A two-dimensional, dynamic-stochastic model presented in this study is used for short-term forecasting of vertical profiles of air temperature and wind velocity orthogonal components in the atmospheric boundary layer (ABL). The technique of using a two-dimensional dynamic-stochastic model involves preliminary estimation of its coefficients using the
(ProQuest: ... denotes formulae omitted.)
The problem of increasing the accuracy of short-term (from 0.5 to 6 h) meteorological forecasts by integrating numerical weather prediction models with observation data has incompletely been solved and remains urgent now. It is especially significant for forecasting of the atmospheric boundary layer (ABL) state with a lead time from 0.5 to 6 h. In particular, results of shortterm forecasting of the ABL temperature and wind velocity can be useful for aviation meteorology. Joint application of the prognostic data and/or the Terminal Aerodrome Forecast Product allows for the safety of aircrafttaking offand landing to be improved.
Numerous investigations devoted to short-term forecasting of the ABL have been performed recently using one-dimensional (1D) models, surface observations, and an ensemble
The short-term air temperature and wind velocity forecast in the ABL is hindered by the lack of reliable data on their vertical distribution obtained with high temporal resolution and of reliable forecast methods. Indeed, data of radiosonde and spaceborne observations used in the production of weather forecasts are characterized by either low resolution or are not accurate enough to comprehensively describe the state of the boundary layer and its evolution. However, newmethods of remote sensing based on the application of modern lidar, radiometer, and sodar systems have been introduced recently that allow vertical profiles of the ABL temperature and wind velocity to be estimated with high altitude and time resolution.
One of the possible applications of the method suggested in the present paper is associated with mobile meteorological systems that can be rapidly deployed and operated at any arbitrary point of the globe under different climatic conditions. The mobile meteorological systems can be used to estimate current meteorological situation and to forecast it for short periods, for example, during such important events as the summer or winter Olympic Games (e.g., Schroeder et al. 2006; Stauffer et al. 2004). In addition, these systems can be used to estimate and to forecast local atmospheric pollution and to predict the speed of spreading natural fires that can cause significant damages to life, health, and property. This is especially important for systems used in remote areas where meteorological observations are rare in time and distributed in space.
Based on the dynamic-stochastic approach, which is not entirely novel in meteorology, the technique of short-term forecasting of vertical profiles in the ABL implemented using the complete version of the
2. Formulation of the problem and suggested method
The main problem in the application of the KF algorithm to real prognostic models is the high computational cost. The proposed method allows achieving an acceptable forecast quality at minimum computational cost in the absence of additional sources of observations. From this standpoint, the work of Rostkier-Edelstein and Hacker (2013) should be mentioned in which problems of simplification of the prognostic model and data assimilation method were considered together with decreasing requirements to computing resources to provide comprehensible short-term forecasting. The approach suggested here follows these principles. In our previous studies (Komarov et al. 2004, 2007) devoted to spatial extrapolation of fields of meteorological variables on the local scale, low-order dynamic-stochastic models were used that require small computing cost on a personal computer and provide acceptable extrapolation accuracy.
a. Two-dimensional dynamic-stochastic model
The two-dimensional dynamic-stochastic model used in a short-term forecast algorithm that describes the vertical and temporal evolution of the ABL state variables was first proposed by Lavrinenko et al. (2005). The expression for the two-dimensional dynamic-stochastic model is written in the following form:
where jz,k is the observed value of meteorological variables at the altitude with subscript z at the kth moment of time, jm,k2j are values of meteorological variables at altitudes with subscripts from z 2 i to z 1 i observed at time moments from k 2 1 until k 2 K, dm,j are the unknown model parameters, and ez,k is the model random error caused by the stochasticity of the atmospheric processes. The subscript k 5 0, 1, 2, . . . designates the absolute discrete current time with sampling interval Dt and tk 5kDt. In the present article, we restrict ourselves to consideration of the forecast time frame equal or multiple to the sampling interval t5Dt to simplify verification of forecast results. Hereafter, m denotes the serial number of the altitude level in the atmospheric layer whose data are taken into account in the forecast. The number of altitude levels in this layer is 2i 1 1. The subscript j designates the serial number of the discrete time interval in the initial sequence used as a predictor of the model in (1); it changes from 1 to K.
