(s) occurring before the break are designated as composing the first biofacies and are then deleted from further consideration. The data set now consists of the re- maining stations. The analysis is repeated and the first change in slope is recognized as a second biofacies bound- ary. The station(s) occurring before the break are designated as biofacies two and are then deleted from the data set. The procedure continues until we run out of stations and the entire traverse is demarcated into ecological zones or bio- facies. 6. The SHEBI procedure differs from all other approaches because it identifies the zones or biofacies by examining the structure of multispecies populations. ACKNOWLEDGMENTS We thank J. Jett for her assistance with computation and graphics. J. A. Comiskey for the QUATTRO program and R. S. Carney, S. J. Culver and P. J. D. Lambshead for helpful comments on the manuscript. 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