An analogue dynamical model for forecasting fog-induced visibility: validation over Delhi

Transcription

1 METEOROLOGICAL APPLICATIONS Meteorol. Appl. 24: (217) Published online 26 April 217 in Wiley Online Library (wileyonlinelibrary.com) DOI: 1.12/met.1634 An analogue dynamical model for forecasting fog-induced visibility: validation over Delhi Prashant Goswami a * and Sumana Sarkar b a CSIR National Institute for Science, Technology and Development Studies, Delhi, India b CSIR Centre for Mathematical Modelling and Computer Simulation (repositioned as CSIR Fourth Paradigm Institute), Bangalore, India ABSTRACT: Accurate forecasts of fog and visibility are important for many applications; while prolonged fog can adversely affect many crops, even a short duration of dense fog can lead to disruption of air and highway traffic. The genesis and dynamics of fog are a result of many processes; accurate forecasting of fog thus continues to be a challenge. A forecast model of the occurrence of fog, measured in terms of visibility, is presented. The model is formulated as an analogue model; thus the merit of the model is primarily based on its validation against observation. Two forecasts using two sets of meteorological fields are considered: one as the benchmark forecasts with visibility calculated from observed meteorological fields and the other based on meteorological forecasts from an atmospheric mesoscale model (Weather Research and Forecasting). While the benchmark (perfect) forecasts from observed meteorological fields provide the potential skill of the model, the mesoscale forecasts provide an assessment of realizable skill in an operational setting. The validation was carried out against hourly visibility data recorded at Indira Gandhi International Airport over Delhi during the winter months (December and January) for the period Error statistics show that the analogue fog model can capture a significant part of the observed variability of fog. The forecasts have more success in forecasting intense (visibility < 5 m) and persistent (duration > 4 h) fog events. The model provides a useful forecasting tool, as shown by measures such as average error, number of false warnings and the number of misses. KEY WORDS visibility; analogue fog model; benchmark simulation; forecast skill; forecast validation; sensitivity analysis Received 4 April 216; Revised 23 September 216; Accepted 6 October Introduction Fog is a high-impact weather event over many locations worldwide; it has considerable socio-economic impact over the Indo-Gangetic plains of India (Bhowmik et al., 24; Jenamani, 27).When fog prevails, the atmospheric visibility is reduced, imposing an adverse impact on people s lives and the socio-economic activities in the locality. The occurrence, development and dissipation of fog result from multiple processes (thermodynamic, radiative, dynamic and microphysical) through a wide range of conditions (Menut et al., 213). A frequently occurring fog type, especially for relatively flat areas, is radiation fog, which occurs in clear-sky conditions with relatively low wind speed. In these conditions air close to the surface cools from the evening transition onwards because the absence of clouds leads to strong radiation loss from air close to the Earth s surface (Steeneveld et al., 2). Radiation fog formation depends on a complex combination of boundary layer and synoptic-scale conditions that often have a diurnal and seasonal nature (Meyer and Jiusto, 1986; Meyer and Lala, 199). Attempts at numerical simulation and forecasting of radiation fog began in the early 196s (Fisher and Caplan, 1963; Brown and Roach, 1976) and continue today (Choularton et al., 1981; Meyer and Jiusto, 1986; Dynkerke, 1991; Guedalia and Bergot, 1994; Nakanishi, 2; Clark and Hopwood, 21; Bott and Trautmann, 22). Fog simulations with a dedicated dynamic * Correspondence: P. Goswami, CSIR National Institute for Science, Technology and Development Studies, Delhi, Pusa Gate, K. S. Krishnan Marg, New Delhi 1112, India. pgoswami@nistads.res.in model (COBEL-ISBA) and an ensemble system showed the forecasts to be very sensitive to local initial conditions and mesoscale forcings (Roquelaure and Bergot, 28). Despite the major economic impact of radiation fog on human society, numerical weather prediction models have relatively low skill in forecasting both the onset and the development of radiation fog (Teixeira, 1999; Gultepe et al., 27; Tudor, 21; Zhou et al., 211; Román-Cascón et al., 212). Forecasting radiation fog is still somewhat problematic because local topography, moisture availability, vegetation and soil conditions introduce spatial variability into the model results and forecast products (Golding, 1993; Meyer and Rao, 1999). In addition, the analytical precision required to diagnose humidity levels, condensation rates and radiative exchanges adequately is very demanding in a forecast context (Bergot and Guedalia, 1994). From a numerical modelling point of view, models usually lack accuracy due to the horizontal (Pagowski et al., 24) and vertical (Tardif, 27) resolutions and physical parameterizations (Musson-Genon, 1987; Gultepe et al., 27). Many of the cited studies underline the importance of applying a sufficiently high vertical resolution to resolve the major processes, while Bergot and Guedalia (1994) and Bergot et al. (25) in particular found that successful fog forecasting requires an accurate model initialization (e.g. Rémy and Bergot, 29). In most cases, fog forecasts are not adequate for direct operational guidance and fog prediction is still much less skilful than the forecasting of precipitation (Zhou et al., 211). A multivariable-based diagnostic fog forecasting method based on five basic model variables (lowest model level liquid water content, cloud top, cloud base, 1 m wind speed and 2 m relative humidity) was explored over eastern China (Zhou and Du, 21), but the method is for prediction of 217 Royal Meteorological Society

