Comparing two methods to estimate the sensitivity of regional climate simulations to tropical SST anomalies

Authors

  • Wei Li,

    1. Department of Meteorology and Earth and Environmental Systems Institute, Pennsylvania State University, University Park, Pennsylvania, USA
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  • Chris E. Forest,

    Corresponding author
    1. Department of Meteorology and Earth and Environmental Systems Institute, Pennsylvania State University, University Park, Pennsylvania, USA
    2. CIRES, University of Colorado Boulder, Boulder, Colorado, USA
    • Corresponding author: C. E. Forest, Department of Meteorology, Pennsylvania State University, 507 Walker Bldg., University Park, PA 16802, USA. (ceforest@psu.edu)

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  • Joseph Barsugli

    1. CIRES, University of Colorado Boulder, Boulder, Colorado, USA
    2. Earth System Research Laboratory, NOAA, Boulder, Colorado, USA
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Abstract

[1] We perform ensemble simulations using NCAR CAM3.1 T42 forced by perturbed SST fields to estimate the sensitivity of regional climate change at seasonal scales to tropical SST anomalies. We compare the sensitivity and linear reconstruction of regional climate change to tropical SST anomalies from the patch method and the random perturbation method (RPM). The patch method adds one SST anomaly patch at a certain location of the tropical ocean to the prescribed SST field at one time. The RPM method randomly perturbs the climatological SST field with spatially coherent anomalies and estimates the anomalous response with respect to the climatological equilibrium state. The two methods provide generally consistent sensitivity information and similar reconstruction of the regional response over the global scale and tropical regions. If only the dominant sensitivity information is desired, the RPM method is about twelve times more computationally efficient than the patch method due mainly to the larger area-integrated amplitude of the SST forcing used.

1. Introduction

[2] Global climate models are the primary tools for climate prediction and policy-makers rely heavily on their simulations. However, predictions at regional scales (e.g., sub-continental) show significantly larger uncertainties than at global scales [Christensen et al., 2007]. These uncertainties are associated with initial conditions, forcings, climate model structural choices or parameter settings, and/or internal unforced variability [Cash et al., 2005; Giorgi and Francisco, 2000; Murphy et al., 2004; Räisänen, 2001; Tebaldi et al., 2005]. These uncertainties lead to lower confidence in obtaining robust results required for impacts, adaptation, and vulnerability assessments. To improve predictions, we require both a methodology to assess the predictability of climate models and systematic approaches to explore the complex interactions in the climate system. As these tools become available, we can begin to assess the relative contributions of the various sources to the overall uncertainty in climate predictions.

[3] Among the uncertainties in regional climate change simulation from GCMs, uncertainty related to sea surface temperature (SST) anomalies is a primary source SST patterns affect the tropical and extratropical climate primarily through Walker circulations and atmospheric teleconnection patterns due to the propagation of Rossby waves [Bjerknes, 1969; Horel and Wallace, 1981; Lau and Nath, 2001; Trenberth et al., 1998; Walker, 1923]. Errors in SST patterns over the tropical Ocean can alter a climate model's seasonal response [Goddard and Mason, 2002; Goddard et al., 2009] and affects the regional climate change on longer time scales [Shin and Sardeshmukh, 2011]. Although it is hard to reduce the biases of simulated SST by improving model formulation in the short term, we can improve our understanding of the sensitivity of regional climate change to tropical SST anomalies to help assess and hopefully reduce the uncertainty in both SST prediction and simulation of regional climate variability.

[4] So far, both statistical analysis and modeling work have indicated that regional climate response is related to SST anomaly patterns over the tropical and extratropical oceans. For example, using observational data and CMIP model output, Perlwitz et al. [2009] showed that the cooling over North America in 2008 was related to the tropical Pacific SST anomaly pattern for that year. Reason and Mulenga [1999] used an AGCM to show that warm SST anomalies over the Southwest Indian Ocean could lead to significant changes of the seasonal rainfall over southeastern Africa. In addition to the impact of large SST anomalies over specific oceans, other works [Barsugli and Sardeshmukh, 2002; Barsugli et al., 2006] (hereafter BS02 and BSS06) examined the impact of small SST anomalies over broad ocean domains. Using an AGCM, BS02 and BSS06 place individual SST anomaly patches over the tropical oceans (“patch method” hereafter) and estimate the sensitivity of the linear component of the regional climate change to the tropical SST anomaly patches.

[5] Although the equations for modeling the climate system are highly nonlinear, it is well established that certain aspects of the climate system can be approximated by a multivariate linear statistical model. The El Nino-Southern Oscillation (ENSO) phenomenon is remarkably well approximated by a linear stochastic model where the nonlinearities of the real system are all subsumed into the additive noise forcing or into a modification of the linear operator [Penland and Sardeshmukh, 1995]. A linear stochastic model with both additive and multiplicative noise can be used to approximate atmospheric dynamics and fluxes [Sura et al., 2005] and for medium-range forecasting [Newman and Sardeshmukh, 2008; Pegion and Sardeshmukh, 2011]. In all these cases, the complexity of the nonlinear system precludes the computation of the linear approximation from first principles, motivating empirical or semi-empirical methods.

