This study determined the theoretical time-to-detect (TTD) global climate model (GCM) precipitable water vapor (PWV) 100-yr trends when realistic measurement errors are considered. Global trends ranged from 0.055 to 0.072 mm yr−1 and varied minimally from season to season. Global TTDs with a 0% measurement error ranged from 3.0 to 4.8 yr, while a 5% measurement error increased the TTD by almost 6 times, ranging from 17.6 to 22.0 yr. Zonal trends were highest near the equator; however, zonal TTDs were nearly independent of latitude when 5% measurement error was included. Zonal TTDs are significantly reduced when the trends are analyzed by season. Regional trends (15° × 30°) show TTDs close to those in the 15° latitude zones (15° × 360°). Detailed case study analysis of four selected regions with high population density—eastern United States, Europe, China, and India—indicated that trend analysis on regional spatial scales may provide the most timely information regarding highly populated regions when comparing detection time scales to global and zonal analyses.
The Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report stated, with high certainty, that the continued release of greenhouse gas emissions will cause a warming of 0.2°C decade−1 (Solomon et al. 2007). This rise in greenhouse gasses could also create changes in the overall climate, altering wind and precipitation patterns and leading to more extreme weather events. In the next 100 years, the sea level is expected to rise, droughts may become more frequent, and the intensity of rainfall may lead to more flooding (Solomon et al. 2007). These extreme weather events have secondary consequences, including severe socioeconomical impacts, increases in waterborne diseases, and escalating cleanup costs (Solomon et al. 2007; IPCC 2014). Understanding the projected changes in the climate is a necessity and attempting to quantify and monitor the current atmospheric state is fundamental to making future policy decisions.
Flooding may cause some of the greatest socioeconomic burden in the future. Geography plays a key role in the potential impact: for example, Asia and the Pacific are prone to flooding due to their low-lying areas and large population density (IPCC 2014). Coastal flooding is predicted to affect millions of people by 2100 primarily because of displacement from land loss. Flooding has been found to affect the health of a person through infectious diseases, vector-borne diseases, and injuries (Jakubicka et al. 2010; Schnitzler et al. 2007). In addition, the economic cost is expected to be severe and many studies have shown the benefits of flood mitigations; prediction of precipitation is key for adaptation policies (IPCC 2014; Kousky and Walls 2014; Pielke and Downton 2000). The climate is a complex system to emulate and global climate models (GCMs) do not agree in regards to simulated trends of PWV and precipitation parameterizations (Roman et al. 2012; Sun et al. 2006). Sun et al. (2006) found that, although some models are able to simulate land precipitation, most were unable to reproduce spatial patterns of precipitation and frequency, a key to understanding regional changes. Furthermore, most models underestimated the intensity of heavy precipitation, making it difficult to characterize flash flooding events (Sun et al. 2006; Stephens et al. 2012). Measurements of precipitation also have large uncertainties. Rain gauges and radar systems are costly and not readily available around the world. In addition, biases exist in both datasets (Seo and Breidenbach 2002; Ciach et al. 1999). Satellites provide the best way to measure precipitation over all terrains, but the measurement uncertainties can be considerable. One study by Tian and Peters-Lidard (2010) found that the uncertainty for the ensemble of six different Tropical Rainfall Measuring Mission (TRMM)-era datasets was 40%–60% over the ocean and 100%–140% over high latitudes.
Sea surface temperatures (SSTs) are much easier to measure and predict. Precipitable water vapor (PWV), over the ocean, is highly correlated with SSTs through the Clausius–Clapeyron relationship. A study by Trenberth et al. (2003) found that the water-holding capacity of the atmosphere increases by 7% °C−1. In addition, another study found that the relative humidity in GCMs remain constant, implying that the PWV will rise with rising air temperatures (Soden et al. 2005). Furthermore, Roman et al. (2012) found a null trend in the PWV in models from 2000 to 2010, which was consistent with observations suggesting that PWV is better stimulated in models than precipitation. Previous studies have shown correlations between PWV and precipitation. For example, Song and Grejner-Brzezinska (2009) showed that, over the Korean Peninsula, maximum GPS PWV was highly correlated with maximum rainfall during typhoons. Another study examined the PWV from a GPS network in the Chengdu Plain, China, and found that rainfall occurs mainly during high levels of GPS PWV and suggested the use of PWV data in nowcasting of severe weather (Li and Deng 2013). Huang et al. (2013) found that there were sudden high increments of PWV before convective precipitation and suggested the use of PWV data to monitor severe rain events and high moisture fluxes. Kunkel et al. (2013) showed that the dominant factor in modeled probable maximum precipitation (PMP) change is increases in higher levels of atmospheric moisture content and PMP is not correlated to changes in winds. Karl and Trenberth (2003) also found that the increase in heavy precipitation events is primarily due to increased water vapor instead of other microphysical processes. PWV, although not a direct means to precipitation, is a requirement for rain and therefore a change in PWV can lead to a change in precipitation.