2 Dynamical model for fog 361 fog occurrence rather than fog intensity. A challenge in fog simulation or forecasting is that it is a threshold process (Tardif and Rasmussen, 27). There are in general two approaches to fog prediction: diagnostic and prognostic. In the diagnostic method, fog is diagnosed from forecasts of other variables, mainly meteorological variables. For such an approach, it is true that improvements in the meteorological forecast models alone can lead to improvement in the fog prediction. However, fog is also governed by intrinsic dissipation mechanisms and other non-meteorological variables such as land surface characteristics. This necessitates the use of an independent prognostic model for fog prediction. Besides, fog is not a part of all standard meteorological forecast models. In the formalism presented, the dynamics of fog is represented as having its own driving processes, some of which are driven by meteorological processes. While the meteorological processes provide the major drivers involving incorporation of the meteorological thresholds, overall improvement in the forecasts from the prognostic fog forecast model is necessary. A major challenge is to develop models that embody the various processes related to fog to generate skilful forecasts. However, for effective application such forecasts should have sufficient accuracy in parameters such as the time of onset, intensity and duration. In this study a dynamic fog forecast system is presented with a new algorithm for the computation of fog (visibility) driven by meteorological fields from observations and forecasts. Here, the megacity Delhi (28.38 N, E) is the focus. Delhi, the capital of India, is characterized by a high frequency of fog events (De et al., 25; Jenamani, 27). Fog most frequently occurs during peak winter from December to January over this region. There are only a few specific studies (Mitra et al., 28; Mohan and Payra, 29; Saraf et al., 21) over India, for fog simulation and prediction. Our earlier study over Delhi (Goswami and Sarkar, 2) reveals that the temperature and the relative humidity have 1 C and 5% contrasts, respectively, near the ground, implying an accuracy of at least 1 C and 5% to predict fog skilfully with a numerical weather prediction model. Here, the model is presented as an analogue model that represents the key processes of fog; however, the emphasis is on the forecast skill and not on basic physics of fog. The paper is organized as follows. Section 2 describes the observed meteorological and visibility data, while Section 3 provides details on the simulation methodology and the atmospheric mesoscale model used to drive the analogue dynamic model in forecast mode together with a detailed description of the analogue fog model. Section 4 describes the analysis and validation strategy. Results and discussion including skill evaluation are described in Section 5. Section 6 contains a sensitivity analysis of the model and Section 7 gives the conclusions. 2. Observed meteorological and visibility data A big challenge in the modelling and forecasting of fog is the highly region-specific nature of the dynamics. Thus, fog forecasts need validation against local observations. Multi-source and multi-scale data were considered for the formulation and validation of our model. The meteorological data were adopted from both observations and simulations (mesoscale forecasts); the visibility (proxy for fog) data were adopted from station observations. Delhi is considered as a test bed for the model Meteorological data Near-surface observed data were adopted from four stations in and around Delhi under the Climate Observation and Modelling Network (COMoN), which includes a mesoscale network of four multi-level profilers established by the Council of Scientific and Industrial Research (CSIR), India (Goswami et al., 212b; Rakesh and Goswami, 214). The observations include high frequency ( 3 min) data on temperature, relative humidity, wind speed and wind direction at three levels (2, 2 and 3 m), and soil moisture and soil temperature at four sub-surface levels (1,, 5 and 1 cm). In addition, surface observations include rainfall, air pressure and net solar radiation. Besides, the COMoN provides long wave radiation, which is also required to obtain net radiation. The data from each COMoN profiler is received telemetrically and passed through quality control software before archiving for analysis. The establishment of the COMoN has provided the first opportunity for examining the relationship between dynamic meteorological variables and the occurrence of fog, on an hourly scale Visibility data and identification of foggy days The hourly visibility (proxy for fog) data, recorded at the Indira Gandhi International Airport (IGIA, 77.5 N, 27 E), were adopted from the India Meteorological Department (IMD) archived at fogvis1.html. The hour of onset is defined as the hour at which visibility falls below 1 m for 2 h or more at a stretch; the duration is defined as the number of hours for which visibility stays less than 1 m from onset. Fog and rainfall are both responsible for the deterioration of surface visibility, but the physical mechanisms behind them are different. Thus, the days identified as foggy were checked to be rain free based on the Tropical Rainfall Measuring Mission (TRMM) 3 h rainfall obtained from ftp://disc2.nascom.nasa.gov/data/trmm/ Gridded/3B42_V6.html and COMoN rainfall data (Table S1). The precipitation estimates from the TRMM (Huffman et al., 27) were interpolated to station scale. However, to avoid any observational bias station (COMoN) data were also used to identify rainy days. The following procedure was adopted to distinguish the cases with only rain from those with rain and fog: if hourly (COMoN) and 3 h (TRMM) rainfall data showed rainfall but observed visibility was >1 m, it was categorized as an only-rain case; if hourly (COMoN) and 3 h (TRMM) rainfall data showed rainfall but observed visibility was <1 m, it was considered as a potential fog + rain day. However, at this stage there remains the possibility that reduction of visibility was only due to rain, although heavy rains causing serious loss of visibility are rare in Delhi in the winter (fog) season. So, the observed (hourly) visibility prior to the onset of rain and in the post-rain hour were also examined; if the visibility was found to be less than 1 m in either of the hours, the case was considered as a fog + rain event. Certain ambiguity remains, however, due to comparison involving different datasets at different resolutions. The analysis was carried out for December and January during three consecutive winters (29 212). Each of the 3 years is characterized by a large number of foggy (visibility < 1 m) days in both December and January.