[6] Based on this linear approximation, in this paper, we estimate the sensitivity of the regional climate to the anomalies of tropical ocean through two methods. The structure of the paper is as follows: Section 2 describes the patch method and the random perturbation method (RPM) that we use in this work and how we apply it to the model. The computational efficiency of the two methods is also discussed in this section. Section 3 presents some applications and comparisons of the two methods in aspects such as teleconnection patterns, sensitivity maps for continental regions, and linear reconstruction of the seasonal response to SST anomalies. Section 4 provides the conclusions of this study and discusses the uncertainties of the regional climate simulation.

2. Methods

2.1. Model

[7] We use the National Center for Atmospheric Research (NCAR) Community Atmospheric Model (CAM) version 3.1 [Collins et al., 2004] at T42 resolution as our simulation tool to demonstrate the method. The base climatological SST and sea ice boundary data set (12 months) for all sensitivity experiments is HadOIBl data at the same resolution as the atmospheric component [McCaa et al., 2004].

2.2. Patch Method

[8] As shown by multiple works, the mean state of the atmospheric response (e.g., seasonal mean) to large scale forcing can be considered as a linear process [Schneider et al., 2003; Deser and Phillips, 2006; BS02; BSS06]. Specifically, the atmospheric response at a given region can be approximated as the sum of the response to each localized SST forcing (i.e., a Green's function as shown in BS02 and BSS06). For a scalar response R this reduces to (see details in Appendix A):

display math

where for this application, R is the mean (e.g., seasonal or annual) atmospheric response, K(xk) is a linear operator relating the response to the SST anomaly field T(xk) on the model grid xk (see Figure 1). Ak is the area of the grid cell, and is included so that K is independent of the grid size and has units of the response divided by the product of temperature and area. We have assumed that systematic nonlinearities are small and that the variance of ε in equation (A3) in (i.e., equation (A3)) is σ02 (following notation in Appendix A).

Figure 1.

Sketch interpreting the SST anomaly patch T at location i and regional response R over region of interest (ROI) j. Blue crosses refer to the location of SST anomaly patches defined by BSS06.

[9] To estimate K, BSS06set 43 idealized small SST anomaly patches across the tropical Indo-Pacific and Atlantic oceanic regions and estimated the response to the SST anomaly by running a GCM. The anomaly patches are added to the prescribed SST field used in the control run and an ensemble of 2 M runs are performed for each of these patches (M runs each for positive and negative amplitudes for each anomaly patch). The patch patterns are cosine-squared functions (seeBS02) with central amplitudes set to −2°C and 2°C with a size (1°C anomaly contour) of 45° × 11° (longitude x latitude) over the Indo-Pacific ocean and 28° × 11° over the Atlantic Ocean. The patches are the same size as inBSS06 (n.b. BSS06 lists an incorrect latitude dimension as 22°.) The patches are small compared to the scale of the remote dynamical response [Sardeshmukh and Hoskins, 1988] while being large enough to provide a statistically significant response. Patches are also chosen to be small enough in amplitude so that the large-scale circulation response to an individual patch is dominated by linear effects in most cases.

[10] As in BSS06, we created 43 tropical SST anomaly patches over the tropical ocean (Figure 2a) and added them to the twelve-month background SST field with M=16. To be consistent withBSS06, we run 20 months and 32 months for Pacific and Atlantic patches respectively. Following BSS06, and noting that the patches act to smooth the Green's function, the sensitivity of the seasonal response R for region j is calculated as:

display math

where math formula is the smoothed Green's function; 〈Rj〉 is the ensemble mean of the seasonal response over a region of interest (ROI) denoted by index j, for warm or cold patches; and math formulais the area-integral of the SST anomaly over a given patchi on the model grid xk. Assuming that the random noise is independent and identically distributed, which is to say that the SST forcing does not appreciably change the variance of the response, the variance of η is approximately:

display math

and represents our uncertainty in the estimate of K. Later we will compare a similar estimate for the RPM method to determine the computational efficiency of each method in certain idealized cases. By moving the anomaly patch to different locations, the sensitivity of climate change for the region of interest can be mapped out across the oceans.

Figure 2.

(a) SST anomaly patches as used in BSS06 and (b and c) two examples of SST perturbation fields over the tropical region. The range of SST perturbations in Figures 2b and 2c is −2K to +2K.

2.3. Random Perturbation Method

[11] Because the mean state of the atmospheric response to tropical SST forcing can be considered an approximately linear process as shown in BSS02 and BS06, inspired by the techniques used in data assimilation [Hawblitzel et al., 2007; Sippel and Zhang, 2008; Zhang, 2005], we propose to estimate K (e.g., regression coefficient) using a random perturbation method (RPM) as described below.

