Motivation

Much of the work in the rapidly growing field of pa-leoclimatology has emphasized the need to put recently observed secular climatic trends and shifts within the context of natural low-frequency variability (Martinson et al., 1995). For example, evidence for (1) global warming since the middle of the last century (Hansen and Lebedeff, 1988); (2) variations in the amplitude, duration, and frequency of El Niño / Southern Oscillation (ENSO) events (Trenberth and Shea, 1987; Trenberth and Hoar, 1996); (3) apparent state changes in the Pacific basin climate (Ebbesmeyer et al., 1991) and the timing of the Indian Monsoon (Parthasarathy et al., 1991; Krishna Kumar et al., 1999); and (4) severe mountain glacier retreat (Diaz and Graham, 1996) suggest changes in the base state of the global climate since the beginning of the Industrial Revolution. However, the processes underlying these observations are not very well understood, and some may indeed be part of quasi-periodic or low-frequency phenomena.

It has been hypothesized that many of these observations are tied to a small number of preferred, natural modes of spatial and temporal variability within the ocean-atmosphere system. Together these modes may explain much of the modern large-scale, low-frequency variance in the climate system (Wallace, 1996a,b). Examples of large-scale climatic processes are ENSO, the Pacific-North American Oscillation (PNA), and the North Atlantic Oscillation (NAO) patterns of standing atmospheric pressure waves, decadal variability in surface and intermediate circulation of the subtropical and tropical oceans (Chang et al., 1997), and globally homogeneous trends (Wallace, 1996b; Cane et al., 1997). Since these phenomena have long temporal and spatial scales, variability associated with them may be successfully retrieved from sparse observations (Miller, 1990; Cane et al., 1996). For instance, limited observations in the eastern equatorial Pacific are sufficient to crudely indicate the occurrence of an ENSO event, albeit with large uncertainties (Kaplan et al., 1998). In this chapter, we test implicitly the hypothesis that a sparse observational network composed of proxy paleoclimatic data (Epstein and Yapp, 1976; Fritts, 1976; Evans et al., 1998) may be used to reconstruct the major features of large-scale climate variability. The focus of this chapter is the development of a methodology by which we may seek skillful and verifiable reconstructions of these dominant modes using proxy data sources.

4.1.2. Proxy Climate Data and Their Interpretation

In the absence of direct physical observations, the science of paleoclimatology provides climate information through the measurement of proxies. In a broad sense these are parameters that vary in some understood way with the climatic variable of interest. Proxy measurements made on tree rings, ice cores, reef corals and sponges, sediment cores, and other geological and biological archives provide qualitative to semiquantitative estimates of climate for decades to millennia prior to the rise of direct observations (Bradley and Jones, 1992; Jones et al., 1996). Ideally, proxy observations may be used to produce what we term here a climate field reconstruction (CFR): a near globally complete, verifiable, and temporally continuous estimate of climate variability over some period of interest. Once we have the CFR, it is straightforward to obtain temporal indices averaged over some subarea of the global spatial domain.

The broader implications of CFR from paleoclimatic data are difficult to overestimate. Traditionally in pale-oclimate research, "reconstruction" indicates recovery of a single time-varying climate index, usually of a local nature (such as seasonal air temperature from tree-ring-width chronologies) or having an immediate physical connection (such as NINO3 reconstruction from any coral site affected by ENSO). The interpretation of proxy reconstructions, usually in terms of a linear or linearized function of desired instrumental quantities, is generally verified by means of comparison with local instrumental data over a short, recent common period, although verification is increasingly made via comparison with regional or synoptic-scale observations over longer time periods. When the proxy data are contemporaneous with direct observations, opportunities are provided for intercomparison of proxy and instrumentally observed data. For example, intercomparison of independently derived proxy time series may be used to determine common features of cross-site variability. Potential biases in proxy-based records of sea surface temperature (SST) variability (age uncertainties, sampling artifacts, or other noncli-matic information) are independent of those in other proxy data or those in the bank of historical observations (instrument calibration, precision, accuracy, measurement method, frequency and spacing of sampling, and analysis procedures). In both cases, we hypothesize that common features observed across proxy realizations and in the direct observations are indicative of large-scale climatic phenomena and not small-scale or nonclimatic effects; if the instrumental and proxy data describe climatic phenomena, then they should agree (Jones et al., 1998; Evans et al., in revision; Villalba et al., 2000; Briffa and Osborn, 1999). Such intercompari-son efforts are essential for development, calibration, validation, and interpretation of individual data sets, but they may also be used to determine the potential for the development and interpretation of CFRs from proxy data (Evans et al., in revision). They provide an important link between research in climate dynamics and paleoclimatology.

