## 1. Introduction

Monthly and seasonal mean time series of meteorological variables are widely used for analyzing interannual climate variability and predictability, using either observed time series or the output of general circulation models (GCMs). Studies of predictability and potential predictability are usually based upon decomposition of temporal variability into a part called the “weather noise” variability that is fundamentally unpredictable on seasonal timescales and another part assumed to be at least potentially predictable (Madden 1976). The potential predictability is often measured as the fraction of the total variability accounted for by the latter part. Such measures are clearly sensitive to how the separation of variance is performed. In many cases, temporal filtering techniques are employed on the assumption that weather noise operates mainly on timescales much shorter than that of the potentially predictable variability (e.g., Basher and Thompson 1996). However, weather events include not only high-frequency, day-to-day fluctuations but also low-frequency intraseasonal fluctuations that give rise to chaotic, unpredictable fluctuations in seasonal-mean time series. Therefore, it is not possible to completely isolate the potentially predictable variability through temporal filtering.

A framework useful for analysis of climate variability and predictability is to consider seasonal mean time series to be made up of the following three components:a forced component, an internal source component, and a residual weather noise component (e.g., Lorenz 1970;Leith 1973; Zwiers 1996; Zheng and Frederiksen 1999). The forced component represents the atmospheric response to gradual changes in the atmospheric composition, particularly in greenhouse gas concentration (e.g., Manabe et al. 1991, 1992) and to slowly varying boundary conditions such as sea surface temperature (SST) anomalies (Bjerknes 1969; Rowntree 1972; Horel and Wallace 1981; Lau and Nath 1994; Shukla 1998) and anomalous sea-ice cover (Honda et al. 1999). The internal source component represents low-frequency atmospheric variability induced by atmospheric internal dynamics. The equatorial stratospheric quasi-biennial oscillation is a particularly striking example (Holton 1992). Interannual variability in the extratropics associated with stationary circulation anomalies, such as the North Atlantic oscillation (NAO; van Loon and Rodgers 1978; Wallace and Gutzler 1981; Kushnir and Wallace 1989) and the Pacific/North American (PNA) pattern (Wallace and Gutzler 1981; Kushnir and Wallace 1989), may be influenced by internal dynamics including persistent feedback forcing from synoptic-scale eddies traveling along anomalous storm tracks (Lau 1988; Lau and Nath 1991; Hurrel 1995). Extratropical remote response to the El Niño–Southern Oscillation (ENSO; e.g., Rasmusson and Carpenter 1982; Cane 1986; Kleeman and Moore 1997) may also be modulated by such a feedback from transient eddies (Hoerling and Ting 1994).

The forced and internal source components of the atmospheric variability are, by nature, (nearly) constant within a season, whereas the weather noise component includes fluctuations over a wide range of frequencies. The weather noise component should average to zero if sampled over a very large number of realizations, but not necessarily so if averaged over a single season. Since the power spectrum of large-scale atmospheric motions is fundamentally red, a nonzero seasonal mean of the weather noise component is largely contributed to by intraseasonal weather events with monthly or longer timescales, including persistent blocking events in the extratropics and the Madden–Julian oscillation (MJO) in the Tropics (Madden and Julian 1971, 1972). Transient eddies along storm tracks, whose characteristic timescales are less than 10 days, tend to exert only a small direct influence upon the weather noise component in monthly or seasonal means (Blackmon et al. 1984), although, as mentioned earlier, their indirect influence may be significant through their feedback on disturbances with lower frequency. An important difference between the forced and internal source components is that the latter are very sensitive to initial conditions while the former are not.

The above variance decomposition may apply in most areas of climate variability research. An important application may be found in the assessment of potential long-range predictability of the atmospheric circulation, that is, the predictability of seasonal means a season or more in advance (Madden 1976, 1981). As mentioned above, the potential predictability may be measured as the fraction of the total variability in seasonal means accounted for by the potentially predictable forced and internal source components. Dominance of the seasonally unpredictable variability associated with the weather noise component therefore lowers the potential predictability.