[12] We randomly generate a 16 × 16 random matrix that are equally spaced in longitude (22.5° resolution) and latitude (11.25° resolution) across the entire globe, and interpolate them to the AGCM model grid resolution (e.g., T42 corresponds to 128 × 64) using bilinear interpolation. Each point on the 16x16 grid is drawn from a uniform distribution with a range of −2 K–2 K (Tmax = 2 K). By design, the perturbations are thus random in magnitude and uncorrelated in space and across ensemble members. We also consider the fact that after bilinear interpolation, local maxima and minima must occur at the original perturbation grid points (the 16 by 16 grid). As such, we randomly shift the initial coordinate on the 128 × 64 grid by 1∼8 grid points zonally and 1∼4 grid points meridionally after doing the interpolation from the coarse coordinate (Note: for T42 resolution, each 16 × 16 size boxes is composed of 8 grid points in the zonal direction and 4 grid points in the meridional direction). The spatial autocorrelation function of these random perturbations is derived in Appendix A, equation (A11).

[13] We generate 200 sets of SST perturbation fields and add them to the model climatological SST field as forcing for the AGCM ensemble simulations. We force the model with the updated SST field and perform a branch run starting from year 10 of a 160-year control simulation (we use a 9-year spin-up for the control run to get an equilibrium state). To include uncertainty in the initial conditions, we perform 200 branch runs from different starting points with a one year-interval (note: 50 of the initial conditions have been used twice) and run 20 months for each perturbation field. The model output from months 9–20 is used to calculate the seasonal mean response.

[14] Because we plan to compare the sensitivity of the regional response to tropical SST anomaly as estimated from the patch method, we extract the SST perturbations from the RPM fields over the tropical ocean (30°S-30°N) from the global domain. To avoid a sharp decrease at the boundaries, we apply a tapering within 20°-30° (both South and North) with weights 0.7, 0.5, 0.3, 0.1 to ensure a smoother transition in the SST perturbation field (Figure 2b).

[15] The random SST fields exhibit spatial autocorrelation on roughly the same length scale as the spacing of the coarse lattice. As a result the autocorrelation function, like the patches, acts as a smoother on the Green's function. The estimate of the smoothed Green's function is derived in Appendix A. By construction, the spatial correlation length scale of the RPM field is smaller than that of the patches (Figure 3). In fact, for the Pacific, the length scale of the RPM is about half that of the patches. In the Atlantic, the disparity is somewhat smaller because the patches are smaller. We could use equation (A15) to estimate the Green's function at the scale of the RPM field, however the differences in spatial scales would complicate the comparison with the patch method.

Figure 3.

Point-correlation between SST perturbation and other points over the ocean at two different locations (Niño3.4 region and grid point (160°W, 15°N)). Only the significant correlations (at the 95% confidence level) have been plotted.

[16] Instead, we use the patch function as a spatial filter on the SST anomalies in the RPM fields, and then use simple linear regression of the response against these filtered SST anomalies. The regression coefficient K for a region of interest (ROI) j and patch i can be estimated as:

display math

where math formulais the area-integral of the SST anomaly over a given patchi on the model grid xk(i.e., masking the RPM field over the region where patch amplitude is greater than 0.05 and then taking the area-integral of the RPM field on model grids), andσ is the standard deviation. We note that the values of math formulacan be estimated using the patch area as described above or using the area of a single grid-cell corresponding toxk. We refer to these estimates of K as patch-weighted and grid based, respectively, and they represent the sensitivity to different scales of the SST anomaly forcing fields (as discussed inAppendix A).

[17] Because the regression coefficients reflect the global teleconnection relation between SST anomaly patches and the entire response vector over the whole globe, we define the matrix form of Kji as the global teleconnection operator (GTO).

2.4. Error Estimate of Patch Method and RPM Method in the Linear Framework

[18] The estimate of the GTO operator from either method is limited in precision by the size of the ensemble and the length of the model runs over which the time averages are taken. Here we discuss how the two methods give the same sensitivity information and estimate their error with specific attention to the computational efficiency of each method. We first give a heuristic explanation, followed by a theoretical calculation based on an idealized case, and finally refer to actual error variance estimate from the GCM runs themselves.

[19] First, we expect the RPM method to have lower linear error variance in the estimate of K simply because the magnitude of each RPM SST anomaly field at patch size, with amplitudes at many locations, is larger than in the case for a single patch. We consider the P = 43 patch locations. In the RPM fields, the variance of SST at each of these locations is 1/3 that of the patches, so that the area-integrated mean square amplitude of the SST field in the patch method is roughly 3/P that of the RPM method (compare to the factor of 3/P in the more complete estimate inequation (12) below). This is because a larger amplitude forcing is used in the RPM method. We also note that it may be more sensitive to nonlinearities in the responses to SSTs when compared with the patch method. A subtlety worth noting is that the SST amplitude at any given location in the RPM method is typically less than in the patch method, reducing the effect of local nonlinearity in the response of the atmosphere to SSTs. Indeed, applying modest forcing at many locations is consistent with the multivariate linear approximation that is the basis for this analysis.