3 362 P. Goswami and S. Sarkar Table 1. Model (Weather Research and Forecasting) configuration used in the study. Dynamics Number of domains Horizontal resolution Integration time step Number of gridpoints Map projection Horizontal grid Vertical co-ordinate Time integration Spatial difference scheme Microphysics Radiation Cumulus parameterization PBL parameterization Land surface parameterization Non-hydrostatic Three nested domains 36 km (outermost domain), 12 km (inner domain) and 4 km (innermost domain) 9 s (outermost domain), 3 s (inner domain) and 1 s (innermost domain) X-direction 26, 325 and 271, Y-direction 26, 325 and 271 points for outermost, inner and innermost domains respectively Mercator Arakawa C-grid Terrain following hydrostatic with 38 sigma levels up to 5 hpa 3 rd order Runge Kutta 6 th order centred difference WSM6 scheme RRTM long wave and Dudhia short wave New Kain Fritsch scheme YSU scheme Thermal diffusion 3. The forecast methodology and model configuration: atmospheric mesoscale model and analogue fog model As a rationale for the formalism, to select the key variables as predictors the association between visibility (fog) and the meteorological variables was first examined through the frequency distribution histogram during the fog and prior to fog formation hours. This is described in Section 5.1. Two forecast models were employed in the present work: the meteorological variables were generated using a mesoscale atmospheric model; the forecasts of fog are from an analogue fog (visibility) model. The performance of the analogue fog forecast model was evaluated by comparing with benchmark (perfect) and test forecasts. The benchmark simulations were generated by driving the analogue fog model with observed hourly meteorological fields from the COMoN. The test forecasts of fog (visibility) were generated by driving the analogue fog model with meteorological forecasts from an atmospheric limited area model (Weather Research and Forecasting ()). The purpose of the benchmark (perfect) simulation is to create a reference set of simulations against which the prediction of visibility driven by the atmospheric mesoscale forecasts, with a lead time of 24 h, can be tested. Hourly observations ( 24 h) starting from UTC through a 3 day period for each of the winter months of December and January during three consecutive winters (29 212) were used for benchmark simulation Atmospheric mesoscale model () The basic 24 h forecasts were generated using a mesoscale model (Version 3.3, Skamarock et al., 25) having multi-nest capability and options for various parameterization schemes. The performance of the model has been widely tested over the Indian sub-continent for a variety of applications (Goswami et al., 212b; Rakesh et al., 2; Rakesh and Goswami, 214); however, optimization of the model configuration can significantly improve skill. A configuration was adopted that has been found optimal over the Delhi region (28 35 N; E) in terms of 24 h forecasts of regular meteorological variables. It is a nested configuration with three domains, with horizontal resolution of the outer, intermediate and innermost domain as 36, 12 and 4 km, respectively. The number of vertical levels is 38, with the top fixed at 5 hpa. Hourly model outputs from the third domain (innermost high resolution) were considered for validation. The model dynamics and physics options deployed in this study are summarized in Table 1. The initial conditions for the outer domain were extracted from the 3 h data archived in real time from the National Centers for Environmental Prediction Global Forecasting System at a horizontal resolution of 5 km. The 1 min ( 19 km) datasets from the US Geological Survey (USGS) were used to create the surface boundary conditions for the outer domain, such as model topography, land use, soil types and monthly vegetation fraction. The initial, lower and lateral boundary conditions for the inner domain were obtained by interpolating the fields from the outer domain. For topography and land use 3 s USGS datasets were used for the inner domain. The time step of integration was adopted as that for the high resolution simulation to avoid any numerical bias. A set of 3 simulations of 55 h (2 days + 7 h) was carried out at each winter month of December and January starting at UTC. Three consecutive winters (29 212) were considered; accordingly, a total set of 18 (3 6) simulations were carried out, from each of which only second forecasted day composition (corresponds to 24 h from model initializing time + spin-up time) constitute the test forecasts The analogue fog (visibility) model The dynamics of fog depends on certain thresholds and regimes of the meteorological variables. This perhaps implies that while the meteorological variables determine conditions for genesis and intensity of fog, other local conditions such as long wave radiative cooling at the fog top, atmospheric stability, the presence of deep inversion, turbulent intensity, fog microphysics and aerosol distribution govern the persistence. Thus, the meteorological variables need to be used through a process model to describe the dynamics of fog; an analogue model was used for this purpose. The primary difference between an analogue process model and a general process model is that the former mimics the physics and the dynamics of a process; the acceptability is based on its skill and performance. Based on the physical and dynamical considerations, the analogue model has been formulated as the combination of genesis, persistence and dissipation mechanisms of fog leading to change in visibility. The reduction in visibility due to the formation of fog depends not only on water content but on the ambient environmental conditions. Visibility with reference to clear sky (V R ) was used as a measure of fog (V F ); thus the reference visibility is reduced due to fog as expressed by the following