With high uncertainty in GCMs, it is essential to detect predicted PWV trends with observations on decadal time scales (Wielicki et al. 2013; Ohring and Gruber 1982; Ohring et al. 2005, Tian et al. 2013). Currently, the satellite era has provided several new advanced IR sounders that provide or are expected to provide high quality measurements of PWV. These sounders have the important characteristic of providing daily global coverage of Earth at spatial resolutions much finer than required for climate studies. However, this nearly complete spatial coverage is obtained only at fixed local times as the satellites are in sun-synchronous orbit. From these daily products, a climatology of PWV can be produced at monthly time scales and for latitude/longitude grids matching the GCM model output. The National Aeronautics and Space Administration (NASA) Atmospheric Infrared Sounder (AIRS) on board the Aqua satellite is part of the NASA Earth Observation System and is meant to support climate research and facilitate better weather forecasting (Chahine et al. 2006). The Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E), also on board Aqua, was created to support observations of the hydrological system on Earth (Kawanishi et al. 2003). One of the newest instruments is the Cross-Track Infrared Sounder (CrIS), launched on Suomi National Polar-Orbiting Partnership (Suomi-NPP) (Han et al. 2013). CrIS is designed to provide high-resolution information on the atmospheric state, which in turn will provide more knowledge about the climate and help improve forecasting systems. Observations have measurement errors created through instrument noise and algorithm assumptions. Theoretically, the time to detect (TTD) a trend depends on natural variability, measurement error, and sampling error in space and time (Wielicki et al. 2013; Leroy et al. 2008; Weatherhead et al. 1998; Whiteman et al. 2011). Since model fields, by definition, integrate over all space and time, they are assumed to have zero sampling error in the context of this paper. However, by inclusion of a range of measurement error in the predicted climate trends we can provide a better understanding of the effectiveness of current satellite instruments and the requirements of any future climate observing system.
The objective of this paper is to understand the effect of measurement error on the TTD a PWV trend and determine the required spatial scaling to help in mitigation of societal impact. The paper will be broken into four main sections: model output (section 2); methods (section 3); results (section 4), which will discuss the differences in spatial scaling and seasonal versus annual TTDs; and conclusions, with an appendix that quantifies the autoregressive character of the GCM PWV time series.
2. Model output
This paper makes use of GCM output relevant to the assessment of trends in atmospheric water vapor, retrieved through the World Climate Research Programme (WCRP) phases 3 and 5 of the Coupled Model Intercomparison Project (CMIP3 and CMIP5) (Meehl et al. 2007a; Taylor et al. 2012). The GCMs used here are a selection of those studied in Roman et al. (2012). The intent of this paper is to illustrate the importance of including measurement error for investigation of any model; thus, we limited the number of models used for this study.
The CMIP3 output was accessed through the Earth System Grid (ESG) portal. Two models from the CMIP3, used in the IPCC Fourth Assessment Report, were included in this study: the National Center for Atmospheric Research (NCAR) Community Climate System Model, version 3 (CCSM3) and the NASA Goddard Institute for Space Studies Model E, coupled with the Russell ocean model (GISS-ER; hereinafter GISS) (Collins et al. 2006; Schmidt et al. 2006). The Special Report on Emissions Scenarios (SRES) A2 run 1 experiment was selected, which is described as “a very heterogeneous world with continuously increasing global population and regionally oriented economic growth that is more fragment and slower than in other storylines” (http://www.ipcc-data.org/ar4/scenario-SRA2-change.html). Each file contains latitudes, longitudes, and time. The native grid varied: CCSM3 has 128 × 256 grid points and GISS has 46 × 72 grid points. Each model data type spans the 100-yr period 2000–2100, although the models have different spinup periods. This study used monthly averages of PWV (https://esgcet.llnl.gov:8443/home/publicHomePage.do).