[20] Second, we present an idealized calculation that sheds light on the various factors that influence the estimation of K, including the ratio of the patch size to the RPM correlation length-scale, the areal coverage of the dominant sensitivity, the magnitude of noise variance in the region of interest, and the choice of the method itself. We note that both methods can be formally reduced to univariate linear regression on smoothed SST anomaly fields (seeAppendix A). Because the overlap of patches and offsets of RPM fields complicate the analysis, we proceed by defining a simpler arrangement in which the comparison can take place. We consider the standard error variance in the estimates of the sensitivity as a function of the total number of runs for each method and ask whether the same level of standard error variance can be achieved with fewer model runs. We do not investigate the effect of nonlinearity in this analysis; based on results in later sections, the linear approximation is appropriate for many combinations of forcing and response locations.

[21] For the patch method, consider P patches with 2 M ensemble members per patch, for a total of Np = 2MP runs. From equation (3), the variance for this case is:

display math

[22] Contrast this to an RPM ensemble of Nrpmmembers over the same domain. We make two simplifying assumptions for this comparison: 1) the statistics of the internal variability is well approximated by the unforced variability that is uncorrelated from run to run and is approximately Gaussian, and 2) no offsets are used, so that each RPM field, with n, is a sum of local patch-like functions (seeAppendix A):

display math

hi is the structure of each individual patch, and bni is a random amplitude that is generated from a uniform distribution on the interval (−1,1). The linear responses over a region of interest are then:

display math

[23] Without loss of generality we have singled out the first lattice point as the one for which we want to estimate the sensitivity. In this idealized version of the RPM, the h-functions act like patches and smooth K, and we can write:

display math

where math formula and math formula are defined analogously to math formula and math formula in equations (A5) and (A6).

[24] We can then estimate K by ordinary least squares regression. We will drop the “hat” notation, with the understanding that in the rest of this section, K refers to the smoothed values at the spatial scale of the h-functions.

[25] The main differences from the patch method are that the forcing has smaller amplitude on average at any point (the absolute value of bi is on average less than one), and that, in addition to the response to h1,, we must deal with the responses to SST anomalies at other locations with random amplitude. By construction, these remote SST anomalies are uncorrelated with the SST at the location of h1, but with a finite sample size they will act as an additional source of noise in the response Rn. We can approximate the second term on the RHS of equation (7) as a random variable with variance:

display math

[26] If we define math formula, then math formula. S2is the spatial variance of K at the spatial scale of the RPM fields and we have assumed that the h-functions are all the same size for convenience of notation in this idealized calculation. Note that the variance ofbni is 1/3, and the variance of a sum of random independent variables is the sum of their variances.

[27] To estimate the standard error variance in estimate of the sensitivity Kj, we note that the standard error formula for ordinary least squares estimate yields:

display math

Insert the expression of σ12 to equation (10) we have:

display math

and the ratio of the two variances (i.e., equation (11)/equation (5)) can be written as:

display math

[28] To say that we have the same power in estimating K, we require that the variance of the estimates be the same. Consider the idealized case where the patches and the RPM fields have the same length scale and magnitude of forcing ( math formula). Note that the σ2 of the K in patch method is directly controlled by math formula (see equation (5)), while the σ2 of the RPM is controlled by math formula and the ratio of S/K1. Typical response signals are on the order of 1/4 of the unforced standard deviation, i.e., math formula, and typical area of strong sensitivity is about 10 patches with other patches contributing much less, i.e., math formula. By inspecting results from both methods, typically, math formula is about 1, and P = 43. After inserting these numbers into equation (12) and assuming σ2(KRPM) = σ2(Kpatch), we have:

display math

[29] Therefore, in the idealized case of a relatively small area of sensitivity and equal patch and RPM amplitudes and scales, the RPM is approximately 12 times more efficient than the patch method.

[30] What does this mean for the RPM and patch methods as implemented, with their different spatial scales, different number of runs (NRPM/Np = 200/1376) and the spatial filtering used in the estimation? For the RPM fields, we have math formulain the Indo-Pacific, and math formula in the Atlantic. Plugging these values into equation (12) yields an error variance ratio ranging from roughly 0.9 to 2.3. However, in using the spatially filtered estimation method, we have effectively averaged over about two times the RPM covariance length scale. This leads to a reduction in average SST amplitude of the square root of two, and a doubling in the effective area of SST forcing for the filtered RPM fields. Therefore, math formula for the filtered RPM fields is effectively twice as large as for the unfiltered RPM fields. These simple scaling arguments lead us to expect that the RPM method as implemented will have roughly the same error variance in the estimate of the sensitivity as the patch method, with a factor of seven fewer runs.

[31] Finally, the standard error variance can be evaluated for the two methods empirically from the runs themselves (see auxiliary material). We find that the RPM in fact performs similarly to what is expected based on the idealized calculations, when all the factors in equation (12) are considered. The RPM, despite using a smaller spatial scale than the patches, is still able to achieve a generally comparable level of error variance with a smaller total number of runs. However, the results are not completely consistent across variables and seasons, with greater gains in efficiency for temperature than for precipitation and for winter than for summer. Nonetheless, this analysis provides a significant basis to compare results for the two methods.

