## Info

Tropospheric Temperature Signal

Water Vapor Signal

Stratospheric Temperature Signal

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Fig. 1 Spectral infrared signals corresponding to calculus of feedbacks. The tropospheric temperature signal shows how the troposphere cools itself; the carbon dioxide signals shows the spectral fingerprint of radiative forcing by carbon dioxide; the water vapor signal shows the spectral fingerprint of the water vapor-longwave feedback. The units for the radiance trend for all plots in this figure are W cm-2 (cm-1)-1 sr-1 decade-1

carbon dioxide increase. Because water vapor inhibits outgoing longwave radiation with time, as can be seen by the sign of the integral of its signal over frequency in Fig. 1, it is associated with a positive longwave feedback. In clear skies, we seek to model the trend in the emitted infrared spectrum as a linear combination of these four signals while allowing for some uncertainty in the modeled shape of these signals. The quality of the posterior fit is determined by comparison of post-fit residuals to the natural variability in the emitted infrared spectrum on interannual time scales. The mathematical technique is just the same as that used in climate signal detection and attribution studies (Allen et al. 2006) with an allowance for signal shape uncertainty (Huntingford et al. 2006). The "model" for the linear trend in the emitted infrared spectrum dd/dt = dFvLW/dt is

where the si are the spectral shapes given in Fig. 1, the dai /dt are scalar estimators of the trends of outgoing longwave radiation associated with individual variables, and the 5n are realizations of interannual variability in the tropics (30°S to 30°N) as they would appear in the annual average emitted infrared spectrum. The solution for the trend estimators dai /dt is given by da d FLW

dt dt where the columns of the matrix F are the components of the contravariant basis to the fingerprint basis established by the s;. As a consequence, FTS = I where the columns of S are the s;, and so we call F the set of contravariant fingerprints. Consistent with Bayesian inference (Leroy 1998), optimal methods (Bell 1986; North et al. 1995), and a geometric approach (Hasselmann 1997), the contravariant fingerprints are given by

where £ is a covariance matrix describing the statistics of natural variability and uncertainty in the shapes s;:

The contributions of natural variability and signal uncertainty must be evaluated differently because of their different natures. Natural variability influences a measured trend in the emitted infrared spectrum simply because any timeseries of a random phenomenon yields a nonzero residual trend. If the covariance of natural variations in the annual average emitted infrared spectrum is £$n, then the residual trend has zero expected mean but an uncertainty of £dn/dt:

where N is the number of years in the continuous timeseries and t = lyr for no serial correlation (see Eqs. (6) and (7) in Leroy et al. 2008b). The covariance of natural interannual variability is evaluated using a long control run of a climate model in conjunction with a forward model for the emitted infrared spectrum. On the other hand, the covariance of signal shape uncertainty must be evaluated using a large ensemble of climate models each of which can be used to determine its own set of signal shapes s;. Because we are interested in trends of spectrally integrated outgoing longwave radiation, for each model used to derive s;, the signals are normalized such that the spectral integral of s; multiplied by n (to account for integration over solid angle) is unity. Then the signal shape uncertainty covariance is

where the {■ ■ ■) denotes an ensemble average over a large number of models and the da; /dt are prior estimates of the trend in outgoing longwave radiation associated with signal i. The contravariant fingerprints are then obtained by substituting the expressions for £dn/dt and £S in Eqs. (9) and (10) into Eq. (8) and in turn into Eq. (7). When the contravariant fingerprints are multiplied by annual average infrared spectral anomalies, the result will be the outgoing longwave radiation (OLR) anomalies associated with the prescribed feedbacks.

Ordinary error estimation (for just one signal s instead of multiple signals S) dictates that the posterior uncertainty covariance for the OLR trends associated with the feedbacks should be

but too often prescriptions of natural variability are grossly different from reality. Consequently, a better estimate of the posterior error should be obtained from the data alone. This is done by ordinary linear regression on a detector timeseries a(t) (cf. Eq. (15) below). With the timeseries a(t), the error is determined first by estimating the natural variability in the detectors which is the variance of the a(t) after removal of a best linear fit afit(t). The uncertainty in the trend da/dt due to natural variability becomes

where (At)2 = J2ilitt - t)2/N is the variance of the coordinate times in the time-series. The timeseries a(t) contains only variability related to natural variability and no uncertainty due to signal shape uncertainty, so the latter must be added separately. By standard error propagation techniques, aja/At(signal shape) = FT£sF (13)

and thus the error in the forecast trend is al*/dt = ad2a/di(natural variability) + CTdl/di(signal shape). (14)

If natural variability were correctly prescribed by that used in composing the con-travariant fingerprint F, then CTd2a/di (natural variability) = FT£dn/diF and the result becomes exactly that in Eq. (11).

To demonstrate the viability of this approach to linear regression, we have computed the contravariant fingerprints F using the output of several models of the World Climate Research Programme's (WCRP's) Coupled Model Intercomparison Project (CMIP3) multi-model data set, subjected to SRES-A1B forcing. Case A1B of the Special Report on Emission Scenarios (SRES) predicts radiative forcing of climate in a world of rapid economic growth, rapid technological growth, increasing social interaction, and decelerating population growth (Intergovernmental Pabel on Climate Change (IPCC) 2000). It features approximately 1% yr-1 CO2 increase to « 720 ppm and radiative forcing by sulfate aerosols peaking in year « 2020. We take annual averages of emitted infrared spectra based on monthly average output and average over the tropics. The signals si are estimated based on the first 50 years of output. We then computed 20 years of emitted infrared spectra from a climate model independent of those used to construct the contravariant basis.