The CCSM3 was selected because this model represented the lower end of SST change with a smaller 50-yr trend compared to the observations and other models, while GISS better matched the observations for SST trends from 1950 to 1999 (Bader et al. 2008). In addition, the CCSM3 showed the greatest change in globally averaged surface air temperature and precipitation from 2000 to 2100, while GISS was closer to the multimodel mean in a study done by Meehl et al. (2007b). Their CMIP5 counterparts were selected for consistency (Collins et al. 2014; Flato et al. 2013). In addition, these models were the subject of a comparison against measurements for a decade of data in Roman et al. (2012).
The CMIP5 output was obtained through the new portal, the Earth System Grid–Center for Enabling Technologies (ESG-CET) portal. The model counterparts to the CMIP3 were used in this study: the NCAR Community Climate System Model, version 4 (CCSM4) and the NASA Goddard Institute for Space Studies Model E2, coupled with the Russell ocean model (GISS-E2-R, hereinafter GISS-E2) (Gent et al. 2011; Schmidt et al. 2006). These models are available starting January 2004 (CCSM4) and January 2006 (GISS-E2) and run through December 2100; however, each model has different spinup periods. The scenario used was the representative concentration pathway (RCP) 8.5, which is described as “rising radiative forcing pathway leading to 8.5 W m−2 in 2100” (http://sedac.ipcc-data.org/ddc/ar5_scenario_process/RCPs.html). Each file contains latitudes, longitudes, and time and the native grid boxes varied: CCSM4 has 192 × 288 grid points and GISS-E2 has 90 × 144 grid points. This study used monthly averages of PWV (http://pcmdi9.llnl.gov/esgf-web-fe/).
a. Global analysis
Monthly mean GCM PWV covering the 100-yr period 2000–2100 over the globe was extracted at the model’s native resolution. These monthly averaged values were interpolated to a 1° × 1° grid using bilinear interpolation. The 15° × 30° (regional) and 15° × 360° (zonal) grids were created using a moving average to smooth the 1° × 1° data with a boxcar window of size (2M + 1) × (2N + 1), where M is the number of latitudes (M = 7 for the regional and zonal analysis and 90 for the global analysis) and N is the number of longitudes (N = 15 for the regional analysis and 180 for the zonal and global analysis). For the global 180° × 360° grid an area-weighted average was applied to correctly account for PWV values on a sphere. To do this the PWV is multiplied by the cosine of the latitude and then normalized. PWV values for a given model were calculated for 12-month time periods to create an annual PWV time series. Seasonal time series were also computed using a 3-month average [December–February (DJF), March–May (MAM), June–August (JJA), and September–November (SON)] extracted from each calendar year. The appendix lists (see Table A1 in the appendix) the number of samples in the original monthly time series and the derived annual and seasonal time series.
b. Compute GCM global and zonal trends
For each grid [15° × 30° (regional), 15° × 360° (zonal), and 180° × 360° (global)], a 100-yr trend was calculated using the annual anomaly time series. The trend was computed at each grid box using a least squares fit with equal weighting for each annual mean PWV anomaly value, a total of 100 points. To calculate seasonal trends, the PWV was divided into four seasons; DJF, MAM, JJA, and SON. The months were averaged yielding one value per year, creating a 100-yr seasonal time series from which the trend would be calculated using the same method as above. Case study region trends were calculated by extracting the smoothed PWV values over the desired area (United States, India, China, and Europe; shown in Fig. 5) from the regional (15° × 30°) grid and averaging over the region.