We note that when choosing the value of the predictor K for the
b. Model application for short-term forecasting
1) ESTIMATION OF MODEL PARAMETERS
In the first stage, according to Sage and Melsa (1971), a discrete model of the dynamic system
must be assigned to estimate the model parameters dm,j, where xt k 5[d0,1, d0,2, . . . , d2i11,K] T is the true state vector with dimensionality n 5 (2i 1 1)K including the unknown state parameters of dynamic system (1) to be estimated at the kth discrete time (here T is the transposition operator). For K and i values indicated above, the dimensionality of the covariance matrices was n 5 33. Following Ide et al. (1997), we have used united notation of superscripts for data assimilation systems and omitted the notation of the altitude level z in subscripts for simplification. Here Ck21 is the (n 3 n) transition matrix of the discrete system (or discrete propagator) that takes into account the spatiotemporal evolution of the state variables, vt k21 5[vt 1, vt 2, . . . , vt n]T is the vector column of the model error (the noise state vector) with dimensionality n, where hvi50, hvvTi5Q, Q is the model error covariance matrix, and h i denotes the mathematical expectation operator. Under the assumption that the unknown parameters xt k, on the average, do not change with time for the examined term of forecast, we replace the transition matrix Ck by the identity matrix I.
The mathematical model of measurements used in the KF algorithm to estimate the system state is generally described by an additive mixture of the useful signal and measurement error:
where jok is the observation vector, in our case, a scalar, Hk is the forward operator with dimensionality n that determines the functional relationship between true values of the state variables and their actual measurements, eok is the observation error at the kth moment of time (observation noise) specified by a number, hei50, heeTi5R, hevTi50, and R is the covariance matrix of observation errors.
Acomparison of expressions for the prognostic model (1) and the mathematical model of measurements (3) demonstrates that the matrix Hk can be written in the form Hk 5[jo z2i,k21, jo z2i,k22, . . . , jo z,k21, . . . , jo z1i,k2K]; the first subscript here designates the altitude level, and the second subscript designates the moment of time in the initial sequence used as a predictor of the model (1).
After determination of all the elements involved in (2) and (3), the problem of estimation of dm,j is solved using the linear KF that provides the estimate of the state vector components with minimum root-mean-square errors (Cohn 1997). In this case, the equation for the optimal estimate of the state vector has the following form:
where xak is the vector of analyzed values (estimate of the state vector) at the kth moment of time, xf k is the vector of forecasted values at the kth moment of time, and Kk is the (n 3 n) matrix of weighting coefficients.
The weighting coefficients in the linear KF are calculated from the recurrent matrix equations:
where Pak 5h(xak 2xtk)(xa k 2xt k )Ti is the (n 3 n) a priori covariance matrix of forecast errors, Pf k 5[left angle bracket](xfk 2xt k) (xf k 2xt k )T[right angle bracket] is the (n × n) a posteriori covariance matrix of forecast errors, and Fk21 [?Ck21(x)/?x is the (n 3 n) constant matrix.
2) FORECAST AT ONE VERTICAL LEVEL
After estimation in the first stage of the vector of unknown parameters, in the second stage the meteorological variable j is forecasted for the (k 1 1)th moment of time using the forecast model of the following form:
Here ... is the estimate (forecast) of the meteorological variable at the zth altitude level at the (k 1 1)th moment of time, dm,j are the model parameters estimated in the kth time step.
Thus, (1)-(8) determine completely the procedure of application of the two-dimensional dynamic-stochastic model and the KF algorithm to short-term forecasting of the meteorological variable j at the preset altitude z by one step into the future. If the forecast period exceeds the sampling interval, the estimate obtained at the preceding time step is used as "observations."