Another useful application may be found in validation of the skill of atmospheric GCMs, in particular in terms of the model’s ability to reproduce the forced component. Model skill is often evaluated as the correlation between raw seasonal anomalies in the observation and a model simulation. Yet, their correlation based only on the forced component, if successfully estimated, should become a more precise measure of the model’s performance. Furthermore, ensemble GCM experiments subject to an identical prescribed forcing with various initial conditions should, in theory, enable separation of the variability due to the forced component from that due to the internal source component, which may lead to deeper understanding of the influence of (nonlinear) internal dynamics on climate variability (Zwiers 1996; Zheng and Frederiksen 1999).

Correctly estimating the variability due to the weather noise component is crucial to the two applications mentioned above (Zheng and Frederiksen 1999). Typically, the estimation requires a daily time series and an assumption of its normality (Jones 1975; Madden 1976; Trenberth 1984; Zwiers 1987; Zheng 1996). These requirements, however, are subject to two main limitations. First, daily time series of many meteorological variables cannot be regarded as normally distributed. For example, the frequency distribution of daily wind speed or rainfall is highly skewed. Second, availability of daily time series is limited in comparison with that of monthly time series. For example, surface observations were recorded on a daily basis rarely during the early instrumental period, and many modeling groups record monthly statistics only. Therefore, it would be very useful to develop a method for estimating the variability due to the weather noise component based only on monthly mean time series.

In this paper, analysis of variance approaches are proposed for estimating the potentially predictable and weather noise variability separately, only from normally distributed monthly means. A method is also described for subsequent testing of potential predictability based on Zheng (1996) and Zwiers (1996). Monthly means were used by Zwiers (1996) for estimating interannual variability due to the weather noise component. Daily data were still required, however, to evaluate the correlation of the component between two adjacent months. The proposed method is then applied to assessing the potential predictability of global surface temperatures and the mid- and upper-tropospheric circulation. An analysis of synthetic red noise data demonstrates that the proposed estimation approach is comparable to the popular approach proposed by Zwiers (1987), which requires daily data.

The present paper begins with a description of the proposed variance estimation method and evaluation of the potential predictability. Section 3 is devoted to applications to monthly fields of observed meteorological data. A summary and conclusions are presented in section 4. Mathematical details and a comparative analysis of synthetic data and meteorological data are discussed in the appendixes.

## 2. Methodology

In the discussion to follow, it is assumed that the mean annual cycle has been removed by a harmonic equivalent analysis (e.g., Epstein 1991). Any uncertainties introduced by the estimation of the mean annual cycle are ignored. In this study, a season is assumed to be 3 months long, but the method described may easily be applied to “seasons” of any length.

### a. Statistical model

*x*) for a particular season obtained over a number of years (

*Y*) may be expressed in the following linear regression form:

*x*

_{ym}

*μ*

_{y}

_{ym}

*y*(=1, . . . ,

*Y*) denotes the year,

*m*(=1, 2, or 3) a month within a given 3-month season,

*μ*

_{y}a seasonal anomaly in year

*y*due to boundary forcings, radiative forcings and slowly varying internal dynamics, and ε

_{ym}a residual monthly departure of

*x*

_{ym}from the seasonal value

*μ*

_{y}. The residual weather noise component, which consists of {ε

_{y1}, ε

_{y2}, ε

_{y3}}, is assumed to comprise a three-dimensional stationary stochastic process in time with zero means and to be statistically independent and identically distributed with respect to year

*y.*The model (1) is similar to the one described in the introduction, but in (1) the forced and internal source components are combined into

*μ*

_{y}. This combination is necessary because one cannot isolate the internal source component from the boundary-forced component in the observed time series (a single realization of the climate system). Still, ensemble GCM simulations should allow such a separation to be achieved, by utilizing the strong sensitivity of the internal source component to initial conditions [see Zwiers (1996) and Zheng and Frederiksen (1999) for more details].

*m*or

*y*) by replacing that variable indicated as subscript with “

*o.*” For example,

*x*

_{yo}indicates the 3-month average of

*x*

_{ym}in year

*y*and

*x*

_{oo}, the average of

*x*

_{ym}, taken over all 3 months and all

*Y*years. With this notation, a seasonal mean can be expressed as

*x*

_{yo}

*μ*

_{y}

_{yo}

As in Zheng and Frederiksen (1999), a symbol *V* denotes the variance of a single variable or the covariance between two variables and another symbol *C* the correlation coefficient between two variables, all of which are evaluated over all *Y* years. Estimated quantities are indicated with a circumflex “ ;p0 .” For example, *Ĉ*(ε_{y1}, ε_{y2}) represents the estimated correlation coefficient between ε_{y1} and ε_{y2}.