3. Results

3.1. Teleconnection Pattern Response to an Individual SST Anomaly Patch

[32] We first examine the ensemble model response for a specific SST anomaly patch over the central Pacific Ocean to compare the two methods. For the seasonal large-scale (global or hemispheric spatial scale) response, the two methods are roughly consistent in terms of the sensitivity of the temperature at 850 hPa to the SST anomaly patch at that location (Figures 4 and 5). The linear fit provides the estimate of Kj in equation (2)and we see good agreement from the patch and RPM response. Additional comparisons indicate similar results (e.g., summer (JJA) season, a second patch localized over the Indian Ocean, and other variables (e.g., precipitation, wind)) (not shown). The winter response of the northern extratropical region shows more variability from the initial conditions than other regions. The resulting similarity in the sensitivity estimates implies that to explore the sensitivity of the large-scale seasonal response to a localized SST anomaly patches, we now have two options. Either we can set single fixed patches and run individual ensembles for each one or set random patches globally and only run a single ensemble to estimate sensitivity for all patch regions.

Figure 4.

Scatterplot of the winter (DJF) regional mean T850 against SST anomaly integrated over one patch region (centered at 187.5E, 5.6S with dimension of 45° × 11°) for Global, Tropics, Northern extra-tropics and Southern extra-tropics. Red dots denote the 200-ensemble response from RPM, blue dots are the 32-ensemble response from patch method. Black and blue regression lines are added for both cases. Black open circles are the T850 at initial conditions for 200-ensemble runs of RPM. Black solid dot is the mean T850 of the 200 initial conditions for RPM. Blue solid dot is the mean T850 of the 32 initial conditions for patch run. The T850 temperature range for all cases is 2K.

Figure 5.

Same as Figure 4except showing results for Amazon Basin (20°S-12°N, 82°W-34°W); Central America (10°N-30°N, 116°W-83°W); Western North America (30°N-60°N, 130°W-103°W) and South Africa (12°S-35°S, 10°W-52°E). The T850 temperature range for all plots is 8K.

[33] We further investigate the seasonal response for continental regions as defined by [Giorgi and Francisco, 2000]. We verify the RPM at all 21 Giorgi regions but illustrate the results for four representative regions for DJF (see Figure 5). Several additional regions (e.g., Australia, other different parts of America, Northern Europe, different parts of Africa, east, south and central Asia) also indicate consistency between the two methods. However, not all Giorgi regions show consistent results for the two methods (e.g., Mediterranean Basin). Inconsistency between the methods can arise for two reasons: noise and nonlinearity. In regions of small linear signal, the noise may dominate the response, even for the relatively large ensemble sizes that are used. Nonlinearity in the response to SSTs may also be dominant over the linear signal. Given the nature of the RPM fields compared to the patches, the potential non-additivity of responses to SST anomalies in different locations is of particular concern. Given the multiple stages of the causal chain leading to teleconnection patterns, we expect nonlinearities in the response would be related to specific physical processes such as Rossby wave-mean-flow interaction, tropical precipitation dynamics, or local nonlinearities in variables in the regions of interest. Whether due to nonlinearity or to unrelated noise in the system, inconsistency between the patch and RPM method usually occurs where the linear signal is only a small contributor to local variations. Thus, the disagreement is not likely caused by a fundamentally different sensitivity originating from each method.

[34] We estimate the teleconnection pattern between the mean SST anomaly for the Niño4 region and the simulated temperature at 850 hPa for the global domain (Figure 6) for both methods. The correlation map between the regional mean SST perturbation over the Niño4 region (Figures 6b and 6d), although obtained by setting random perturbation across the entire tropical region (RPM), shows strong similarity to the result obtained from the ensemble mean of the response to a single patch over same Niño4 region (i.e., patch method) (Figures 6a and 6c). The patterns indicate consistency of the teleconnection relation between a local SST anomaly and global response from the dynamical perspective and statistical point of view. The patch method is a direct way to see the teleconnection response, while the RPM provides similar features estimated from the 200-member ensemble. The critical design component in the patch method is that the SST perturbation at each grid point (or patch-like region) is independent of the perturbations at other grid points. On average, the response to the other SST perturbations is zero and we can analyze the teleconnection pattern focusing on SST anomalies over specific tropical regions.

Figure 6.

Comparison of Temperature at 850 hPa forced by SST anomaly patch over ∼ Niño4 through (a and c) patch method and (b and d) RPM for different seasons: DJF (Figures 6a and 6b) and JJA (Figures 6c and 6d). The units for the plots are K.