4 Dynamical model for fog 363 expression: V = V R αv F (1) where α is the visibility reduction co-efficient or fog constant. Following convention, a value of 1 m was adopted for the reference (clear sky) visibility (V R ). The dynamics of fog on an hourly scale is represented as: dv F = F dt G + F P F D (2) The term on the left-hand side represents the time variation of the intensity of fog represented by a proxy such as visibility (a result of the number of cloud droplets at a point). The terms F G, F P and F D represent, respectively, the processes that trigger, maintain and dissipate fog, also varying in time. A primary requirement for the formation of fog is also the availability of adequate cloud condensation nuclei (CCN). However, for an urban environment such as the Delhi air basin it can be assumed that an adequate amount of CCN is always present in the atmosphere (Goswami and Baruah, 28, 211; Srivastava et al., 212). Thus, the dynamics of fog is considered to be essentially controlled by the meteorological conditions. The initial condition is given as the observed visibility The genesis process It is known that the genesis of fog requires favourable meteorological conditions simultaneously in wind, temperature and humidity (Menut et al., 213). The term F G is composed of parameters reflecting the state of local meteorological conditions. It is a nonlinear term which implicitly incorporates the synchronous processes leading to fog formation. The main mechanism driving radiative fog formation is infrared radiative cooling, modulated by the influence of upward and downward heat flux (soil, sensible and latent) but also by turbulent mixing during the stable conditions in the boundary layer as well as the warming effect and moisture losses through dew deposition (Duynkerke, 1991). These simultaneous requirements were incorporated in the genesis term through F G expressed as: F G = F q F T F Ts F w F rad (3) The multiplicative nature of the terms in Equation (3) represents the necessity of each atmospheric variable to fulfil certain conditions. The decrease in visibility as the humidity increases is a somewhat continuous process as the haze particles grow in size and condensation begins on the less hygroscopically active nuclei. Assuming that the air contains the entire spectrum of CCN, the dependence of the genesis of the fog on ambient humidity can be expressed as: {( ) rh rhc αq for rh > rh F q = c (4) otherwise where rh c is a threshold value of humidity below which the formation of fog does not occur. The formation of fog is also known to depend critically on the range of variation in temperature near the surface. This dependence on near-surface temperature is expressed as: ( Tch T ) α th for T < Tch ( ) F T = T Tcl αtl for T > Tcl (5) otherwise where T ch and T cl are thresholds of temperature that determine the triggering of fog formation due to thermal processes. The co-efficients in Equations (3) (5) are chosen to ensure dimensional homogeneity. Fog formation at temperatures > C can occur in high humidity situations with other favourable conditions such as clear skies, radiative cooling, weak wind and strong moisture in the lower layers, mixing of masses of moist air of different temperatures, replacement of cool surface air by a warm current, and supply of moisture through evaporation from warm moist soil. The sensitivity of fog to turbulence, and specifically to wind speed, makes forecasting of fog quite challenging. During turbulent conditions, the exchange of sensible heat and latent heat (water vapour) play important roles for fog to form and to evolve (Rodhe, 1962; Welch et al., 1986). Later Zhou and Ferrier (28) showed that there exists a critical turbulence threshold to control the balance and persistence of radiation fog. Only when turbulence intensity near the ground is weaker than this critical threshold inside a fog can it be stable and persistent; otherwise fog cannot form or soon disperses after formation. Because of the close relationship between wind speed and intensity of turbulence, three regimes were considered for the wind speed for fog formation. Case 1: w > w b, no fog can form, i.e. F w =, where w b is the high wind threshold. Case 2: w c < w < w b, fog can form and be maintained, i.e. F w >, where w c is the low wind threshold. Hence, w c w b is the wind speed range where fog is most likely to form and persist. Case 3: w < w c no fog can form or persist, i.e. F w =. These effects are included in the following equation: {α F w = w ( ) w w c for w < wc (6) otherwise α w is the wind constant. An explicit inclusion of the case w c w b is not considered in the present formulation but can be added with a speed dependence of the co-efficient α w. Another important controlling factor is the gradient in temperature between the soil and the air. This difference scales with the soil heat flux minus the fraction of incoming solar radiation (Dynkerke, 1991). In the presence of a temperature gradient, water tends to be distilled from warmer regions to condense in cooler regions. Because of the heat storage and relative constant temperatures of the deeper soil layers, the moisture flow due to temperature gradients is usually upward in winter. This upward movement of moisture can then increase the moisture just above the ground which can lead to subsequent cooling and stronger thermal inversions with the aid of radiational cooling (Cox, 27). The effect of soil temperature (T S ) on fog formation is given by: {( ) T TS αts for ( ) T T F Ts = S < TSc (7) otherwise α Ts is the soil temperature constant and T Sc is the soil temperature threshold. Clear-air radiative cooling is very important for fog formation. Radiative cooling together with turbulent mixing causes the air to become saturated. The effect of net radiation is expressed as: {( ) RN R F rad = NC αnr for R N > R NC (8) otherwise α nr is the radiation constant and R NC is the radiation threshold for the genesis of fog.