*V*(ε

_{yo}). Since its daily time series within a season is, in general, assumed to be stationary, so are the monthly statistics. Specifically, its interannual variance is assumed to be independent of months, that is,

*V*

_{y1}

*V*

_{y2}

*V*

_{y3}

*V*

_{y1}

_{y2}

*V*

_{y2}

_{y3}

*V*

_{y1}

_{y3}

_{y1}− ε

_{y2}and ε

_{y2}− ε

_{y3}are assumed to be normally distributed. This assumption is weaker than the assumption that all ε

_{ym}are normally distributed (Rao 1973). Here, since month-to-month fluctuations arise entirely from the weather noise component (i.e., ε

_{y1}− ε

_{y2}=

*x*

_{y1}−

*x*

_{y2}), we can check the normality assumption directly from observed monthly data. We confirmed that the normality condition is approximately satisfied for monthly means of such variables as temperature, pressure, and wind speed, but not for monthly rainfall.

### b. Variance estimates

*Ĉ*(ε

_{y1}, ε

_{y2}), which is the key statistic in calculating others, has been derived through a maximum likelihood estimation similar to that documented in Zheng (1996) and Zwiers (1996), with a constraint that the correlation lies between 0 and 0.1. For such a continuous meteorological variable as pressure and temperature (but not precipitation), its daily time series can roughly be regarded as AR(1), a first-order autoregressive process (Trenberth 1984; Zwiers and von Storch 1995; Zwiers 1996), characterized by the first-order autocorrelation coefficient

*α,*which is generally between 0 and 0.9. For a daily AR(1) time series, the correlation coefficient of its mean between two adjacent months is approximately given by(Zwiers 1996). Substituting 0 and 0.9 to

*α*in (6) yields 0 and 0.1 as the lower and upper bounds for

*Ĉ*(ε

_{y1}, ε

_{y2}), respectively.

### c. A test for potential predictability

In the statistical model (2), a seasonal mean *x*_{yo} conceptually consists of *μ*_{y} and *ε*_{yo}. As mentioned earlier, the potential predictability of the former may be measured as the ratio *V*(*μ*_{y})/*V*(*x*_{yo}). For assessing the level of its statistical significance, however, it is convenient to consider *V*(*x*_{yo})/*V*(ε_{yo}) rather than *V*(*μ*_{y})/*V*(*x*_{yo}). A high significance level implies high potential predictability of observed seasonal means. Otherwise, they are dominated by the weather noise component and hence considered unpredictable.

_{yo}as given in Table 1, the statistical significance of the potential predictability of seasonal means can be assessed as follows. Since

*C*(ε

_{y1}, ε

_{y2}) is constrained between 0 and 0.1 and

*C*(ε

_{y1}, ε

_{y3}) is zero,

*V̂*(ε

_{yo}) is approximately distributed as

*χ*

^{2}(2

*Y*)

*V*(ε

_{yo})/(2

*Y*), where

*χ*

^{2}(2

*Y*) is the chi-square random variable with 2

*Y*degrees of freedom (DOF; appendix A). Under the null hypothesis of no interannual variability arising from boundary forcings and slowly varying internal dynamics, that is,

*μ*

_{y}is identical for every

*y*= 1, 2, . . . ,

*Y,*the statisticis distributed as

*χ*

^{2}(

*Y*− 1)

*V*(ε

_{yo})/(

*Y*− 1). The statistic in (7) becomes asymptotically independent of

*V̂*(ε

_{yo}) as

*Y*increases. Then the probability density distribution of the statisticis approximated by F(

*Y*− 1, 2

*Y*), the F distribution with DOFs of

*Y*− 1 and 2

*Y,*which has the mean

*Y*/(

*Y*− 1) ≈ 1. If

*R*differs significantly from 1 at the

*α*percent level, we claim the potential predictability is significant at that level.

## 3. Results

### a. Surface air temperature

The temperature data used here are monthly mean surface temperature records archived on a regular 5° latitude by 5° longitude grid by the Climate Research Unit (CRU), the University of East Anglia (Jones 1996). They cover 106 yr since 1890, based on daily station data of surface air temperature (SAT) over land and in situ ship measurements of SST over the ocean. The record length is unevenly distributed in space, from 106 yr over Europe to less than 30 yr over the polar regions. Any grid point at which the observed record is available only for less than 30 yr has been excluded from our analysis. Most of these points are located poleward of 70°N or 80°S.

