Summary of Algorithm of Mann et al (1998). The summary here follows details
provided in Mann et al (1998), including the "Methods" appendix and
supplementary information. Some additional methodological details that were
not possible to include in the "Methods" section of the original article, due
to space restrictions, are provided here.
Note that an additional, independent application of the methodology of MBH98
can be found in the following publication:
Zorita, E., F. Gonzalez-Rouco, and S. Legutke, 2003: Testing the Mann et al.
(1998) approach to paleoclimate reconstructions in the context of a 1000-yr
control simulation with the ECHO-G Coupled Climate Model, J. Climate, 16,
1378-1390.
STEPS IN MANN ET AL (1998) ANALYSIS
1. Initial Processing of Instrumental and Proxy Data
Prior to analysis, small gaps in proxy series during the latter part of the
calibration period (between 1972 and 1980) were filled by Mann et al (1998).
The analysis is insensitive to this step, as nearly identical results are
obtained through the use of a 1902-1971 calibration period (Mann et al,
in review).
Surface temperature anomaly [Jones et al (1995) version of the CRU
instrumental surface air temperature dataset, available from 1854-1993,
represented as anomalies from 1951-1970 base period] with nearly continuous
monthly sampling were used. For the verification data,
gridpoints with a single gap greater than 24 months in duration, or with
more than 120 total missing months of data over the period 1854-1980 were
eliminated. This yielded 219 gridpoints back to 1854. For the calibration
data, gridpoints with a single gap greater than 36 months over the period
1902-1980 were eliminated. This yielded 1082 calibration gridpoints back
to 1902. Remaining temporal gaps within these monthly anomaly
series were filled through linear interpolation.
The Northern Hemisphere mean series through 1998 used in Mann et al (1998)
was based on updated values of the Northern Hemisphere mean series through
1998 from the CRU website.
All predictors (proxy and long instrumental and historical/instrumental
records) and predictand (20th century instrumental record) were standardized,
prior to the analysis, through removal of the calibration period (1902-1980)
mean and normalization by the calibration period standard deviation. Standard
deviations were calculated from the linearly detrended gridpoint series, to
avoid leverage by non-stationary 20th century trends. The results are not
sensitive to this step (Mann et al, in review).
2. Dimensional Reduction of Data Sets Through Principal Component Analysis
A conventional Principal Component Analysis (PCA) was performed (through
Singular Value Decomposition of the data matrix) on the monthly
instrumental surface temperature gridpoint data over the interval 1902-1993
(1104 months), standardized as described in "1." Only the leading 16
eigenvectors of the instrumental record (resolving roughly 50% of the total
annual mean variance in the instrumental surface temperature field, but a
considerably greater fraction of roughly 95% of the Northern Hemisphere
annual mean variance) were retained for subsequent analysis.
PCA was also performed on certain proxy sub-networks (spatially dense
regional networks of tree-ring data available separately in different
continents) as means of dimensional reduction of the predictor network. In
this case, the procedure was performed separately for each independent step
of the stepwise calibration/reconstruction procedure described in "3"
below. A decreasing number of PCs of these sub-networks are retained
increasingly further back in time, as dictated by application of objective
selection criteria (consideration of results of both Preisendorffers Rule N
and Scree test). PCs were no longer calculated back in time once a given
network contained fewer than 7 available series (with the exception that PCs
were calculated for the 'Stahle Southwest U.S./Mexico network' with 6 series
available). Thus, although some series may be available further back in time,
they may not have been used to calculate PCs. For example,there are 110
series available back to 1400, but only 95 are used because PCs were not
calculated on 6 Australian and 6 South American ITRDB series and 3 'Vaganov'
series. In total, 415 individual (proxy, and long historical or instrumental)
series were used as predictors, but represented, through the above process,
as a smaller network of indicators (112 available back to 1820, and
successively fewer available further back in time).
3. Calibration/Surface Temperature Pattern Reconstruction
The statistical calibration procedure is described in detail in the "Methods"
section of Mann et al (1998). The procedure involves a calibration procedure,
at annual resolution, in which each predictor is expressed in terms of an
optimal linear combination of the retained instrumental PCs (annual means of
the monthly mean instrumental PC series described in "2" were used for this
purpose).
This calibration procedure is followed by a reconstruction procedure
involving the solution of an inverse problem to determine the loadings on the
subset of retained annual PCs of the instrumental surface temperature record
most consistent with the values of the various standardized proxy indicators.
This inverse problem is performed on a yearly basis, back in time. This latter
procedure yields optimal estimates of the time histories of the retained
instrumental surface temperature PCs (which are termed reconstructed PCs or
'RPC's) back in time, allowing an (implicitly spatially smoothed)
estimate of the surface temperature field to be reconstructed through an
appropriate expansion in the retained eigenvectors. In this process, the RPCs
were scaled to have same variance as the annual mean instrumental PCs over
the calibration period (1902-1980). However, the results are insensitive to
whether or not this latter step is performed (Mann et al, in review). The
proxy series were multiplied, prior to PCA, by a weighting factor that
attempts to account for redundancy of predictors (for example, we
downweighted multiple series from a common site to preserve the relative
weight of estimates from different sites). Our results are insensitive to
whether or not these weighting factors are employed in the analysis.
This procedure was performed iteratively, in a stepwise matter back in time,
involving the separate use of 11 successively less widespread networks of
indicators available back to 1820, 1800, 1780, 1760, 1750, 1730, 1700, 1600,
1500, 1450, and 1400 as described in Mann et al (1998) [see supplementary
information therein] to make as full use as possible of the predictors
available back in time.
A decreasing number of PCs were retained (i.e., RPCs reconstructed)
increasingly further back in time, as dictated by application of objective
selection criteria as described in "2" above. A maximum of 11 of the total
(16) instrumentalsurface temperature eigenvectors were retained (see
supplementary information of Mann et al, 1998), decreasing back in time.
Prior to AD 1450, only a single eigenvector (PC#1) was retained.
It should be noted that the hemispheric mean reconstructions of Mann et al
(1998) relies largely on the PC#1 of the instrumental surface temperature
record, and is relatively insensitive to use of higher order PCs. By contrast,
the spatial details of the reconstructions exhibit a greater dependence on
the particular combination of PCs used.
4. Statistical Verification
An essential step in the procedure of Mann et al (1998), as described therein,
was the use of conventional verification procedures to establish the level of
skill in the proxy-based surface temperature reconstructions. Verification
estimates based on correlation and Reduction of Error ('RE' or, 'beta' in the
language of Mann et al, 1998) were established for each of the 11 separate
procedures contributing to the stepwise reconstruction procedure, based on
comparison of the proxy reconstructions. It should be noted that the
verification estimates for the "Northern Hemisphere mean" series were based
on a 'sparse' hemispheric mean calculated from the 219 gridpoints available
back to 1854, and not the full hemispheric mean of the 1082 instrumental
temperature gridpoints only available back to 1902. Longer verification skill
estimates were performed for a smaller number of (11) temperature gridpoints,
based on long instrumental surface temperature records available back to at
least 1820.
It should be stressed that reconstructions that did not pass statistical
cross-validation (i.e., yielded negative RE scores) were deemed unreliable.