c. Model implementation with real observations
Real observations used in our study represented sequences of vertical profiles of U and V wind components measured with a sodar and vertical temperature profiles measured with a radiometer deployed in the region of Tomsk. The data of aerologic station
Since in the expression for the model (1) there is no explicit dependence on the meteorological variables, we performed estimation and verification of the meteorological variables-air temperature and U and V wind components-consistently and independently. As can be seen from the expression describing the model, to perform forecasting for an altitude level, we need observations at this level and at the nearest neighboring levels. For each altitude level, the special KF should be constructed together with the special model (1). Thus, the number of independent models/
For simplicity, theKalman filter was initialized using the same values for all altitude levels of the vertical profile. The initial state vector was xa0 5[0, . . . , 0], the unique element of the observation error matrix was R0 5, and the diagonal elements of the covariance matrix of forecast errors were jpa ii j5100; for the covariance matrix of model errors, they were jqiij51. These values were obtained based on the results of preliminary analysis of sodar, radiometric, and radiosonde data in various meteorological situations (Lavrinenko 2006). We note that in our case, the elements of the covariance matrix of model errors represent dimensionless quantities related to the model parameters dm,j.
In addition, the restart procedure was envisaged according to which the algorithm was reinitialized through the fixed time period between observations with the initial conditions described above. This allowed the significance of the last observations in the final forecast to be increased. The time period between restarts was set equal to about (1.5-2)K. For a shorter time period, the error caused by the KF transition periods increased, and for a longer time period, the sensitivity of the filter to sharp changes in observations decreased. The restart was performed when the condition
was satisfied, where Dlim is the restart threshold, jz,k11 is the estimate of the meteorological variables at the (k 1 1)th moment of time, and jz,k is the value of the observation at the kth moment of time. Investigations of cases with various threshold values Dlim for the temperature and orthogonal wind components allowed us to choose 38S and 3ms21 as criteria for minimal total forecast errors. The threshold was introduced primarily to minimize the impact of erroneous observations that fall into the data assimilation system.
The model and the assimilation scheme were implemented on a nonexpensive compact dual-processor 2.33-GHz desktop PC. It took no more than 5 s to forecast each individual vertical profile with initial conditions and observations considered in these experiments.
The originality and novelty of the algorithm lies in the facts that
* the forecast is implemented in two steps: the observations are first assimilated and the coefficients of the mathematical model are estimated, and then the meteorological variable is forecasted for a short term at a preset level using the estimated coefficients;
* the state vector of the dynamic system consists of the unknown coefficients of the two-dimensional dynamicstochastic model;
* the short-term forecast procedure in the suggested algorithm relies on observations at one aerologic station.
3. Results of experiments and statistical evaluation
In the present work, two cases were considered. In the first case, experiments were based on the data of the "MTP-5" radiometer and the "
a. Experiment (case) with sodar and radiometer data
Sodar observations represented 30-min-averaged 10-s measurements of orthogonal wind components at altitudes of 100, 150, 200, and 250 m. Four measurement periods are considered. The two longest periods are from
The radiometer provides retrievals of temperature up to 600m with resolution of 50m and a temporal resolution of 5 min. Then, by analogy with sodar, the measured values of temperature were averaged over 30-min periods. The experiment was performed from 2 to
Table 1 gives the RMS errors (RMSEs), MAE, and relative errors of short-term forecasting of the temperature and zonal and meridional wind components for three altitude levels. From the table it can be seen that the best results were obtained for the predicted values of the air temperature. Indeed, even for lead time t53 h, the RMSEs at all examined altitudes did not exceed 0.98S, and for lead time t51:5 h, they were already within 0.48-0.68S. The quality of forecast of zonal and meridional wind components, however, deteriorated, but even then the RMS errors for t53 h did not exceed 1.3-1.7ms21 irrespective of the wind component and altitude level.
In addition, as can be seen from the table, MAE is less than RMSEs for all meteorological variables at all altitudes for lead time from 0.5 to 3 h. This is due to the fact that MAE is more stable to outliers. If we compare the RMSEs of short-term forecasting of vertical profiles with that of the persistence forecast, a gain in the accuracy of the suggested method for the air temperature can be seen for all altitudes and lead times of 0.5-3 h. The errors for the persistence forecast algorithm often used to compare different methods of short-term weather forecast (Wilks 2006) are also shown in the table. For the wind components, the result is not so unambiguous, especially for altitude of 100m where the RMSEs of the persistence forecast are close or equal to those of our model. This result can be explained partly by the fact that measurements were performed on the edge of the city and were influenced by buildings and noise background at low altitudes.