c. TTD trends with measurement uncertainty
Following Weatherhead et al. (1998), the equation to calculate the number of years of data needed to detect the 100-yr trends at a 95% confidence level with a probability of 0.90 and no measurement error is given by
where natural variability σN is the monthly standard variation of the detrended time series anomaly of PWV, also known as the natural variability; ω0 is the 100-yr trend; and Φ is the AR autocorrelation. The dimensionless decorrelation time (1 + Φ)/(1 − Φ) is the time scale factor that multiplies the variance. A random process, Φ = 0, has a time scale factor of 1, while the factor is infinite for an autocorrelation coefficient of 1. The absolute value of the AR autocorrelation was used in this paper so that all correlation time scale factors are greater than 1. The primary assumption of Eq. (1) is that an autoregressive process of order 1 (AR) can approximate the time series. An appendix to this paper provides a test of the AR process through comparison of the monthly averaged, 3-month seasonal averaged, and 12-month annual averaged time series to a simulation of a pure AR process. The PWV time series from the GCM model runs are seen to roughly follow the expectations of a first-order autoregressive process, thereby justifying the use of Eq. (1) for estimating GCM PWV TTDs.
To introduce measurement error εm into the time series analysis we use a simple extension of the variability, total standard deviation of σN = (σ2NatVar + ε2m)1/2. Unlike the formalism of Leroy et al. (2008), here the measurement error is assumed to have the same correlation time scale as the natural variability. The assumption that the time scale of the measurement error correlation is the same as the natural variability is justified by the result of Roman et al. (2012) that showed GCM monthly PWV time series analysis had the same standard deviation and autoregression coefficient as a time series of ground-based PWV observations. The verification that the error in satellite-derived PWV behaves as a fractional percentage of the PWV amount from the tropics to the Arctic was shown in Bedka et al. (2010) for a 5-yr period of NASA AIRS data. The AIRS has lasted longer than the nominal 7-yr lifetime so it is unusual to have such a long continuous record from a single sensor. For the next 15 years, the products from CrIS on the Joint Polar Satellite System (JPSS) series (Suomi-NPP, JPSS-1, and JPSS-2) and the Infrared Atmospheric Sounding Interferometer (IASI) on the Meteorological Operation (MetOp) series (MetOp-A, MetOp-B, and MetOp-C) will be used to piece together a global PWV climatology. The sounder data have global daily coverage, which means that they will track the day-to-day signal of PWV down to the 1° × 1° scale.
For this study a range of measurement errors from 0% to 5% were used, representing the error from both measurement noise and algorithm uncertainties. The first was a measurement error of 0%, which represents an ideal sensor with no sampling error. The second was a measurement error of 1% to represent a nearly ideal climate sensor. Finally, measurement errors in the 2%–5% range were used to represent the accuracy of current satellite sounders, which have been shown to be within 5% when compared against selected ground-truth data (Bedka et al. 2010). Trends, variability, and TTDs were calculated for each grid size. Finally a TTD range was calculated using the equation for uncertainty from Weatherhead et al. (1998),
4. TTD results
Global annual trends among the four models ranged from 0.055 mm yr−1 for GISS-E2 to 0.072 mm yr−1 for the CCSM4, as shown in Table 1. Seasonal trends do not vary much; the models differ by at most 0.009 mm yr−1. Autocorrelation time scale does not vary much; CCSM3 had the smallest annual autocorrelation time scale of 1.18 and GISS-E2 had the largest time scale of 1.68. Seasonal autocorrelation factors range from 1.18 in MAM to 2.09 in SON for the CCSM3 while the CCSM4 ranged from 1.10 in DJF to 1.57 in SON; SON has the largest seasonal autocorrelation factor for each model. The smallest annual total standard deviation was 0.07 mm for GISS, with 0% measurement error, and the largest annual total standard deviation was 1.46 mm for the CCSM4, with a 5% measurement error. Generally, the standard deviation does not vary much by season or model. Similarly the TTDs do not vary much seasonally and range from about 3.6 yr for the GISS during DJF to 6.8 yr for the GISS-E2 for 0% measurement error. The highest TTDs occurred with a 5% measurement error and ranged from 17.3 yr for the CCSM4 during DJF to 22.5 yr for the GISS-E2 during SON. GISS-E2 shows the highest TTD, 22.5 yr, for 5% measurement error in SON, possibly because of the extremely high autocorrelation factor of 1.79 for this season in this model.