Tropospheric Temperature Signal

Tropospheric Temperature Signal

Water Vapor Signal

Water Vapor Signal

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Stratospheric Temperature Signal

Stratospheric Temperature Signal

Carbon Dioxide Signal

Carbon Dioxide Signal

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S 10 15 20

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Fig. 2 Detection amplitude timeseries for four signals. The solid squares show detection amplitudes for each of four detected signals and the open squares show true OLR anomalies for each of the four signals. The thin solid line is the best linear fit to the detection amplitudes. The "truth" data set is taken from the first 20 years of output of an SRES-A1B run of GFDL CM2.1

We multiplied the contravariant fingerprints by tropical average, annually averaged emitted infrared spectra from that climate model. The result is a timeseries of detectors a(t):

The result is shown in Fig. 2. In both the "truth" (open squares) and analysis (solid squares) there is variability from year to year. This variability contributes in large part to the length of time required to elapse before useful climate model testing can take place. In the case of greenhouse forcing by carbon dioxide, it is evident from the small fluctuations associated with interannual variability that direct observation of anthropogenic radiative forcing of the climate should be detected and strongly constrained within just a few years. After 5 years of observation, in fact, an estimate of radiative forcing by carbon dioxide with just 20% uncertainty should be obtained. Detection of tropospheric temperature trends (climate response) and longwave suppression by water vapor requires more time because of the large fluctuations associated with interannual variability. After 20 years of observation, an estimate of the water vapor-longwave feedback in the tropics with ~ 50% uncertainty can be obtained by trend analysis.

Figure 2 suggests a different analysis as well. The year to year anomalies of the tropospheric temperature and water vapor signals are strongly anticorrelated. This is related to the simple fact that tropical tropospheric water vapor increases and blocks surface radiation in years when the tropical troposphere is warm following the Clausius-Clapeyron equation. The slope of this correlation then can be used to estimate the water vapor-longwave feedback. In fact, such an anomaly correlation analysis can be used to estimate the water vapor-longwave feedback in the tropics with 7% uncertainty in ten years. The uncertainty scales as (At)-1/2 for anomaly correlation analysis whereas the uncertainty scales as (At)-3/2 for trend analysis, with At the time baseline of the continuous timeseries of data.

Actual spectral longwave data, though, are dominated by clouds, and thus the use of GNSS radio occultation (RO) is likely to be necessary. Leroy et al. (2006) have shown that the optimal fingerprint of climate change in upper air dry pressure—dry pressure is the atmospheric pressure derived from GNSS RO data under the assumption of a completely dry atmosphere—is poleward migration of the mid-latitude jet streams in both the Northern and Southern Hemispheres. In Fig. 3 we show the results of the application of the methodology described by Eqs. (6) through (9) when applied to zonal average, annual average log-dry pressure (instead of infrared spectra Flw) as might be obtained from GNSS RO data normalized by the surface air temperature trend dT/dt.

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Fig. 3 The top plot shows the contravariant fingerprint (F) for log-dry pressure as an indicator of surface air temperature trends. The lower plot shows the result of the application of this approach using detectors, with a 20-yr timeseries of dry pressure "data" taken from the output of a model not used in the construction of the contravariant fingerprint. The black curve shows actual global average annual average surface air temperature, and the red curve shows the detectors a(t). The red-shaded area from years 2020 to 2050 shows the forecast trend of surface air temperature based on the simulated upper air dry pressure data from 2000 to 2020, and the gray curve shows the actual evolution of the surface air temperature

2010 2020 2030 2040 2050 Year

Fig. 3 The top plot shows the contravariant fingerprint (F) for log-dry pressure as an indicator of surface air temperature trends. The lower plot shows the result of the application of this approach using detectors, with a 20-yr timeseries of dry pressure "data" taken from the output of a model not used in the construction of the contravariant fingerprint. The black curve shows actual global average annual average surface air temperature, and the red curve shows the detectors a(t). The red-shaded area from years 2020 to 2050 shows the forecast trend of surface air temperature based on the simulated upper air dry pressure data from 2000 to 2020, and the gray curve shows the actual evolution of the surface air temperature

The contravariant fingerprint F and the detector timeseries a(t) are used to infer past and predict future surface air temperature trends given zonal average, annual average log-dry pressure data from GNSS RO only. The contravariant fingerprint is the map by which one convolves the trends in data to obtain a trend in surface air temperature:

where 0 is latitude and h is height, the coordinates of the map. The contravariant fingerprint has dimensions of surface air temperature per log-dry pressure per height interval per solid angle interval on the Earth's surface. The intervals are determined by the data grid. The slope of the detector timeseries is used to infer and predict the surface air temperature trend and its related uncertainty.

As can be seen in Fig. 3, when upper air log-dry pressure is used as an indicator of trends in surface air temperature, the fingerprint searches for poleward migration of the mid-latitude jet streams, a tropical contribution that involves subtraction of upper tropospheric temperature trends from lower tropospheric humidity trends, and a possible weakening of the southern stratospheric polar vortex. Poleward migration of the mid-latitude jet streams is the leading indicator of climate change in the tropospheric upper air (Leroy et al. 2006). In this application, the detectors capture the interannual fluctuations of global average surface temperature with accuracy < 0. l K, meaning GNSS RO can be relied upon to obtain an accurate estimate of the dT/dt that is necessary for estimating radiative feedbacks. This is an improvement over the use of in situ meteorological stations which are largely restricted to land.

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