Before calculating subsequent variance estimates, a linear trend has been removed at each grid point since detection or analysis of trends is not a focus of this paper. A trend estimate depends rather strongly on the record length available, implying that the trend removal may leave significant uncertainties in the estimated forced/internal source variability in regions where the available record is relatively short. However, the trend removal should have no significant effect on the estimation of weather noise variability.

The estimated standard deviations of ε_{yo} and *μ*_{y} and the potential predictability of the latter are mapped in Fig. 1 for boreal summer (JJA) and winter (DJF) seasons. For presentation, most of the high-latitude data gaps have been filled in using a local regression interpolation. Variability in surface temperature due to the weather noise component tends to be stronger over land than over the ocean, which reflects the smaller heat capacity of the land surface. Over the mid- and high-latitude continents in the Northern Hemisphere, as is well known, the standard deviation is nearly twice as large in winter as in summer. This wintertime variability is attributed to strong intraseasonal transient activities such as blocking events that not only enhance warm and cold temperature advection but also act to form and destroy near-surface stratifications over cold continental surfaces. The opposite seasonal tendency is found in the weak weather noise component of extratropical SST.^{1} In winter, *V*(ε_{yo}) is suppressed due to the large heat capacity of the oceanic mixed layer that is developed through the thermal and mechanical forcing by vigorous atmospheric intraseasonal disturbances, whereas the mixed layer shallows in summer acting to enhance the variability (Alexander and Deser 1995).

The overwhelming weather noise variability in SAT over the extratropical continents renders the potential predictability of SAT substantially lower than that of SST, despite the forced component being weaker in SST than in SAT, particularly over the wintertime continents. The potential predictability is particularly high in the equatorial Pacific, where the ENSO signal dominates over the weather noise component. The predictability is also particularly high in the wintertime extratropical North Pacific, where the prevailing upper-level westerlies allow effective teleconnection from the Tropics that enhances the remote influence of ENSO (Horel and Wallace 1981; Lau and Nath 1994). The relatively low predictability over the Southern Hemisphere maritime regions is primarily due to the relatively large *V*(ε_{yo}), which possibly reflects uncertainties in the data due to insufficient observations.

### b. Tropospheric circulation

We use monthly mean global fields of 500-hPa height (Z500) and 300-hPa wind components (U300 and V300) over the period 1958–96, based on the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalyses (Kalnay et al. 1996). The data were subsampled on a regular 5° × 5° latitude–longitude grid. Artificial climate “jumps” are minimized in the reanalyses by the use of a fixed assimilation/modeling system, but jumps and/or trends due to changes in observation systems may still be present (e.g., Kidson 1999). As above, a linear trend component fitted to the data time series has been removed at each grid point.

The estimated standard deviations of ε_{yo} and *μ*_{y}, and the potential predictability for Z500, U300, and V300 are mapped in Figs. 2–4, respectively. Intraseasonal variability in mid- and upper-tropospheric circulation is, in general, influenced by thermal conditions at the underlying surface to a much lesser degree than that in SAT or SST. Its distribution also reflects the land–sea thermal contrasts to a much lesser degree and hence should be more uniform in the zonal direction than that in SAT or SST. Still, weak zonal asymmetries are evident in weather noise variability in all fields, as described by Blackmon et al. (1984) for the Northern Hemisphere (NH) and by Trenberth (1981, 1982) for the Southern Hemisphere (SH). The variability itself and its zonal asymmetries both tend to be stronger in the winter hemisphere than in the summer hemisphere (Blackmon 1976;Trenberth 1982). The maximum over the far southeast Pacific is in the predominant blocking region identified by Sinclair (1996) and Renwick (1998). The weather noise variability is quite weak in the Tropics throughout the year.