Each zone number represents a 15° latitude band, starting with zone 1, which represents the band 90°–75°S, and ending with zone 11, which represents the 75°–90°N band. Figure 1 shows the zonal annual trend, autocorrelation factor, total standard deviation, and TTD. The trend maximum of 0.13 mm yr−1 occurs near the equator and generally the Northern Hemisphere has larger trends than the Southern Hemisphere. The zonal average, an average of all 11 zones, is around 0.08 mm yr−1. Peaks of high autocorrelation factor occur in the mid latitudes while the smallest factor, about 1, is at the equator. Similar to the trends however, the autocorrelation factor and total standard deviation is generally larger in the Northern Hemisphere. The total standard deviation ranges from 0.2 to 3 mm depending on measurement error; although the greatest natural variability occurs in the midlatitudes, the largest total standard deviation with a 5% measurement error occurs at the equator, most likely due to the exceedingly high PWV amounts in the tropics. Despite similar latitude dependence in trends and natural variability, the latitude dependence of the TTD is more similar to that of the autocorrelation factor, with the lowest TTD of 10 yr occurring at the equator, where the smallest autocorrelation factor and largest trend was observed. The TTDs ranged from 10 to 40 yr and are generally larger in the Northern Hemisphere. There is a smaller range of TTDs by latitude zone for each measurement error than seen in the global average, suggesting the measurement error hinders the global TTD more than the zonal TTD by overwhelming the extremely small natural variability of the global annual average PWV.
Figure 2 shows the monthly PWV time series from 2000 to 2030 for three different zones—equator, tropics, and the Arctic—and shows the relative importance and magnitude of the seasonal cycle. Seasonal trends (shown in Fig. 3, by latitude zone) show the shift of water vapor from one hemisphere to the other. For example, the Southern Hemisphere has relatively higher trends in PWV during DJF while the Northern Hemisphere has higher trends in JJA. The Clausius–Clapeyron theory suggests that warmer temperatures create more water vapor in the atmosphere because of greater evaporation. Therefore, it would be expected to see a latitude shift of high PWV trends from the Northern Hemisphere to the Southern Hemisphere during season changes. MAM and SON show symmetry around the equator, with MAM showing slightly higher trends in the Southern Hemisphere and SON showing slightly higher trends in the Northern Hemisphere. Figure 4 shows the TTD for each season, which does not change much from season to season, and ranges from 5 to 20 yr depending on measurement error. Generally, zone 1 (Antarctica) shows larger TTDs than the other zones, most likely due to the very low trend of about 0.01 mm yr−1.
Figure 5 shows the annual average PWV for the year 2004 and suggests that certain areas will experience higher PWV than others. In addition, regional variability will affect the TTD. Thus, it is imperative to understand the trends in PWV at a regional level to identify regions that will be more heavily impacted. The PWV trend is greatest (larger than 0.1 mm yr−1) near the equator, as shown in Fig. 6, while the autocorrelation factor tends to be higher in the Northern Hemisphere. Natural variability is highest in the tropics. Generally the TTD for 0% measurement error is less than 20 yr, with some regions, like Indonesia, with TTDs less than 5 yr. The greatest TTD occurs in the South Pole region where smaller trends (less than 0.05 mm yr−1) and high autocorrelation factors are observed. In addition, the region over northern China shows high TTDs, exceeding 50 yr, because of the larger variability and autocorrelation factor. A 1% measurement error generally does not increase the TTD significantly, as shown in Fig. 7, but a 5% measurement increases the TTD to the point where no TTD less than 20 yr exists. Seasonal regional TTDs (not presented) show similar results to seasonal zonal TTDs, in that the overall TTDs did not change much but a shift in the location of the trends and total standard deviation was observed.
d. Case study regions
As observed in the regional analysis, particular areas that had higher predicted trends may have shorter TTDs. Figure 5 shows the region boundaries used for each case. This case study analysis will further investigate annual and seasonal dependencies at these smaller scales, and all results are shown in Table 2.
Europe is a highly populated continent making the country susceptible to climate change (EEA 2012; Kelemen et al. 2009). Europe is widely separated from tropical moisture; thus, the trends in this region are small, varying from 0.02 to 0.05 mm yr−1, depending on the season (Dore 2005; Wibig 1999). The autocorrelation factor ranges from 1 to 1.87, with the smallest autocorrelation factors seen in the seasonal breakdown. Because of the generally low trends in Europe, the TTDs are much larger compared to the other case studies, never lower than 14 yr; however, the seasonal breakdown shows a positive effect in lowering the TTDs by at least a few years.