3.2. Sensitivity to Tropical SST Anomalies: The Global Teleconnection Operator

[35] We present the GTO (i.e., the sensitivity of regional response to tropical SST anomaly, dR/dTpatch) of the T850 and precipitation for three Giorgi regions from both methods. (Figures 710). Eastern North America represents a continental region over extratropics, while East Africa and Southeast Asia are two regions largely affected by monsoon activity. A common feature revealed in the plots is the sensitivity to the SST anomaly over the Indian Ocean and the western Pacific. The winter low level temperature over Eastern North America and the summer temperature and precipitation for all three regions are also sensitive to the SST anomaly over the eastern Pacific. Another feature revealed in the result is the Indian Ocean - western Pacific dipoles in the sensitivity of winter temperature over Eastern North America and winter and summer precipitation over East Africa and Southeast Asia. Such dipole patterns can also be found in the sensitivity map of other Giorgi regions. For example, the dipole patterns exist in the sensitivity map of winter T850 over the western North America, Central North America, and Alaska. It also exists in the sensitivity map for other high latitude regions of Northern Hemisphere such as Greenland, Mediterranean Basin and Northern Europe (figures not shown). Because the sensitivity maps shown inFigures 710 reflect not only the correlation between the tropical forcing and the response but also the amplitude of the response, the sensitivity map can be used as a metric to quantify the regional climate change to SST patterns. The information is independent of the strength of the forcing, however, it does depend on the background climatological SST.

Figure 7.

Comparison of sensitivity map of winter (DJF) T850 over three different target regions: Eastern North America; East Africa, and Southeast Asia as estimated from the (a, c, and e) patch method and (b, d, and f) RPM. RPM ensemble size is 200. Shaded regions denote where the sensitivity is significant using a two-tailed t-test at 20% significance level. (Units: K per (K*km2*109).)

Figure 8.

Same as Figure 7 but for summer (JJA) season.

Figure 9.

Same as Figure 7 but for precipitation. (Units: (mm/day) per (K*km2*109).)

Figure 10.

Same as Figure 9 but for summer (JJA) season.

3.3. Linear Reconstruction of the Regional Response

[36] To examine how well the regional climate anomaly will be represented by the linear regression of the tropical SST anomaly, we multiply the sensitivity map for each Giorgi region from both the RPM and patch methods by the historical SST anomaly weighted by the patch SST anomaly to estimate the reconstructed linear component of the regional climate anomaly based on the linear reconstruction method from BS02 and BSS06:

display math

where ΔSST(xk,t) is the historical SST anomaly from HadOIB1 during 1950–2000 at location xk, ΔSSTi(xk) is the SST anomaly defined in the patch method and is used as a weighting function. We defined α = 1.25 considering the overlap of the patches as BSS06. Kji is the sensitivity of the interested target region j to SST anomaly patch i, and Ā is the average grid-box area as discussed below. For RPM, we used SST anomaly on the raw model grid to avoid distorting the linear reconstruction from using the patch-weighted SST anomaly. In that case,

display math

where N is the total ocean grid for the model, and β ≈ 10 for global scale region but varies from 8 to 56 for T850 reconstruction and 8 to 48 for precipitation after an extensive examination of β for all Giorgi regions. We recognize that β is related to the length scales of both the forcing and response for a given target region based on the dynamics. Considering that β is trivial in the correlation result that presented here, we leave further investigation of β to future work.

[37] In this work, we use a uniform grid area, which is the average grid box area across the globe. Similar to BSS06, we then correlate the reconstructed response anomaly with the result from ensemble mean with the full CAM model runs forced by the observed SST during 1948–2000 (i.e., AMIP runs [Deser et al., 2006]). To first illustrate the impact of the size of AMIP runs and RPM ensembles on the correlation, we tested the variation of the averaged correlation coefficient of tropical and extratropical Giorgi regions with the change of the number of AMIP runs and size of the RPM ensembles (figures not shown). There is dependence of the correlation on both AMIP and RPM ensemble sizes. In the examination of correlation as a function of AMIP ensembles, the results in extratropical regions converge to a stable state more slowly than for tropical regions. In the examination of correlation as a function of RPM ensembles, the correlations increase quickly after 100 ensembles of RPM runs and then increase slowly or vary only within a narrow range of correlations.

[38] The correlation coefficients between the reconstructed response from the two methods and the full model response during 1950–2000 are shown in Figure 11. To demonstrate the impact of the RPM ensembles on the correlation results of individual regions, we included the reconstructions using 200, 400 and 600 ensembles for RPM for comparison. If we only focus on the regions with significant correlation, the two methods show consistent results for global scale regions and most of the tropical continental regions. Compared to these regions, the correlations for most extratropical regions are either low or have larger discrepancies between two methods. For instance, a discrepancy occurs in the lower level temperature (T850) over Southern South America (SSA), North Western America (WNA), and certain parts of Asia. A discrepancy also occurs in precipitation over the American continent and South Asia (SAS). Average correlations from seven tropical regions (AMZ, CAM, WAF, EAF, SAH, SEA, SAS) and fourteen extratropical regions (i.e., the rest Giorgi regions) indicate that the two methods are more consistent with each other in reconstructed lower level temperature than for precipitation. Note: as an alternative to the Giorgi regions, we have also examined the quality of the grid point based reconstruction as smoothed using a T15 filter and see strong consistency between the methods (see auxiliary material).