5 364 P. Goswami and S. Sarkar The genesis term is calculated from observed values of the previous time step. In addition, the model fields are forced to the observed values for the first time step to provide a spin-up The persistence process As the radiative cooling persists, a fog layer may grow vertically and horizontally with time. The horizontal formation and spread of the fog is initially a function of the radiative properties of the surface. In the presence of favourable meteorological conditions (such as high pressure and light winds), a fairly uniform fog in terms of intensity and duration may form. The persistence of the steady stage implies that there exists a certain self-sustaining mechanism or a balance in its liquid water budget (Choularton et al., 1981). In general, it is a complex process and depends on the presence of the fog itself in a nonlinear manner to bring about the self-regulation. In our model the simple representation of a nonlinear self-sustained persistence term is: F P = α P F G (t 1) F G (t 1) (9) where α P is the fog persistence constant or self-regulating parameter and F G (t 1) is fog of the previous hour The dissipation process The dissipation of fog is a function of the processes that act against cooling. Eventual dissipation of fog occurs when the effect of surface heat flux on the relative humidity exceeds the effect of the supply of moisture from the surface or mixed downward from the fog top. The effects of cooling are mitigated or overcome through direct solar heating of the ground surface or potentially the heating of fog droplets and the air layer in which the fog is found. In general, the destruction of a stratified or inversion layer is through turbulent mixing. The dissipation process is thus represented as a direct function of near-surface wind and solar heating and is given by: F D = α D w (1) where w is the wind speed and α D is the fog dissipation constant. There is a critical turbulence exchange co-efficient that defines the upper limit of the turbulent intensity for persistent fog. Several factors may cause the turbulence intensity to exceed the critical turbulence exchange co-efficient leading to dissipation of the fog. One of the factors is the reduction in cooling rate due to solar heating. As solar heating is not explicitly included in Equation (1), the co-efficient α D is assumed to incorporate the effect of solar heating implicitly. 4. Model calibration and forecast validation As already emphasized, fog is a complex process and it is difficult to ascertain the completeness of a model; thus, the merit of the model has to be assessed in terms of its forecast skill. The forecast skill of the analogue model in an operational setting will therefore be focused on, i.e. without using any observational information beyond initial data for forecasting. The observed visibility is used for validation; benchmark forecasts are used for evaluation Calibration of the analogue model As it is not possible to assign precise values to the set of parameters that define the analogue model, a calibration was adopted Table 2. Parameters used in the analogue fog model (standard case). Serial No. Parameters for fog model Symbol Value Unit 1. Air temperature: low cut-off T ch 6 C 2. Air temperature: high cut-off T cl C 3. Low temperature co-efficient α tl 1.1 C 1 4. High temperature co-efficient α th 2.5 C 1 5. Threshold relative humidity (%) rh c 77 % 6. Relative humidity co-efficient α q Wind cut-off (m s 1 ) w c 1.5 ms 1 8. Wind co-efficient α w 1 ms 2 9. Dew depression threshold T dw 1.7 C 1. Dew depression co-efficient α dw 4.5 C Soil temperature threshold T Sc 12 C 12. Soil temperature co-efficient α Ts 2.7 C Net radiation threshold R NC 95 W m Net radiation co-efficient α nr 2.9 W 1 m 2. Fog persistence co-efficient α P.7 s Fog dissipation co-efficient α D 2.5 C 1 m Fog constant α.89 m 18. Reference visibility for clear sky V R 1 m to determine the optimum values. The set of parameters (i.e. co-efficient and thresholds) that define the analogue model was thus determined empirically through a search procedure for an optimum fit of the simulated value of the hourly visibility to that observed over the IGIA, Delhi. In particular, each parameter was allowed to vary over a range and the optimum value was adopted for which the average error in the forecasts was minimum for the period December 29 January 21. This set of parameters (Table 2) was kept constant for the remaining periods, namely December January , Validation of atmospheric mesoscale forecasts To ensure reliability of the mesoscale forecasts, time series between daily values of the meteorological variables at the 2 m level from the COMoN network and the forecasts for different time windows within the period of analysis were considered. The mean absolute error in the forecasted daily average 2 m temperature (T), relative humidity (RH) and wind speed (W) were computed for the six winter months, where the forecast was from the atmospheric model and the observed data were taken from the National Physical Laboratory (NPL) (COMoN), Delhi. Other statistical measures include the mean, standard deviation of observations (σ O ) and prediction (σ P ). The correlation co-efficients of the daily average temperature, humidity and wind at the 2 m level from the observed data (COMoN) and the forecasts from the over the NPL were also computed Evaluation parameters The validation was carried out against hourly observed values for visibility recorded at the IGIA in terms of the distribution of errors in visibility. These validation parameters were computed twice based on the persistence of fog, one for persistence of 2 h and the other for persistence 4 h. A number of parameters were used to quantify the skill of the forecasts, outlined below Absolute error The absolute error in the forecast of a variable X is considered to be given by: e AX (i, n) = X F (in) X O (i, n) (11)

6 Dynamical model for fog 365 Table 3. Summary of evaluation parameters for the forecast skill of the analogue fog model. Sl. no. Parameters Decision thresholds of visibility (m) for objective evaluation of forecast skill Observed visibility (V O ) Forecast visibility (V F ) 1. False warnings V O 1 (observed clear sky) V F 7 (predicted fog) 2. Over warnings 7 V O < 1 (observed fog) V F 5 (predicted dense fog) 3. Misses V O 5 (observed dense fog) V F 1 (predicted clear sky) where X F and X O represent, respectively, the values of X from forecast and observation at hour i on the n th day. The daily average absolute error in the forecast of X F is defined in the standard manner as: e A (n) = 1 N N=24 i=1 e AX (i, n) (12) where N is the number of forecasts or observations on an hourly scale for day n. Similarly, the monthly average forecast is defined as: e A (m) = 1 N m =31 e N A (n, m) (13) m n=1 where N m is the number of days (31). In addition, for overall evaluation of forecast skill, the skill of forecasts in terms of predicting false warnings, over warnings and misses was also examined (defined in Table 3). To define these, ranges of thresholds in visibility were considered to allow certain measures of error in the forecasts Percentage of