Over NH, the wintertime weather noise variability in Z500 is strongest within the main jet exit regions, in good agreement with Blackmon (1976) and Blackmon et al. (1984). The strongest variability in summer appears in the polar region. In contrast to the weather noise variability, the “forced” variability in Z500 is distributed more or less uniformly. In the extratropics, the variability tends to be larger during winter in both hemispheres, but the seasonality is much more pronounced in NH. The prominent maxima over the far North Atlantic and central North Pacific correspond to the NAO and PNA, respectively, two of the most dominant anomaly patterns in the interannual timescales (Kushnir and Wallace 1989). The potential predictability of Z500 exhibits a pronounced tropical–extratropical contrast with much higher predictability in the Tropics than in higher latitudes, which reflects the suppression and predominance of the weather noise component in those two regions, respectively. Modest wintertime maxima in the predictability over the North Atlantic and North Pacific correspond to the NAO and PNA-like signals, respectively. The latter may manifest remote influence of ENSO via atmospheric teleconnection (Horel and Wallace 1981). The high predictability around Antarctica may be artificial due to the lack of observations, but it may in part reflect the interannual signal of the so-called Antarctic circumpolar wave (White and Peterson 1996).

Geographical and seasonal characteristics similar to those for Z500 apply to the estimated variance of the weather noise component in U300 and V300. Weaker tropical–extratropical contrasts in the wind variability than in the height variability are attributed to the breakdown of geostrophy in the Tropics. In the deep Tropics the influence of MJO leads to the dominance of U300 over V300 in the weather noise component (Madden and Julian 1971, 1972). Since extratropical fluctuations in U300 are associated primarily with quasi-stationary height anomalies with meridional dipole-like structure in jet exit regions (Blackmon et al. 1984; Kushnir and Wallace 1989), pronounced U300 variability due to the weather noise component is found in the central North Pacific and Atlantic. In contrast, extratropical V300 variability is associated primarily with wavelike anomalies. Distribution of the weather noise variability in V300 seems to reflect the seasonal characteristics of the mean flow in the extratropical SH with a single, zonally extended jet in DJF and double Pacific jets in JJA (Trenberth 1982).

Since the forced component of variability is much more pronounced in U300 than in V300, the potential predictability is generally higher in U300 than that in V300. Although somewhat less marked than in Z500, the potential predictability of U300 and V300 exhibits a considerable tropical–extratropical contrast with higher predictability in the Tropics. As for Z500, the meridional contrast is attributed, in part, to a stronger signal of the weather noise component in midlatitudes. In the Tropics, the pattern of the potential predictability of U300 is particularly reminiscent of ENSO signal, with a broad maximum across the equatorial Pacific. Since the forced seasonal anomalies associated with ENSO are particularly evident in the Walker circulation, the potential predictability in the Tropics is higher for U300 than for V300 in spite of the stronger weather noise component in the former. The remote influence of ENSO through the anomalous Walker cells contributes to the high predictability in the tropical Atlantic and Indian Oceans. Teleconnection of the ENSO signal results in the strong forced variability in U300 over the extratropical Pacific of the winter hemisphere, which leads to the modestly high predictability in U300 there.

## 4. Conclusions

A reliable method for estimating the intermonthly correlation using monthly data is proposed. This leads to a new estimation of the weather noise variability based only on monthly data. By applying the proposed method, it is apparent that monthly time series can be used successfully to partition seasonal-mean variability into the weather noise and forced and internal source components without any information derived from daily data (cf. Zwiers 1987; Zwiers 1996).

Compared with techniques based on daily time series, the method described here exhibits not only higher computational and data-handling efficiencies but also broader applicability. It was previously not possible to estimate the interannual variance due to the weather noise component either of surface temperature over a multidecadal or centennial period, where daily data are not available for much of the period, or of wind components (e.g., U300, V300), whose frequency distribution based on daily sampling is far from normally distributed. Many studies address the potential predictability of regional surface air temperatures, for example, over North America (Madden and Shea 1978; Shea and Madden 1990), Australia (Nicholls 1983), and New Zealand (Madden and Kidson 1997), but to date none have addressed global predictability because long-term global daily temperature fields are not available.

Recognizing that the weather noise component is, by nature, unpredictable on seasonal and interannual timescales, we attempted to assess the potential predictability as the ratio of the forced and internal source components to the weather noise component. Based on the multidecadal records, we found the potential predictability of surface temperature to be high over the tropical oceans and subtropical and midlatitude oceans in the winter hemisphere. Otherwise, the predictability is low, particularly over wintertime continents. Much stronger latitudinal dependencies appear in the potential predictability of the mid- and upper-tropospheric circulation. It is high only in the Tropics, where the weather noise components are weak and where seasonal anomalies tend to be coupled strongly with the underlying SST. The overall characteristics of the potential predictability of the variables studied, such as the geographical and seasonal dependencies, seem physically reasonable and consistent with current understanding.