Figure 1a shows the time behavior of individual realizations of observed values for sodar and radiometer at z 5 200m and for 0.5-h forecasts. From the figure it can be seen that the efficiency of forecasting of the temperature and orthogonal wind components is sufficiently high judging by the degree of agreement of these curves. Considerable forecast errors arise only in the initial stage of algorithmic implementation. This is due to the fact that under conditions of lacking of a priori information on the statistical properties and behavior of the unknown model parameters, the initial conditions for the
b. Experiment (case) with radiosonde data
In this experiment, data of observations at the
Figure 1b shows the time behavior of the temperature and wind components observed and forecasted for 6 h. As for the radiometer and sodar data, a high degree of agreement of the sequences presented here testifies to the high forecast quality. The cold start of the algorithm with zero initial estimation vector was used. The time period between restarts was 72 h, and the time period of observations used as a predictor in the KF algorithm was 66 h. The initial conditions for initialization of the
Figure 2 shows the vertical distributions of the rootmean- square forecast errors and relative errors (top axis) for January and July and t56 h. An analysis of Fig. 2 demonstrates that first, the RMS forecast errors for the suggested algorithm, irrespective of the month and altitude level, lie in the range 0.48-1.18S and 0.7-1.3ms21, respectively. Second, they are much smaller than the RMSEs for the persistence forecast. For example, the RMS errors of the suggested method are approximately by a factor of 2.5-3.0 smaller for the air temperature at all altitude levels. We do not consider the altitude levels close to the surface in summer, when the influence of the diurnal behavior is significant, and the persistence forecasting yields incorrect results. For the wind component, the gain in accuracy is smaller: by a factor of 1.8 for the V component in summer at altitude of 400m and by a factor of up to 2.8 for the U component in winter at altitude of 1600 m.
4. Conclusions and subsequent work
Based on the results of our study, the general conclusion can be made that the suggested algorithm based on the two-dimensional dynamic-stochastic model and the
We could not describe in detail all stages of our work on the development and improvement of the given algorithm within the limits of this article. We hope that our future publications will help us to solve this problem. There are numerous directions for subsequent work. Later on we plan to estimate possibilities of application of this algorithm to long-term forecasting of the meteorological variables for 1 year. We also plan to study the possibilities of application of the suggested algorithm in mesoscale systems of data assimilation for generation of the first-guess fields at points on a regular grid.
Acknowledgments. The authors would like to thank the editors and three anonymous reviewers for their valuable comments and for pointing out errors. Their suggestions, whichwere accounted for in the final version, helped us to improve the consistency of the manuscript.
Cohn, S. E., 1997: An introduction to estimation theory. J. Meteor. Soc.
Hacker, J. P., and D. Rostkier-Edelstein, 2007: PBL state estimation with surface observation, a column model, and an ensemble filter. Mon. Wea. Rev., 135, 2958-2972.
Komarov, V. S.,
Lavrinenko, A. V., 2006: Investigation of dynamic-stochastic algorithm for ultra-short-term forecast of meteorological fields. J.
_____, V. S. Komarov, and Yu.
Rostkier-Edelstein, D., and J. P. Hacker, 2010: The roles of surface observation ensemble assimilation and model complexity for nowcasting of PBL profiles:Afactor separation analysis. Wea. Forecasting, 25, 1670-1690.
_____, and _____, 2013: Impact of flow dependence, column covariance, and forecast model type on surface-observation assimilation for probabilistic PBL profiles nowcasts. Wea. Forecasting, 28, 29-54.
Sage, A. P., and
Schroeder, A. J.,
Stauffer, D. R., A. Deng,
Wilks, D. S., 2006: Statistical Methods in the Atmospheric Sciences. 2nd ed.
V. S. KOMAROV,
Corresponding author address:
Most Popular Stories
- Crimean Referendum Violates International Law: Obama
- Justin Bieber Loses Cool Over Selena Gomez
- Fuentes Makes NAHREP's Top 10 List
- Social Media Can Help a Company's Credit Line
- Hispanic Unemployment Eased in February
- Juanes Back to Singing About Love
- Boeing Freezes Nonunion Workers' Pensions
- Alfredo Ramos Martínez, Mexican Muralist, Symposium at Scripps
- Goya Nutritionist Answers Demand for Healthy Hispanic Dishes
- 2 Million Long-term Jobless Have No Benefits