Hence, we believe the most effective use of paleocli-matic data for the study of climate dynamics is via CFR based on consideration of all available proxy information, and for the reconstruction of patterns of variability, rather than point measurements. This is the idea of globality (Kaplan et al., 1997, 1998; Mann et al., 1998, 2000; Evans et al., 1998, 2000). Our view of the role of CFR as a natural link between data (proxy or instrumental) analysis and the study of climate dynamics follows the pioneering work of Fritts et al. (1971) and Fritts (1976, 1991) and is illustrated schematically in Fig. 1. Fritts provided a lucid and comprehensive introduction to the CFR concept within the context of dendroclimatological reconstruction of air temperature, sea level pressure (SLP), and precipitation anomalies over North America. More recently, the potential use of CFR on a global scale has been discussed by Jones and Briffa (1996). Bradley (1996) and Evans et al. (1998) have examined the observational array design problem, considering CFR from proxy data of varying quality and quantity. The successful extraction of variability in large-scale, low-frequency climatic phenomena from single point proxy data sources or small observational networks (Cook 1995; Wiles et al., 1998; Stahle et al., 1998; Cook et al., 2000; Evans et al., 2000; and others) has shown the potential for CFR from subsets of the available proxy paleoclimatic database. In a recent application, Mann et al. (1998, 2000) obtained very encouraging CFR results by applying space reduction and statistical techniques related to Smith et al. (1996) and Kaplan et al. (1998) for the reconstruction of surface temperature fields from several proxy data sources (Bradley and Jones 1992).

4.1.3. Globality and Optimality

The field of inverse modeling provides tools particularly suited for extraction of maximum information from a sparsely sampled, smoothly varying field. The term Objective Analysis (OA) encompasses a set of inverse methods employed in the estimation of the bestfit field (here in a least-squares sense) to both a sparse observational network of data and a description—a

FIGURE 1 Methodology of climate field reconstruction (CFR) from proxy data. Numbered arrows on the diagram correspond to the steps in our CFR procedure (see Section 4.2).

model—of how the field varies. Errors are admitted in both the observations and the model. A cost function, consisting of the appropriately weighted squared errors in the observations and the model, is then formulated. Minimization of the cost function with respect to the field variable produces the analyzed field that is consistent with both the data and the model, given prior estimates of how precisely the model and observations are known. The error in the OA-analyzed field produced in this manner is a function of the observational error, the model error, and the extent to which the field is resolved by the observations and can be compared to either a withheld set of data for validation or to the data used in the assimilation to check the assumed observational error magnitudes.

The OA techniques have become a powerful tool used in the estimation of atmospheric and oceano-graphic fields of interest from spatially and temporally incomplete observations and imperfect models (Bennett, 1992; Wunsch, 1996). Special reduced space analogs of these techniques have been developed to reconstruct fields from sparse sampling in space and time when the fields are sufficiently smooth and have coherent, large spatial, and long temporal-scale patterns (Cane et al., 1996; Kaplan et al., 1997). Here, we apply these same tools to the reconstruction of climate fields from the even sparser but rapidly growing network of annual-resolution paleoclimate records, in order to produce CFRs from proxy data. By the reduced space rationale, we seek reconstruction of only the features of climate variability that are expected to be resolved, given the quality of the observations. Bringing the CFR problem into the context of reduced space OA provides four benefits:

1. The solution is optimal, provided the a priori assumptions hold; that is, if the data are unbiased and the covariance of a priori errors in the observations and model are correctly estimated, then the solution obtained minimizes the squared error in the reconstructed field.

2. Space reduction emphasizes common features expressed across the proxy data set, which we expect to be climatic in origin, and discounts both climatic and proxy variability that explains little variance and is most likely to be dominated by observational errors.

3. Theoretical error estimates are provided for the solution.

4. The a priori assumptions, space reduction choices, solution, and error estimates can be tested for mutual consistency.

We illustrate this approach using the candidate data set described by Villalba et al. (2000) for the reconstruction of gridded SST for the Pacific basin over the time interval 1001-1990. The procedure is given in detail later. We focus here on the methodology of objective paleoclimatic reconstructions; the actual reconstructions shown here, which are based on a very limited number of tree-ring indicators, are not intended for interpretation. Future work will apply the methodology described here to more extensive sets of proxy indicators. Instead this methodological work is intended to complement the work of others, such as Villalba et al. (2000), who seek to characterize and interpret the climate information content of North and South American Pacific coast tree-ring chronologies. In Section 4.2, we describe the reduced space, OA CFR methodology. Section 4.3 describes the application of the technique to the reconstruction of near-global SST anomaly fields from selected Pacific-American tree-ring-width chronologies. We discuss the character of the climate information provided by tree-ring-width chronologies and conclude (Section 4.4) with some general comments on future applications of the procedure.

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