Yet, the statistical model (2) used in our assessment is by no means perfect. For instance, the variability due to the weather noise component in our estimation may include the atmospheric response to SST fluctuations from one month to another within a season. The response should possess a certain level of potential predictability, although it is still unknown what fraction of the observed month-to-month variability in the atmosphere is accounted for by that response (Madden 1983;Shukla 1983). The imposition of a linear trend (perhaps to describe radiative forcing changes) may be somewhat limiting. It is proposed to extend the model in future to include nonlinear trend terms.

## Acknowledgments

This study was initiated during a visit of one of the authors (XZ) to the Institute of Statistical Mathematics (ISM) in Tokyo under the support of a Visiting Fellowship offered by the Japanese Ministry of Education, Science, Sports and Culture. XZ thanks Prof. T. Ozaki of ISM for his hospitality. The study has been continued under the support of the Foundation for Research Science and Technology of New Zealand under Contract CO1628. The surface temperature data we used were obtained from the Climate Research Unit, University of East Anglia and the reanalysis data from the NCAR Data Support Section. We wish to thank Dr. F. Zwiers and the two anonymous reviewers for their valuable comments.

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## APPENDIX A

### Proofs of Formulas in Table 1

_{y1}, ε

_{y2}, ε

_{y3}} may be expressed aswhere

*E*denotes the expectation value based on the all

*Y*years,

*σ*

^{2}the interannual variance of ε

_{ym}, and

*ρ*

_{1}=

*C*(ε

_{y1}, ε

_{y2}) =

*C*(ε

_{y2}, ε

_{y3}) the correlation coefficient of ε

_{ym}between two adjacent months. Likewise, the variance matrix of the intermonthly fluctuations of {ε

_{y1}− ε

_{y2}, ε

_{y2}− ε

_{y3}} may be written asTherefore the −2 ln (logarithmic) likelihood for (A1) becomes (Jones 1993)

*L*

*ρ*

_{1}

*x*

_{ym}

*Y*

*πσ̂*)

**V**

_{dε}

**V**

_{dε}are given byrespectively, we find after some algebraic manipulations that

*L*(

*ρ*

_{1}, {

*x*

_{ym}}) is minimized for

*ρ*

_{1}= 1.5 −

*υ*

_{1}/

*υ*

_{2}, where specific expressions of

*υ*

_{1}and

*υ*

_{2}are given in Table 1. Then, the constraint 0 ⩽

*ρ*

_{1}⩽ 0.1 based on the discussion in section 2b is imposed on

*ρ*

_{1}, which leads to the particular expression for

*C*(ε

_{y1}, ε

_{y2}) (i.e.,

*ρ*

_{1}) shown in Table 1. By substituting

*ρ*

_{1}in (A2) by the estimated

*C*(ε

_{y1}, ε

_{y2}), we obtainboth of which are also shown in Table 1.

*V̂*(ε

_{yo}). The key innovation here is that the intermonth correlation is estimated from monthly data but not daily data. Once the intermonth correlation is known, our estimation for

*V̂*(ε

_{yo}) is essentially identical to that proposed by Zwiers (1996). Especially,

*V̂*(ε

_{yo}) is approximately distributed as

*χ*

^{2}(

*T**

*Y*)

*V*(ε

_{yo})/[

*T**

*Y*], where

## APPENDIX B

### Comparison with Frequency Domain Method

*x*

_{yt}

*μ*

_{y}

_{yt}

*t*

*T*

_{yt}is a weather noise anomaly on day

*t*in year

*y,*and

*T*(=90) is the number of days

*within*a 3-month season (Zheng 1996). If the daily time series associated with the weather noise component {ε

_{yt},

*t*= 1, . . . ,

*T*} is stationary, the interannual variability of the seasonal mean arising from the weather noise component can be estimated from the variance spectrum of ε

_{yt}aswhere

*i*stands for the imaginary unit (Zwiers 1987).

