1. Introduction

In the course of examining model-simulated vertical distribution of heating (Q₁) and moistening (Q₂), several authors expressed concerns about possible modeling errors in the amplitude and in the placement of the vertical levels of maximum heating or moistening (Tao et al. 2007). Such errors may stem from choice of cumulus parameterization schemes in large-scale models. Furthermore, it has also been recognized that error from a cumulus parameterization can be model dependent; that is, the rest of the model physics and dynamics can dictate a particular behavior of the cumulus parameterization scheme within a particular model. Nonhydrostatic models that include microphysical processes can carry sensitivities specific to microphysical processes and can also exhibit similar model dependencies. For hurricane simulations, the modeling of the vertical level of maximum heating and its amplitude were recognized to be fairly important (Ooyama 1969; Ceselski 1974). Convection plays a major role in tropical intraseasonal variability and quasi-stationary circulation. Errors in the vertical placement of heating and the magnitudes of heating do have implications in the modeling of convectively coupled wave propagation such as ITCZ, MJO, and moist Kelvin waves. (A list of acronyms is provided in Table 1.) Climate models have difficulty in the simulation of MJO, and large sensitivity of such simulations to the vertical heating distributions has been noted by several authors (Slingo et al. 1996; Waliser et al. 2003; Zhang 2005; Zhang et al. 2006).

Having observed estimates of heating rates is fairly important for model validations. In a series of papers, Shige et al. (2004, 2007, 2008) have examined the vertical profiles of heating in tropical weather systems. Their studies have shown the limitations in the modeling of the vertical distribution of heating from case studies that covered Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) and Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) events. Lau et al. (2000) and Johnson and Ciesielski (2002) have pioneered examining vertical distribution of heating using field experiment datasets, especially those from South China Sea Monsoon Experiment (SCSMEX). Yuter et al. (2005) have studied the diurnal change of latent heating within the context of the Tropical Rainfall Measuring Mission (TRMM) Kwajalein Experiment (KWAJEX).

2. Superensemble methodology

The superensemble methodology combines a set of multimodel forecasts to construct a single consensus forecast (Krishnamurti et al. 1999, 2000). In this methodology, the models are combined with different weights, which is unlike a simple ensemble construction. The weights of the models are determined based on their past performance. The details of this methodology are as follows.

The time line of the model dataset is divided into two parts: a training phase and a forecast phase. In the training phase, the models are compared with the observation–analysis data. A multiple linear regression is performed in this phase to estimate the relative performance of the member models. The outcome of this regression is statistical weights assigned to each of the models. These weights are then passed on to the forecast phase to construct the superensemble forecast:

View Expanded

where O is the observed mean field during the training phase; a_i is the weight for the ith member model; F_i and F_i are the forecasts and mean forecast fields during the training phase from the ith model. The summation is taken over the N member models of the suite. The calculation of the weights are done by multiple linear regression, which minimizes the error term

View Expanded

where N_tr is the number of time samples in the training phase, and S′_t and O′_t are the superensemble and observed field anomalies, respectively, at training time t. This exercise is performed for every grid point and vertical level in the dataset during every forecast phase. In other words, one weight is given to every model at every grid point in the three-dimensional space for each forecast.

The weights of the superensemble vary geographically and with forecast lead time. This allows the incorporation of the information on the geographical distribution of model performances within their respective weights. The superensemble forecast is different from the conventional bias-removed ensemble mean forecast; in the latter case the weights to the models are 1/N, where N is the number of models in the suite. It was shown by Stefanova and Krishnamurti (2002) and Chakraborty and Krishnamurti (2006) that superensemble performs better than bias-removed ensemble mean forecast for short-to-medium range as well as for seasonal time-scale predictions.

3. Satellite-based heating rates

The observational estimates for the vertical profile diabatic heating rate (Q1) were derived from the TRMM Microwave Imager (TMI) instrument. This version “b” of the convective–stratiform heating rates (CSH) was constructed from 3G68 gridded rainfall obtained from TMI, precipitation radar (PR), and from echo-top height data from the TRMM PR (Tao et al. 2006). The horizontal resolution of the dataset is 0.5° in both longitude and latitude directions, and the data are available from 1 January 1998 to 31 December 2007 on a daily time scale. In north–south, the dataset covers the entire TRMM domain: 37°S–37°N. There are 19 levels in the vertical where this dataset is defined. The vertical levels start at 0.5 km; thereafter the levels are at 1, 2, and 3 through 18 km at intervals of 1 km. The data coverage from TRMM for one day (2 July 2007) of orbit and for two consecutive days (2 and 3 July 2007) is illustrated in Fig. 1. The coverage for two days shows the precession of the TRMM orbit. The one-day coverage leaves some data-void squares within which the heating rates are not directly available. We have validated the model-based heating profiles along the orbital paths and especially at the location of these orbital intersections.

