doi:10.1016/j.gca.2004.12.002 Inverse methods for estimating primary input signals from time-averaged isotope proﬁles BENJAMIN H. PASSEY, 1, *THURE E. CERLING, 1 GERARD T. SCHUSTER, 1 TODD F. ROBINSON, 2 BEVERLY L. ROEDER, 2 and STEPHEN K. KRUEGER 3 1 Department of Geology and Geophysics, University of Utah, Salt Lake City, Utah 84112 USA 2 Department of Integrative Biology, Brigham Young University, Provo, Utah 84602 USA 3 Large Mammal Department, The Toledo Zoo, Toledo, Ohio 43614 USA (Received January 28, 2004; accepted in revised form December 7, 2004) Abstract—Mammalian teeth are invaluable archives of ancient seasonality because they record along their growth axes an isotopic record of temporal change in environment, plant diet, and animal behavior. A major problem with the intra-tooth method is that intra-tooth isotope proﬁles can be extremely time-averaged compared to the actual pattern of isotopic variation experienced by the animal during tooth formation. This time-averaging is a result of the temporal and spatial characteristics of amelogenesis (tooth enamel formation), and also results from laboratory sampling. This paper develops and evaluates an inverse method for reconstructing original input signals from time-averaged intra-tooth isotope proﬁles. The method requires that the temporal and spatial patterns of amelogenesis are known for the speciﬁc tooth and uses a minimum length solution of the linear system Am = d, where d is the measured isotopic proﬁle, A is a matrix describing temporal and spatial averaging during amelogenesis and sampling, and m is the input vector that is sought. Accuracy is dependent on several factors, including the total measurement error and the isotopic structure of the measured proﬁle. The method is shown to accurately reconstruct known input signals for synthetic tooth enamel proﬁles and the known input signal for a rabbit that underwent controlled dietary changes. Application to carbon isotope proﬁles of modern hippopotamus canines reveals detailed dietary histories that are not apparent from the measured data alone. Inverse methods show promise as an effective means of dealing with the time-averaging problem in studies of intra-tooth isotopic variation. Copyright © 2005 Elsevier Ltd 1. INTRODUCTION Intra-tooth isotope variation tracks seasonal patterns of iso- tope variation in animal systems (Koch et al., 1989; Fricke and O’Neil, 1996), and the study of such variation is becoming a fundamental tool for studying ancient seasonality in climate, ecology, and behavior (Fricke et al., 1998; Kohn et al., 1998; Fox and Fisher, 2001; Balasse et al., 2002; Zazzo et al., 2002). The intra-tooth method involves sampling teeth along their growth axes; samples taken near the occlusal surface record an earlier portion of an animal’s life, those taken near the root record a later portion, and a series taken in between records isotopic variation in the animal for the duration of tooth for- mation. Already this method has led to a signiﬁcant body of research in several ﬁelds: selected applications include the reconstruction of herd-management strategies of ancient pasto- ralists and seasonal environments in which ancient humans lived (Balasse et al., 2002), examination of cohort structure of fossil mammals (Zazzo et al., 2002), detection of seasonal dietary variation and migration in fossil mammals (Koch et al., 1998; Hoppe et al., 1999; Gadbury et al., 2000; Feranec and MacFadden, 2000), and estimation of oxygen isotope season- ality associated with climatic and tectonic change (Fricke et al., 1998; Kohn et al., 2002). It has recently become apparent, however, that intra-tooth isotope proﬁles in tooth enamel are time-averaged compared to the actual pattern of isotopic variation—the input signal— experienced by animals during tooth formation (Fox and Fisher, 1998; Balasse, 2002; Passey and Cerling, 2002). Tooth enamel is the material of choice for isotopic studies of fossil mammals because of its resistance to isotopic alteration (Lee- Thorp and van der Merwe, 1987; Quade et al., 1992). It provides a time-averaged isotopic record because, during amelogenesis, each layer of enamel is deposited as a mineral poor matrix that gradually accumulates mineral over an ex- tended period of time. Numerous studies of tooth enamel de- velopment in rats, pigs, steers, humans, and other animals show that the initial enamel matrix contains, by volume, a fraction (20 –30%) of the total hydroxyapatite content of mature enamel, and that there is a spatial maturation zone along the growth axis of the tooth where the enamel accumulates the remaining mineral (Hiller et al., 1975; Robinson et al., 1978; Suga et al., 1979; Deutsch and Pe’er, 1982; Sakae and Hirai, 1982; Suga, 1982; Robinson et al., 1987; Robinson et al., 1988; Passey and Cerling, 2002). This maturation zone often corre- sponds to a large portion of the overall crown height. For example, the lengths of the maturation zone (length of matu- ration) for cow and bison lower ﬁrst molars studied by Passey and Cerling (2002) were at least 15 mm and 30 mm, respec- tively. The total crown height for these teeth is on the order of 35 mm to 45 mm, and taken with a tooth crown formation time of 1y(Brown et al., 1960), any volume of enamel will have isotopic composition that is the time average of several months (Balasse, 2002). Passey and Cerling (2002) developed a series of equations that serve as a forward model allowing prediction of how primary input signals will appear as measured tooth enamel signals for different maturation parameters and sampling ge- ometries. The model is speciﬁc to ever-growing teeth and is a type of running average that predicts that features of the input * Author to whom correspondence should be addressed (bpassey@ mines.utah.edu). Geochimica et Cosmochimica Acta, Vol. 69, No. 16, pp. 4101– 4116, 2005 Copyright © 2005 Elsevier Ltd Printed in the USA. All rights reserved 0016-7037/05 $30.00 + .00 4101