false warnings (P FW ) A false warning is defined as a forecast of visibility (V F )against observed visibility (V O ), for which V F 7 m but V O 1 m (no fog); in other words a false warning is counted if dense fog is predicted against observed clear sky. The percentage of false warnings in N forecasts is then defined as: P FW = N FW 1 (14) N where N FW is the number of false warnings on a daily scale and N is the total number of forecasts on a daily scale Percentage of over warnings (P OW ) In a similar manner, the percentage of over warnings is given as: P OW = N OW 1 () N where N OW is the number of forecasts (days) when V F 5 m while the observed visibility (V O ) lies between 7 and 1 m Percentage of misses (P M ) The percentage of misses quantifies the failure of the forecast to indicate observed cases of dense fog and is defined as: P M = N M 1 (16) N where N M is the number of forecasts (days) for which V F 1 m against V O 5 m. A brief description of the evaluation parameters is provided in Table Percentage correct (P C ) The hit rate represents the fraction of forecasts for fog and non-fog events which are correctly diagnosed within the whole set of observations (N) within a month, defined as: P C = N F + N N 1 (17) N where N F is the number of correctly forecast foggy days and N N is the number of correctly forecast non-foggy days in terms of occurrence and non-occurrence of fog Threat score (P T ) The threat score, also known as the critical success index, is defined as the fraction of all forecasts and/or observed data that were correctly diagnosed: P T = N F N F + N N + N M 1 (18) The forecast bias (B) The forecast bias representing the number of forecasted events compared to the observed events is given by: P B = N F + N FW N F + NM (19) 5. Results and discussion The results on each of the major aspects are presented next, from the rationale for the study to model validation Validation of the atmospheric mesoscale forecasts As the forecasts from the analogue fog model are driven by the meteorological forecasts from the atmospheric mesoscale model (), these mesoscale forecasts of the meteorological variables were first examined against observations. Figure 1 shows that, except for wind speed, the diurnal cycles of temperature, relative humidity and net radiation at the 2 m level from are significantly correlated (99% confidence) with observations for all the winter seasons considered here. In most cases the observed wind speed is recorded as almost m s 1 throughout the day, indicating the presence of very calm weather. However, for the wind speed, the model always overestimates the observed values. Also, the net radiation predicted by the model is positive during the night time. This can be attributed to the model artefacts. A comparison of the mesoscale forecasts of temperature, relative humidity and wind speed at the 2 m level with the COMoN profiler (NPL) shows (Figure 2) the daily forecasts (2 m level) to be significantly correlated (95% confidence) with the corresponding observations for most cases except for some of the months. Several earlier studies (Goswami et al., 21, 212a;

7 366 P. Goswami and S. Sarkar (a) 21 Temperature ( C) (c) 4 Wind speed (m s 1 ) (.85) (.13) (b)1 Relative humidity (%) (d) 6 Net radiation (W m 2 ) 45 3 (.77) (.81) 5:3 8:3 11:3 14:3 17:3 Time in IST 2:3 23:3 2:3 5:3 8:3 11:3 14:3 Time in IST 17:3 2:3 23:3 2:3 Figure 1. Observed and modelled diurnal cycles of (a) 2 m temperature ( C), (b) relative humidity (%), (c) wind speed (m s 1 )and(d)netradiation (W m 2 ) at the National Physical Laboratory (NPL) site for December and January through The numbers in parentheses represent the correlation co-efficient between the diurnal cycles of observed and modelled variables. The 95% significance of the correlation is.4. Mean absolute error Mean absolute error T ( C) Correlation coefficient Correlation coefficient T ( C) Standard deviation Standard deviation WRT Mean absolute error Relative humidity (%) Correlation coefficient Relative humidity (%) Standard deviation WRT Mean absolute error Wind speed (m s 1 ) 29 Dec 21 Jan 21 Dec 211 Jan 211 Dec 212 Jan Correlation coefficient.9 Wind speed (m s 1 ) Dec 21 Jan 21 Dec 211 Jan 211 Dec 212 Jan Standard deviation WRT 29 Dec 21 Jan 21 Dec 211 Jan 211 Dec 212 Jan Figure 2. Validation statistics of daily averaged meteorological fields from the Weather Research and Forecasting () forecast against Climate Observation and Modelling Network (COMoN) observations at the 2 m level over the National Physical Laboratory (NPL), Delhi, in terms of mean absolute error, correlation co-efficient and standard deviation for December and January during The 99% (95%) significance of correlation is.4 (.3). The correlation co-efficients were computed for the daily values for each month and the error averages were computed over a 3 day period for each month.

8 Dynamical model for fog 367 (a) (b) < Rh (%) Temperature ( C) (c) 5 (d) (e) < Soil temperature ( C) 1 11 Dew-depression ( C) (f) ( 11) ( 8) ( 8) ( 5) ( 5) ( 2) Wind speed (m s 1 ) Net radiation (W m 2 ) ( 2) Figure 3. Distribution of (a) relative humidity (%), (b) 2 m temperature ( C), (c) soil temperature ( C), (d) wind speed, (e) dew depression ( C) and (f) net radiation (W m 2 ) during the fog period for December and January during at the National Physical Laboratory (NPL), Delhi. Goswami and Mallick, 211; Goswami and Mohapatra, 214) have also shown the applicability of the model over India. It may be seen, however, that the simulated daily maximum often occurs earlier by 1 or 2 h than the observation. It is believed that this is a part of the general model () bias for the location (Delhi), however, and an unambiguous explanation is difficult Analysis of pre-fog conditions and the predictor variables The characterization of the local conditions leading to radiation fog formation was performed with the objective of identifying the physical processes influencing fog formation. Six key variables for fog formation were analysed: 2 m temperature, relative humidity, dew depression (the difference between air temperature and dew-point temperature), soil temperature, net radiation and wind speed at 2 m. The analyses corresponding to time of fog onset were examined for all foggy days during the analysis period. The distributions of observed near-surface key meteorological variables at fog onset are shown in the form of frequency histograms (Figures 3(a) (f)). Relatively common scenarios (lower temperature, calm wind, large humidity at near surface) are characterized by the meteorological variables. The near-surface relative humidity condition during fog onset is characterized by high values (almost 45% of the events have values in the range 9 1%). During fog onset the cases for wind speed and net radiation are also representative of conditions for radiation fog formation with the distribution centred to lower ( 2ms 1 ) and negative values, respectively, indicating calm and clear sky conditions. For the other parameters the meteorological values are distributed within small intervals, suggesting that these meteorological variables can be used as proxies of fog formation probability.

