Effect of the Form of the Error Correlation Functions on Uncertainty in the Estimation of Atmospheric Aerosol Distribution When Using Spatial-Temporal Optimal Interpolation ^†

Abstract

Spatial-temporal optimal interpolation (STOI) is a data assimilation method based on minimizing the error in an estimate. Error correlations can be modeled with analytical functions. We investigated the effect of the form of the error correlation functions on the uncertainty in an estimate. We applied STOI to the estimation of aerosol distribution over Europe using the results of GEOS-Chem chemical transport model simulations and observations from a ground-based radiometric Aerosol Robotic Network. We used exponential functions to model correlation curves. The results show that a ±15–25% change in the argument of the constructed exponential functions has little effect on the mean square error in the estimate in regions where observations are dense. STOI estimates are sensitive to the form of the correlation curves in regions where observations are sparse.

1. Introduction

A common approach to estimate the spatial-temporal distribution of atmospheric species properties is data assimilation [1,2,3]. This comprises methods to combine information from different sources for obtaining the best estimate of a system state. Data assimilation is based on minimizing the error in the estimate (optimal interpolation and Kalman filtering methods [4,5,6]) or on minimizing the cost function (variational methods [7,8,9]). Under certain conditions, variational methods turn out to be equivalent to optimal interpolation or Kalman filtering. Data assimilation techniques require knowing data error statistics for the observations and background (appropriate prior information).

When performing assimilation in practice, some approximations have to be carried out. The observation error covariance and background error covariance are never exactly known. Numerous works are dedicated to the estimation of the observation and background error covariance matrices using differences between the observations and the background [3,10,11,12]. The common assumption is that the background error field is stationary in time and uniform in space. In this case, correlation curves can be constructed that describe the dependence of the correlation coefficient (or covariance) of the errors at different points on the distance between the points. The observation error covariance matrix is often assumed to be diagonal, and can be estimated empirically. Even under these simplifying assumptions, the estimation of error covariance is challenging [3]. Correlation curves are commonly approximated by analytical functions with tuned parameters [1,13,14,15,16]. The background error covariance matrix should be positive definite, so an exponential function is often chosen to model background error correlations.

The performance of data assimilation schemes can be sensitive to the specification of the observation and background errors [2,3,14]. The error covariances are subject to uncertainties. In [3], a simplified synthetic example showed the importance of the specification and tuning of the error covariance matrices. However, in real data assimilation problems the impact of error covariance matrices is hard to estimate [3]. The goal of the present work was to evaluate if slight differences in error covariance matrices can significantly affect the results of real atmospheric data assimilation.

In the present work, we investigated the effect of the form of the background error correlation functions on uncertainty in the estimate of aerosol optical depth (AOD) when using the spatial-temporal optimal interpolation (STOI) technique. The estimation of error covariance matrices is crucial in various data assimilation methods in application to the estimation of a broad range of atmospheric characteristics. We chose STOI because it is a relatively simple and computationally cheap technique. We chose AOD as it is one of the most important quantities characterizing columnar atmospheric aerosol.

2. Materials and Methods

STOI [17,18] is one of several data assimilation methods. It was developed as an extended optimal interpolation method that uses both spatial and temporal background error covariance. STOI is based on estimation theory equations. A general form of the data assimilation equation is:

$$x^a=x^b+K[y-H(x^b)]$$

(1)

$$K=B{H}^{\mathrm{T}}{(HB{H}^{\mathrm{T}}+R)}^{-1}$$

(2)

x^a is a vector containing estimates at a regular grid, x^b is a vector containing background values at a regular grid, y is a vector containing values of observations, K is a matrix containing weight coefficients, H is an observation operator providing the link between observation space and background space, B is a covariance matrix of background errors, and R is a covariance matrix of observation errors. Weight coefficients are chosen so as to minimize the mean square error in the estimate.

We applied STOI to the estimation of AOD distribution over Europe. To perform STOI, we used an output from the GEOS-Chem chemical transport model [19,20] simulations as a background, and observations from a ground-based radiometric network, AERONET [21,22,23].

AERONET is a global network of ground-based sun–sky–lunar radiometers. We used version 3 AERONET Level 2.0 cloud-screened and quality-assured daily averaged total AOD data at wavelengths of 440, 675, and 870 nm at 86 AERONET sites in Europe.

We obtained the background field using version v12.1.1 of the GEOS-Chem chemical transport model in the classical configuration. We calculated daily mean AOD at wavelengths 440, 675, and 870 nm over the nested European domain (32.75° N–61.25° N, 15° W–40° E) at 0.25° latitude × 0.3125° longitude spatial resolution for 2015–2016.