Figure 11.

Correlation between the reconstructed response and the AMIP ensemble mean during 1950–2000 for Giorgi regions and global scale regions from RPM and patch method. For RPM, 200 (blue bars), 400 (green bars), and 600 (gray bars) ensemble sizes are used respectively in estimating K so that to compare the impact of the RPM ensemble number on the reconstruction. For patch method (black bars), 32 ensemble members are used in estimating K. For both methods, 15 AMIP runs are used in the correlation. The critical value of the correlation coefficient for n = 51 is 0.27 at the significant level of 5%.

[39] The seasonal response over some regions are shown to be sensitive to the ensemble size of the RPM, for example, the winter temperature over EAF, EAS, SAS and summer temperature over AUS, AMZ, SSA, EAS, CAS, NAS. The winter precipitation over CAM, WNA, CAN, ENA, ALA, EAS, TIB and summer precipitation over AUS, AMZ, CNA, WAF, EAF, SEA and SAS are also shown to be sensitive to RPM ensemble size and the correlations are shown to be improved by increasing the RPM ensembles.

[40] A comparison of the internal variability versus forced variability for T850 and precipitation over Giorgi regions during different seasons are shown in Figure 12. Over all, for the 850 hPa temperature over extratropical regions are subject to more internal and forced variability than tropical regions and global scale regions. The extratropical regions exhibit more variability in T850 during winter than in summer. For the precipitation, tropical regions indicate increases in both internal and forced variability than extratropical regions. The internal and forced variability of Giorgi regions are similar during both summer and winter except Central America and South Asia where the forced variability are prominently larger during the summer season.

Figure 12.

Forced variability versus internal variability of the response for Giorgi regions and global scale regions for RPM. The internal variability is defined as the variance of the seasonal-mean regional response for 15 AMIP runs during 1950–2000. Forced variability is defined as the variance of the ensemble spread (ensemble mean is removed) for 200 ensemble members from the RPM.

[41] If we define the above correlation as a measure of predictable skill of the regional climate change, the regions with smaller internal and forced variability (e.g., tropical regions) generally have higher predictable skill of the lower level temperature. The predictable skill for the regional precipitation can be related to the ratio of the forced variability to internal variability. That is, the regions with a larger ratio of forced variability to internal variability have larger predictable skill than those regions with smaller ratios (except global scale regions).

4. Conclusions and Discussion

[42] The assessment of sensitivity of regional climate change usually requires a large number of ensemble runs to obtain reliable results. Previous work estimating the regional climate sensitivity to tropical SST anomalies would set an SST perturbation using a single patch in a given location for individual runs. We introduce a Random Perturbation Method (RPM) that sets the SST perturbations over the entire tropical ocean for each simulation. We estimated the standard error for the two methods for an idealized version of the two methods and compared that to the error of estimation using the GCM data. Under idealized conditions where the largest sensitivities are restricted to an area of around 10 patches, and where the spatial autocorrelation scale of the RPM field is similar to the patch size, the RPM is expected to take about 1/12 of the computational time of the patch method to obtain the same standard error in the estimated sensitivity. Taking into account differences in implementation from the idealized case, we expect the computational time to differ by a factor of 1/7. In practice, the standard errors from the two methods are generally consistent in their order of magnitude with these theoretical estimates (see auxiliary material). The main reason for the increased efficiency is the larger area-integrated mean square amplitude of the SST forcing anomaly field in the RPM method. The two methods provide very similar sensitivity information for many of the teleconnection patterns investigated in this paper. We have tested the robustness of the approach using alternate version of math formula in equation (4) and find the math formula to be stable in providing the sensitivity information.

[43] We further evaluate the RPM to reconstruct the regional anomalies that linearly related to historical SST anomalies. Both the RPM and patch methods show similar correlations between the linearly reconstructed response and the full model response for global scale regions and most of the tropical continental regions. Overall, the global-mean climate changes are reconstructed well by the linear regression of SST anomaly because the noisy signal has been smoothed by the large-scale averaging inherent in the historical SST record. Climate variations in tropical regions generally are better reconstructed than the extratropical regions. The large-scale temperature response has a better reconstruction than the precipitation due to a less noisy signal at smaller regional scales.

[44] The correlation for precipitation change in tropical regions is generally lower than patch method. There are also some regions that the reconstruction using RPM is better than the patch method (e.g., the winter temperature over SSA, MED, EAS, summer temperature over CAS, TIB, NAS, GLB, winter precipitation over WNA, NEU, SAF, EAS, CAS, TRO, and summer precipitation over CAN, MED, EAS, CAS, TIB). For both temperature and precipitation, the correlations between the linear reconstruction and AMIP response are significantly high over some regions (e.g., global scale regions and tropical regions). The response over most of the subtropical and high latitude regions is more difficult to predict due to multiple factors. These include the uncertainty due to the internal variability [Christensen et al., 2007, Branković et al., 1994], uncertainty in forcing, or the nonlinearity of the regional response to tropical SST anomalies.