*α*was set to be 0.4, 0.6, and 0.8 in turn. The innovation variance was held constant at 1 throughout. In total, 100 such independent synthetic datasets were generated. Both analysis methods were applied to each synthetic dataset to estimate

*V*(ε

_{yo}). The theoretical variance of ε

_{yo}isThe mean and standard deviation of the 100 members of

*V̂*(ε

_{yo}) (denoted as

*E*[

*V̂*(ε

_{yo})] and

*V*[

*V̂*(ε

_{yo})]

*V*(ε

_{yo})

*α,*no significant differences are found in either of those two statistics between its estimations based on the (traditional) frequency domain method and the method proposed in this study. The two estimation methods both yield

*E*[

*V̂*(ε

_{yo})]/

*V*(ε

_{yo}) close to unity and

*V*[

*V̂*(ε

_{yo})]

*V*(ε

_{yo}) close to 0.158 (the theoretical value when

*V̂*(ε

_{yo}) is distributed as

*χ*

^{2}(80)

*V*(ε

_{yo})/80), indicating that the bias and error introduced in either of the two methods both tend to be small.

## APPENDIX C

### Constraints on *C*(ε_{y1}, ε_{y2})

The proposed method is again applied to each synthetic dataset in the same manner as in appendix B for estimating *V*(ε_{yo}) but with the following two modifications applied to *Ĉ*(ε_{y1}, ε_{y2}): (a) no constraint: that is, *Ĉ*(ε_{y1}, ε_{y2}) = 1.5 − *υ*_{1}/*υ*_{2}, and (b) constrained as zero:that is, *Ĉ*(ε_{y1}, ε_{y2}) = 0. All results are included in Table B1.

The modification (a) is designed to check the importance of the constraint that limits *Ĉ*(ε_{y1}, ε_{y2}) below 0.1. Table B1 shows that without this constraint the ratio *V*[*V̂*(ε_{yo})]*V*(ε_{yo}) is between 0.4 and 0.5, which is much larger than that based on other methods (around 0.15). Hence, constraining *Ĉ*(ε_{y1}, ε_{y2}) between 0 and 0.1 is an important step toward reducing estimation errors.

The modification (b) is designed to examine the relevance of assuming the weather noise component to be completely independent from one month to another, which could simplify the estimation substantially (Zwiers 1996). Although this constraint acts to reduce estimation errors slightly regardless of the values of the AR(1) coefficient (Table B1), it gives rise to a considerable negative bias, especially for the time series with the relatively large coefficient (i.e., *α* = 0.8). With these reasonable results, we conclude that appropriate estimation of the intermonthly correlation of the weather noise component is important in estimating its interannual variability, especially for such a daily time series with relatively high persistency as that of SST or even atmospheric variables in the Tropics.

## APPENDIX D

### An Example for Meteorological Data

We have analyzed DJF Z500 using both the traditional and proposed methods. The results are shown in Fig. D1. Overall, *V̂*(ε_{yo})*V̂*(ε_{yo})

For the North Pacific, *V̂*(ε_{yo})*C*(ε_{y1}, ε_{y2}) = 0 (bottom) is about the same as that derived from the traditional approach, but less than that derived from the proposed approach. Since the daily autocorrelation coefficients over North Pacific are more than 0.8, with the reasoning given in appendix C, *V̂*(ε_{yo})*V̂*(ε_{yo})

Formulas for parameter estimates.

Simulation results using 40 yr of synthetic data (*Y* = 40). The “traditional” method refers to Zwiers (1987) as shown in (B2) of appendix B. “Proposed” is the method described in this paper. The other two methods are variations of the proposed method as described in appendix C. Column 2 indicates the degree of daily autocorrelation in the time series. Columns 3 and 4 indicate how well the weather noise variability is estimated. The perfect values for them are one (i.e., no bias) and 0.158 [i.e., *V̂*(ε_{yo}) is distributed as χ^{2}(80)*V*(ε_{yo})/80], respectively.

^{1}

Several studies have shown that the total interannual variability *V*(*x*_{yo}) in extratropical SST tends to be larger in summer than in winter (Cayan 1980; Iwasaka et al. 1987; Nakamura and Yamagata 1999), probably for the same reason as mentioned in the text. Our analysis indicates that the same tendency applies separately to each of the forced and weather noise components.