4. Model description

We have used the Florida State University (FSU) Global Spectral Model (GSM) for the calculation of vertical heating rate. This model has a triangular truncation at 170 waves (T170), which corresponds to about a 0.7° grid in both the longitude and latitude directions near the equator. The model has 27 vertical sigma levels with more closely spaced levels near the surface and at the tropopause. The details of this model are presented in Krishnamurti et al. (2007).

Three different cumulus convection parameterization schemes were used within the FSU GSM to construct multimodels. These three schemes have entirely different physical basis for convection parameterization. These schemes are 1) Kuo parameterization (Krishnamurti et al. 1980; Krishnamurti and Bedi 1988), 2) simple Arakawa–Schubert parameterization (Grell 1993; Pan and Wu 1995), and 3) Zhang–McFarlane parameterization (Zhang and McFarlane 1995). Details of these parameterization schemes as used in the FSU GSM are given in Krishnamurti et al. (2008b).

The outputs of the models were obtained at every 3 h from the model start time through 60 h of forecasts. We had monitored the vertical velocity and the sea level pressure every time step, and these two variables had equilibrated by around hour 12 (for the initial day) and their variations became more monotonic at grid points, Therefore, in order to leave out model spinup, we have not used the first 12 h of forecasts to calculate skill scores. Mean fields from the period of 12–36 h are termed as day 1 in this study. Day 2 forecasts consist of the mean fields from 36 through 60 h.

5. Experimental details

Integration of these models was carried out to 60 h using each of the above cumulus convection parameterization schemes starting at 1200 UTC of 23 March 2007 through 17 July 2007 (117 days in total). The first 100 days constituted a training phase and superensemble forecasts were constructed for the last 17 days. Separate sets of regression coefficients [a_i’s of Eq. (1)] were calculated for 12–36 h (day 1) and 36–60 h (day 2) of forecasts for each of the member models at each grid location.

To calculate the vertical profile of heating rate Q₁, we have used the familiar equation (Yanai et al. 1973; Krishnamurti et al. 2008a)

View Expanded

where C_p is the specific heat of air in constant pressure, p is pressure at the level where Q₁ is to be calculated, p₀ is a reference pressure (taken as 1000 hPa), κ is a constant (0.286), θ is potential temperature, V is horizontal wind vector, ∇ is the horizontal gradient operator, and ω is vertical pressure velocity. The values of Q₁ were calculated at every vertical level and at all grid points at each 3 h of forecast interval for day 1 and day 2 forecasts. Averaged values of Q₁ for all the 8 time intervals during day 1 defined the model database for Q₁. Day 2 forecasts follow similarly.

To compare model-generated heating profiles and to construct superensemble forecasts, we have interpolated observed profiles of heating rates to the model grid (∼0.7°) using bilinear interpolation technique. Figure 1 shows that because of the narrow swath of the TMI instrument there is a wide gap between two successive passes. This gap is greater near the equator and decreases toward the poles. Because of the precession of the TMI orbit, the full equatorial belt is covered in about 4–5 days. In other words, two consecutive passes over a grid occur only once in 4–5 days in this region. This reduces the number of effective training days on a single grid near the equator to as low as 20 out of the total 100-day period. It was shown in Mishra and Krishnamurti (2007) that the minimum number of training data points to obtain a stable set of weights is around 75 days. To avoid this problem, we have calculated a single set of weights [of Eq. (1)] for every 2 × 2 (4 in total) grid boxes. All valid observed and model data points belonging to these four boxes during the training period were used in the training phase. This increases the number of training data points. The latitudinal variation of the number of training data points along 180°E at every 2 × 2 grid box is shown in Fig. 2. Note that the total number of training data points near the equator is around 80 (as opposed to about 20). Once a set of weights is calculated for 2 × 2 grids, this is applied to all the four original model grids in the forecast phase to carry out the superensemble forecasts. This procedure allows the construction of a stable set of weights while still maintaining its geographical variations.