[45] Overall, we find that the RPM provides similar information as the patch method in regions of strong linear response and standard errors can be chosen depending on the purpose. For exploring the GTO from a large number of models, we would choose Nr to be smaller such that more models could be explored. If higher accuracy is required for the GTO to be used as linear prediction tool, we would increase Nr appropriately. The patch method is still valuable for more intensive exploration of regions of low signal, or for exploring the nonlinearity of the response because each patch experiment can unambiguously be interpreted as the response to SSTs in a single location. However, the flexibility and ease of implementation of the RPM makes it a useful addition to exploring sensitivity of regional climate variability to SST anomaly patterns.

Appendix A

[46] Here we provide a detailed examination of the estimated error of the sensitivity values, Kji as discussed in section 2 of the main text. We start with the continuous Green's function representation for a linear response:

display math

where T(x′) is the anomalous SST at location x′, R(x)is the (time-averaged) anomaly that occurs in response to the SST forcing,dA′ corresponds to the area integral over x′, and e(x) is an error term that includes nonlinearities and random errors due to the internal variability of the atmosphere. In the bulk of this paper we will assume that the systematic nonlinearities are small and that the error term is dominated by random noise that is approximately Gaussian. In a discretized grid space, this becomes:

display math

where k corresponds to the spatial distribution of the response grid space vector . If we subsume the area factor into fk = AkTk and choose vector notation so that the Green's function G, is independent of the size of the model grid. (Note: Hereafter and where possible, we use vector notation to limit the use of subscripts.) For a scalar response R = wTR, for example the weighted area average of R, we have

display math

where K = wTG, ε = wTe, and K is the linear operator corresponding to a specific region.

[47] For the patch method, let fi be a single anomaly patch centered at xi, as described in the text and in BSS06. Equation (3) becomes:

display math

where we define a smoothed sensitivity

display math

and the integrated patch amplitude for the i-th patch as

display math

[48] Because the anomaly patches are smooth and positive, they can be thought of as acting to smooth the Green's function. Given an ensemble of M model runs for warm anomalies +fi and M for cold anomalies −fi, we can estimate math formula from the difference of the ensemble mean response, 〈R〉:

display math

[49] Assuming (as above) that the nonlinearity is small and that the random errors for each ensemble member are independent and identically distributed with variance σ02 (σ02 refers to the variance of ε in equation (3), which is the variance of the internal variability and nonlinearity then the variance of η, which is the standard error in the estimate of math formula, is approximately equal to:

display math

[50] For the random perturbation method, recall that the RPM field is defined as the bilinear interpolation to the model grid from random temperature values at a coarser lattice of grid points. This is equivalent to the sum of random-amplitude pyramid-shaped patches, with a spatial scale determined by the lattice spacing. Therefore, we define the n-th RPM field as follows:

display math
display math

where bni are uniformly distributed random numbers on the interval [−1,1] at the i = 1,..,P lattice points, Tmax = 2K, λ and ϕ are the longitude and latitude coordinates of x, and

display math

is the triangle, or “chapeau” function that defines linear interpolation. The lattice spacing, (λ0, ϕ0), determines the spatial autocorrelation scale of the RPM fields. The offset Δn = (λn, ϕn) shifts the entire RPM field in space by integer multiples of the model grid, and is randomly chosen for each RPM ensemble member in order to obtain a more uniform coverage of the domain.

[51] The spatial autocovariance of the RPM fields, in the limit of large NRPM can be determined from equation (9), by first calculating the correlation function for a single offset Δn and then summing over all possible offsets. The covariance depends only on the difference in latitude and longitude between the points. It is, however, simpler to replace the sum over Δn by an integral, in which case the covariance can be written as:

display math

where

display math

Note that the integral of this function over the real line is 1.

[52] To estimate K we take the covariance of the SST at a given location (subscript i) with the response defined in equation (A3) across the RPM ensemble (denoted by subscript n),

display math

[53] We can write the finite-sample auto-covariance function as the sum of the infinite ensemble size estimate defined in(A10) and a perturbation covariance, C′, to obtain:

display math

[54] Note that C can be thought of as a smoothing function applied to K. Analogous to the way that the patches behave in equation (A5), equation (A12) can be used to estimate the smoothed Green's function as follows:

display math

where

display math

[55] In this formalism, because C is localized, we assumed a locally planar approximation in equation (A10) and integrated over the domain, with the local length scales for the RPM lattice being Lx = 0 sin ϕiLy = 0, and a the radius of the Earth.

[56] Given the above we can write the estimate of the smoothed Green's function as

display math

where we identify two error terms corresponding to the second and third terms on the RHS of equation (A13), the first due to the random SST amplitude away from xi that contributes spurious responses in the ROI, and the second due to the internal variability in the ROI itself. These error terms are discussed further in section 2.4 of the paper.

Acknowledgments

[57] We appreciate Fuqing Zhang (PSU) for his suggestion and subsequent discussion on constructing the ensemble. We also appreciate the two anonymous reviewers for their valuable comments. This work is supported by DOE grant DE-SC-0005399.