9 368 P. Goswami and S. Sarkar (a) CS1 CS2 CS3 CS4 1 (b) (c) (d) (e) (f) Figure 4. Distribution of observed and modelled hourly changes in (a) visibility (m), (b) 2 m air temperature ( C), (c) dew depression ( C), (d) wind speed (m s 1 ), (e) net radiation (W m 2 ) and (f) soil temperature ( C) during the hour leading to fog onset (CS1), 2 h prior to fog onset (CS2), 3 h prior to fog onset (CS3) and 4 h prior to fog onset (CS4) for December and January through The evolution of both observed and modelled meteorological variable tendencies a few hours prior to fog formation was also examined (Figures 4(a) (f)). The majority of events were characterized by successively decreasing visibility trends equal to or smaller than 4 m h 1 during the few hours leading to fog (Figure 4(a)). The response of hygroscopic aerosols to the increasing ambient humidity is probably a contributing factor in the gradual reduction in visibility. More than 75% of the cases for both observation and model denote successively decreasing trends of about 2 C in temperature, soil temperature and dew depression (Figures 4(b), (c) and (f)). Changes in saturation-specific humidity reflect changes in temperature. Successively decreasing trends in dew depression toward zero are an indicator of the likelihood of fog formation. The successively decreasing trends in wind speed and net radiation show values of 1 m s 1 and 1 W m 1 for most cases prior to fog formation (Figures 4(d) and (e)).the distribution is always centred to more extreme values for 1 h pre-fog periods, which

10 Dynamical model for fog 369 (a) Mean absolute error (m) MAE Hit rate Humidity forcing Average hit rate (%) (b) 172 Mean absolute error (m) Temperature forcing MAE Hit rate Average hit rate (%) (c) Mean absolute error (m) Wind forcing MAE Hit rate Average hit rate (%) (d) 174 Mean absolute error (m) MAE Hit rate Soil-temp forcing Average hit rate (%) (e) Mean absolute error (m) Net-radiation forcing MAE Hit rate Co-efficients Average hit rate (%) (f) Mean absolute error (m) Fog constant Co-efficients MAE Hit rate Average hit rate (%) Figure 5. Calibration of model parameters in terms of average error and hit rate. Variation of average error in visibility (m) and percentage of correct forecasted events for the period December January (29 21) due to percentage change in model parameters from their standard value: (a) humidity forcing; (b) temperature forcing; (c) wind forcing; (d) soil temperature forcing; (e) net radiation forcing; (f) fog constant. indicates an enhancement of their skill as fog predictors when the fog onset is closer Calibration of the analogue model The results of model calibration following the procedure outlined in Section 4.3 show variation of the skill with the parameters of the fog model. To optimize the model, the percentage of correct forecasted events and average error in visibility were examined to check whether ±4% change in the standard value of the co-efficients of the model parameters can significantly affect these two parameters. The results of the calibration show (Figure 5) that the analogue fog model can produce comparatively less error in forecasts when the co-efficients of the model variables have been properly calibrated to optimum values. The model is robust within its ±2% change in the co-efficients of the model variables Diurnal variability in observation and forecasts The monthly average diurnal cycles of observed visibility over the IGIA (dark bars, Figure 6) show generally low visibility in the early hours of local time both in December (Figure 6, left panels) and January (Figure 6, right panels); however, the reduction in visibility is much stronger in January. The benchmark forecasts (shaded bars, Figure 6) capture these trends well in general, with correlation co-efficients between observed and benchmark forecasts above the 99% level of significance in all but one case (December 211); with mesoscale forecasts (crossed bars, Figure 6), the correlation between observations and forecasts is generally significant at about the 95% level Forecast skill in visibility The histogram of errors shows about 6% of the benchmark forecasts to be within the error of 1 m; for mesoscale forecasts, about 5% of the forecasts are in the error bin of 1 m (Figure 7). It is worth noting that for both months the distributions of forecasts in the error bins are nearly Gaussian (Figure 7); it is also worth noting that for both months the percentage of benchmark forecasts with large errors (5 m) is only about 5%.The 24 h forecasts of visibility were then evaluated against hourly observations of onset, duration and visibility. The hour of onset is defined as the hour at which visibility falls below 1 m for