The eastern United States has a high population density and has the potential to be prone to flooding, making this region an important area to understand (De Sherbinin et al. 2007; Hirschboeck 1988; Zhang et al. 2000). The PWV trend is greatest in JJA and SON, around 0.07 mm yr−1. The autocorrelation factor is much smaller when broken down by season ranging from 1 to 1.46, while the annual autocorrelation factor is never smaller than 1.42. Similarly, the total standard deviation is much smaller for each season’s PWV than the annual PWV. The smallest total standard deviation occurs in JJA because of the higher PWV amounts in the region. The small total standard deviation paired with the high trends in JJA creates the lowest TTDs ranging from 11 to 22 yr. The highest TTD occurs in the annual results, which range from 20 to 31 yr. DJF has the highest seasonal TTDs, ranging from 17 to 28 yr, but this is still smaller than the annual TTDs, again suggesting the need for regional seasonal analysis.
China is predicted to have high PWV amounts in the future; paired with the ever-growing population in the country, this region is one example of a potentially high socioeconomic impacted country (De Sherbinin et al. 2007; Piao et al. 2010; Wei et al. 2009). The highest PWV trend occurs during JJA, around 0.1 mm yr−1. This is slightly higher than the annual trend of around 0.07–0.09 mm yr−1. In particular, JJA also has a lower autocorrelation factor and generally a low total standard deviation. This in turn creates small TTDs of 10–20 yr. These TTDs are 2 times smaller than the annual TTDs. Seasonally, the highest TTD with a 5% measurement error occurred in DJF, most likely due to the small trend and high autocorrelation factor.
The final case study is India, which is another example of a high impact country because of the high population density and susceptibility to climate hazards (Binswanger et al. 1993; Brenkert and Malone 2005; De Sherbinin et al. 2007). Overall, the PWV trend is largest in JJA and SON. The autocorrelation factor varies by model but never exceeds 2. Seasonal autocorrelation factors are smaller than the annual autocorrelation factor, which may decrease the TTD. The seasonal total standard deviation for 0% measurement error is almost half the annual total standard deviation, which in turn creates much lower TTDs. Generally the seasonal TTDs are 5–10 yr shorter than the annual TTDs, except for DJF. JJA has the shortest TTDs, ranging from 9 to 22 yr, depending on measurement error. This is primarily due to the exceedingly high trend for this season and low total standard deviation and autocorrelation factor, suggesting that the seasonal analysis at this small regional scale can help lower TTDs.
This paper has shown that measurement error can severely affect the ability of any real sensor to detect a trend in PWV. Global TTDs degrade quickly with measurement error, with TTDs above 13 yr for measurement errors of 3% or higher. Annual zonal TTDs similarly degrade quickly with measurement error with only the tropical zone having a TTD of less than 20 yr for a 3% measurement error. Regional estimation of TTDs offer a chance to measure statistically significant trends in specific regions, which may be important in the future when considering the societal impact of climate change. The global TTDs ranged up to 22 yr depending on the model and measurement error, but certain regions were able to detect these trends within 10 yr for measurement errors of up to 5%. Zonal TTDs varied from 5 to 50 yr, suggesting regional studies may even prove to be an advantage over zonal analysis.
We are grateful for the support from the NASA CLARREO Science Team project under grant NNX11AE70G. We acknowledge the World Climate Research Programme’s Working Group on Coupled Modeling, which is responsible for CMIP, and we thank the climate modeling groups (NASA Goddard Institute for Space Studies and National Center for Atmospheric Research) for producing and making available their model output. For CMIP, the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals.