AERONET provides AOD data with low uncertainty that is much smaller than the uncertainty of AOD data simulated by GEOS-Chem [24,25]. This allows treating the AERONET AOD observations as a “true” field. Under this assumption, we undertook a preliminary bias correction for the AOD background field. The observation error was taken to be zero. We derived background error statistics empirically from background-minus-observation AOD values. The obtained statistics showed a large range in values of background error correlation coefficient. To specify an error correlation curve, we selected areas with the highest graphic density of values of background error correlation coefficient on the scatter plots of spatial and temporal dependencies. Within the selected areas, we modeled spatial and temporal correlation coefficients by exponential, gaussian and polynomial functions using the least-squares method. We chose the functions such that the correlation coefficient is equal to one when distance (for spatial correlation curve) or time interval (for temporal correlation curve) is equal to zero, and the correlation coefficient tends to zero at the correlation length. The exponential function provided the best fit. Therefore, for further research we used the exponential function.

Exponential spatial and temporal correlation functions obtained using the least-squares method are shown in Figure 1. The shape of the spatial correlation curves is almost similar for all three wavelengths; temporal correlation curves lie a little higher with a decrease in wavelength.

Then, we changed the arguments of the exponential functions to make the correlation curves change their shape. The change in the arguments was chosen to be about ±15–25% so that the correlation curves would lie at the upper and lower boundaries of the selected areas on the scatter plots described above. We considered three cases: (1) exponential functions obtained using the least-squares method; (2) exponential functions with artificially decreased arguments (estimates are closer to the observations); (3) exponential functions with artificially increased arguments (estimates are closer to the model output). Correlation curves for these three cases are shown in Figure 2 and Figure 3.

We calculated root mean square errors (RMSE) in the AOD values obtained with the GEOS-Chem simulation and the AOD estimates obtained using STOI for the three cases described above. RMSE were calculated for AERONET sites Granada, Lille, and Minsk. We calculated the reduction in RMSE of the estimate after STOI in comparison with the GEOS-Chem output. We compared the reduction in RMSE for the three cases of different shapes of correlation curves.

3. Results

The results of the comparison of the reduction in RMSE for three cases of different shapes of correlation curves are presented in

Table 1. Constructed correlation functions and reduction in root mean square errors (RMSE) for case 1.

	440 nm	675 nm	870 nm	Averaged
Spatial	exp (−0.00207d)	exp (−0.00215d)	exp (−0.00204d)
Temporal	exp (−0.298t)	exp (−0.384t)	exp (−0.424t)
Reduction in RMSE
Granada	54%	61%	62%	59%
Lille	18%	13%	14%	15%
Minsk	24%	17%	16%	19%

,

Table 2. Constructed correlation functions and reduction in RMSE for case 2.

	440 nm	675 nm	870 nm	Averaged
Spatial	exp (−0.0015d)	exp (−0.0015d)	exp (−0.0015d)
Temporal	exp (−0.25t)	exp (−0.335t)	exp (−0.375t)
Reduction in RMSE
Granada	53%	61%	62%	59%
Lille	18%	13%	14%	15%
Minsk	24%	19%	19%	21%

and

Table 3. Constructed correlation functions and reduction in RMSE for case 3.

	440 nm	675 nm	870 nm	Averaged
Spatial	exp (−0.0025d)	exp (−0.0025d)	exp (−0.0025d)
Temporal	exp (−0.35t)	exp (−0.435t)	exp (−0.475t)
Reduction in RMSE
Granada	54%	61%	63%	59%
Lille	18%	13%	14%	15%
Minsk	23%	15%	12%	17%

. Table 1 refers to case 1, correlation curves modeled by exponential functions obtained using the least-squares method. Table 2 refers to case 2, correlation curves modeled by exponential functions with artificially decreased arguments. Table 3 refers to case 3, correlation curves modeled by exponential functions with artificially increased arguments. In the tables, d is distance in kilometers in the arguments of the constructed exponential spatial correlation functions; t is time interval in days in the arguments.

4. Discussion and Conclusions

Granada, Lille, and Minsk regions differ in the density of observational AERONET sites. Minsk is located in the eastern European region where observations are sparse. Observations are rarer in Lille than in Granada because Lille is located in the northern part of Europe with a smaller number of sunny days than in the southern part of Europe where Granada is located.

Comparison of the results presented in Table 1, Table 2 and Table 3 shows that a ±15–25% change in the argument of the constructed exponential correlation functions did not affect RMSE at the estimation points in regions where observations are dense (Granada and Lille). STOI estimations were sensitive to the shape of the correlation curve at points in the region where observations were sparse (Minsk). This can be because in regions where AERONET site density is high, the weight of observations is large enough for all three cases of correlation curve shapes. By contrast, in regions where observations are sparse, the increased weight given to observations is of significant importance. Therefore, the accuracy of STOI is slightly better in regions with sparse observations when estimates are closer to the observations than to the model output.

The practical importance of the results is that there is no need for fine-tuning of the correlation curves when using AOD spatial-temporal optimal interpolation. We assume that this conclusion may be extended to other data assimilation techniques and other atmospheric characteristics, but this requires further research.