6. Results

7. Concluding remarks and future work

We have shown in this paper that we can reduce the systematic errors of the heating (amplitudes, vertical level of maximum heating, the shape of the heating profile in the vertical, and the geographical distributions) from the construction of a multimodel superensemble of a suite of models. These member models utilized different cumulus parameterization schemes. All of these member models were identical in their dynamical and physical parameterizations (except for the cumulus parameterization). They carry same number of vertical levels and same formulations of surface and other processes including resolution, orography, and initial states. Using nearly 100 experiments from each of these member models, during the training phase of the multimodel superensemble we were able to obtain the systematic biases for their diabatic heating representations. The multimodel superensemble exploits these collective bias errors of the member models and reduces them. We show here that for a large number of forecasts we can drastically reduce these errors in the vertical and geographical distributions and the amplitude of the errors of the member models from the multimodel superensemble. Since the superensemble can provide forecasts for all the variables including the heating and other functions of model variables (only Q₁ superensemble is carried out in this paper), we have useful products here that can be exploited for a variety of postprocessing studies with these datasets.

Examining the vertical profiles of heating (Q₁), we noted that the Kuo scheme consistently provided overestimates for the amplitude of the heating and placed the heating at a higher level compared to the TRMM-based observed estimates. The corresponding values from the AS and the ZM schemes were slightly overestimated and were placed at a slightly lower level compared to the TRMM-based estimates. The RMS errors in these predicted vertical profiles of heating were reduced to values as low as 0.1–0.5 K day⁻¹ as compared to the best model, which carried RMS errors of the order of 1.1–2.2 K day⁻¹. The geographical distributions of the RMS errors for the amplitude of the maximum heating for the superensemble was around 6.6 K day⁻¹ as compared to the best model, which carried an RMS error of around 14.2 K day⁻¹. These RMS errors were generally similar for forecasts between 12 and 60 h. The geographical distribution of RMS errors for the level of maximum heating was reduced to roughly 2.4 km compared to the best model, which carried an RMS error of roughly 8.8 km. While looking at the geographical distributions of these heating errors we noted that over nonprecipitating areas large errors in heating Q₁ arise from the radiative transfer algorithms. The nature of these errors were very persistent, that is, systematic, and the multimodel superensemble was able to reduce these errors quite drastically.

The strength of this study reconfirms the great strength of the multimodel superensemble. In a series of papers (Yun et al. 2003; Chakraborty and Krishnamurti 2006; Krishnamurti et al. 2006; Stefanova and Krishnamurti 2002) we have recently shown that it is possible to reduce the deterministic and probabilistic errors, from the use of the multimodel superensemble, for seasonal forecasts compared to those of as many as 13–15-member models of our suite of global coupled models. These were the state-of-the-art models from Europe, the United States, and Canada. The variables that we had examined in this series of papers include precipitation, surface air temperatures, 850-hPa-level winds, and the SST anomalies. The success of the multimodel superensemble lies in its ability to reduce persistent systematic errors of these member models. In the climate modeling context we use as many as 10⁷ statistical weights for the reduction of errors. That large number comes from a product of the grid points in the three dimensions times the number of models, times the number of variables that are handled. These weights (fractional, negative, or positive) vary geographically, vertically, by the model, and by the variable. The present paper suggests that a further reduction of errors may be possible for the seasonal climate forecasts if we also include a reduction of the systematic errors for the vertical distribution of heating.

The most important aspect of this exercise has been the diagnosis of the model errors on the placement and amplitudes of heating of the member models. That information can be very useful for addressing possible corrections for the various cumulus parameterization schemes used in our study. In this context we also recognize that the behavior of a particular physical parameterization algorithm also depends to a certain extent on the particular model within which it resides. The rest of the model’s physics, dynamics, and resolution also dictate the mode of behavior of a particular physical parameterization scheme. Nevertheless, we feel that the information we can obtain from this type of modeling can still be quite useful. Following Krishnamurti and Sanjay (2003) we can also construct a single unified model that utilizes the statistical weights of the multimodel superensemble for the heating. Such an exercise has been carried out previously by us for a choice of different cumulus parameterization (Krishnamurti and Sanjay 2003), for different planetary boundary layer formulations (Krishnamurti et al. 2008a), and for different cloud radiative transfer algorithms (Chakraborty et al. 2007). In each such design of a single unified model we noted that it is possible to obtain higher skills for the respective processes compared to any of the single member models. It is thus possible to design a single unified model that utilizes the heating functions based on the weights of a multimodel superensemble. Such a model would carry the least errors for the three-dimensional placements of the heating and their amplitudes. Ideally, the design of a single model carrying the highest skill forecasts is necessary. That requires much further work in almost all areas of model physics, dynamics, and data assimilation. This work is in progress and will be reported separately.