Autoregressive Time Series Analysis
a. Expanded detail on global PWV time series analysis
To justify the analysis approach used in this paper, additional details are provided in this appendix for the global PWV time series. Table 1 summarizes the results of analyses on 3-month seasonal averages and 12-month annual averages. Table A1 augments the values shown in Table 1 with statistics of the original monthly time series. Analysis of the monthly time series will be used to test the autoregression assumptions used in this paper. Table A1 contains the number of samples used in each case and the corresponding AR coefficient allowing the calculation of the uncertainty parameter B described in Weatherhead et al. (1998) and used in Eq. (2) in the main text. For values less than about 0.2, B is approximately equal to the percentage uncertainty in the TTD (divided by 100) because of uncertainty in the estimation of the AR coefficient. Despite the fact that the number of samples drops by an order of magnitude between monthly time series and seasonal/annual time series, a corresponding drop in the autoregression coefficient means that the uncertainty in the TTD increases only about 6% from between 7% and 12% to between 14% and 19%. However, the TTD value itself drops significantly for seasonal and annual time series relative to monthly time series due to large decreases in the autocorrelation. For 0% measurement error, the TTD of both seasonal and annual trends is about one-half of the TTD for monthly time series. The lower TTD is one reason why this study has emphasized seasonal and annual time series analysis.
b. Properties of an autoregressive time series
An AR autoregressive time series N(t), composed of an underlying random variable X(t) with autocorrelation coefficient Φ, is defined as
where X is normally distributed. The standard deviation σN of the autoregressive time series N(t) is increased over the standard deviation σX of the independent time series through the autoregression equation. As described in Weatherhead et al. (1998), for an AR process, with coefficient Φ = AR, we expect
c. Underlying random noise of the GCM time series
If one assumes an AR process and takes the values of the monthly standard deviation and monthly AR coefficient from Table A1, then the value of the underlying noise process σX can be computed from Eq. (A2), giving the values 0.195, 0.132, 0.185, and 0.138, for the CCSM3, GISS, CCSM4, and GISS-E2, respectively. We find the remarkable result that the implied standard deviation of the random noise from CCSM4 (CMIP5) is very close to that of CCSM3 (CMIP3) and likewise between GISS-E2 (CMIP5) and GISS (CMIP3). The CCSM model runs have an underlying random σX, which is about 35% higher than the GISS model runs. Figure A1 illustrates the dependence of the standard deviation σN on the AR coefficient assuming a perfect AR process. The symbols in Fig. A1 are the monthly values of σN and AR shown in Table A1 computed from each GCM model run PWV time series. The curves are the dependence of an ideal AR process that is consistent with each GCM model. The dashed curves are the CMIP3 models and the solid curves are the CMIP4 models. We note that a change in the AR coefficient is the main difference between the GISS (CMIP3) and GISS-E2 (CMIP5) model runs. In contrast, the CCSM3 model run is within 10% of the CCSM4 AR coefficient.
d. Testing the autoregressive character of the GCM time series
The simulation of a time series with using Eq. (A1) was performed using a 1200 month time series (100 yr) repeated 1000 times. The values of σX from the previous subsection were used along with the AR coefficient shown in Table A1 computed from the GCM monthly time series. To test the autoregressive nature of the PWV time series from the GCMs, we applied a running boxcar filter of length N months to the monthly AR noise simulation time. The standard deviation of this smoothed time series varies according to both the width of the boxcar and the magnitude of the autocorrelation.
Figure A2 illustrates the validity of the assumption that the GCM PWV time series can be approximated by an AR process. In each model run shown, the reduction in standard deviation of the GCM time series (symbols) with increasing time averaging is comparable to a simulated reduction of an ideal AR process (dashed curves). The 1- and 12-month GCM standard deviation values are those shown in Table A1 for the rows labeled “monthly” and “annual,” respectively. The 3-month GCM standard deviation was estimated from the mean of the four seasonal values shown in Table A1. Since there is a considerable spread in the seasonal standard deviation values this 3-month estimate only roughly approximates the ideal 3-month running average shown by the dashed line.
Inspection of Fig. A2 shows that the deviation of the GCM seasonal and annual time series from an ideal process is within an uncertainty of about 20% of the estimated monthly AR parameter. Note that the solid lines represent a range of AR values from 0 to 0.9 in increments of 0.1 using the value of σX derived in the previous subsection for each model run. In particular, the lowest curve for an AR value of zero is simply the random noise standard deviation reduced by the square root of the number of months used in the boxcar filter time average. The fact that the GCM standard deviation reduction for seasonal and annual averages is close to that of an ideal AR process and is far above that of a purely random process vindicates the use of the Weatherhead et al. (1998) analysis method for the trend analysis of the GCM PWV time series described in the text.