11 37 P. Goswami and S. Sarkar (a) 12 December 29 (.95) ANL_ (.84) (b) 12 January 21 (.88) ANL_ (.89) Visibility (m) (.92) ANL_ (.78) (.87) ANL_ (.95) Visibility (m) (.21) ANL_ (.76) (.85) ANL_ (.97) Visibility (m) :3 8:3 11:3 14:3 17:3 2:3 23:3 2:3 5:3 8:3 11:3 14:3 17:3 2:3 23:3 2:3 Time in IST Time in IST Figure 6. Diurnal distribution of monthly averaged visibility from observation (black bars) and simulation with Climate Observation and Modelling Network (COMoN) data (; dark grey bars) and simulation with the Weather Research and Forecasting () forecast (ANL_; light grey bars) over Indira Gandhi International Airport, Delhi, during the winter months: (a) December ; (b) January The numbers in parentheses represent the correlation co-efficient (CC) between hourly observed and simulated visibility for the respective cases. The 99% (95%) significance of correlation is.5 (.4). 2 h or more at a stretch; the duration is defined as the number of hours for which visibility stays less than 1 m from onset. The month-wise histogram of error in fog onset and duration time is plotted in Figure 8. From the distribution of onset error it is evident that almost 6% of onset error time is less than 1 h. The simulated visibility is validated against the observed visibility in terms of percentage of moderate fog as well as dense fog days. A comparison of forecasts of mild (daily visibility < 1 m) foggy days and dense (daily visibility < 5 m) foggy days for the six winter seasons shows (Figure 9) that the analogue fog model driven by the meteorological fields from forecasts can accurately represent the observations. For both mild fog (Figure 9, lower panels) and dense fog (Figure 9, upper panels), the percentage of foggy days in observation (over the IGIA) and simulations match well, with correlation co-efficients between observation and simulation for the six cases above 95% significance. It needs to be emphasized that a perfect match is not expected as the observed visibility (over IGIA) is only representative visibility over Delhi; indeed, as expected, the forecasts based on area average often underestimate the number of cases Skill scores: false warnings, over warnings, misses, hit rates and false alarms The hit rates for the 6 months during December 29 to January 212 are generally above 8% for the benchmark forecast with an average of 8.5% (Figure 1, top left panel); the corresponding percentage of correct proportion with the mesoscale forecast is 79.5%. There is systematic bias with respect to the months (Figure 1, bottom right panel). The percentage of false warnings, consistently, is generally below 5% for the benchmark forecasts and about 4% for the mesoscale forecast (Figure 1, top right panel). In terms of threat score, both benchmark and mesoscale forecasts are characterized by more than 7%; expectedly the benchmark forecast has a higher threat score (Figure 1, bottom left panel). The summary of performances in terms of various statistical measures such as average error in intensity of

12 Dynamical model for fog 371 No. of Foggy days No. of Foggy days (a) December (b) January >( 5) ( 4)_( 3) ( 2)_( 1) (1)_(2) (3)_(4) Error in visibility intensity (m) >(5) >( 5) ANL_ ANL_ ( 4)_( 3) ( 2)_( 1) (1)_(2) (3)_(4) Error in visibility intensity (m) >(5) Figure 7. Histogram of errors in visibility intensity showing the number of foggy days in different error bins (predicted observed) for the benchmark simulation () and for the test forecast (ANL_) for (a) December and (b) January through (a) No. of Foggy days (b) No. of Foggy days December ANL_ ANL_ January ANL_ ANL_ Error in onset (h) Error in onset (h) Error in duration (h) Error in duration (h) No. of Foggy days No. of Foggy days Figure 8. Histogram of errors in fog onset and duration showing the number of foggy days in different error bins (predicted observed) for the benchmark simulation () and for the test forecast (ANL_) for (a) December and (b) January through visibility, percentage of false warning days, percentage of misses, percentage of under warning days shows (Table 4) considerable skill in each month and for each year. 6. Sensitivity analyses The calibration process, in effect, is an optimization process. Thus, sensitivity analysis is necessary to examine other configurations, within the uncertainties of the semi-empirical constants. The parameters used in the analogue fog model (Table 2) are based on calibration; it is necessary to examine the sensitivity of the results of these parameters. While this requires a very large number of experiments, a few cases are discussed here to highlight certain important points. The first case involves changing the fog constant to 1. and.62 respectively from its standard value (.89). A second case of sensitivity analysis in terms of mean absolute error between observed and simulated visibility was done by driving the analogue model with varying thresholds of relative humidity of 6% and 45% respectively from its standard value (75%).The third experiment was done with and without soil temperature. The analysis was carried out for both benchmark and test simulations for the 2 month period December January during The results showed (Figure 11) that each of these variables plays an important role. An increase in the average error in visibility was observed for the new set of parameters in the fog constant. The average error is also higher for forecasts without soil temperature. Using a lower threshold of relative humidity, there is a significant increase in average error for the test forecast (ANL_) and the error in the benchmark forecast () is decreased. However, there is a significant increase in the number

13 372 P. Goswami and S. Sarkar % of cases (a) Vis ( 5 m) December ANL_ (b) Vis ( 5 m) January ANL_ m < Vis < 1 m 5 m < Vis < 1 m ANL_ ANL_ % of cases :3 7:3 9:3 11:3 13:3 :3 Time in IST 17:3 19:3 21:3 23:3 1:3 3:3 5:3 7:3 9:3 11:3 13:3 :3 Time in IST 17:3 19:3 21:3 23:3 1:3 3:3 Figure 9. Frequency distribution of the average percentage of dense fog (visibility 5 m) cases and mild fog (5 m < visibility < 1 m) cases as a function of time of day from the analogue fog model driven by Climate Observation and Modelling Network (COMoN) data (; shaded bar) and by the forecast from the Weather Research and Forecasting () model (ANL_; grey bar). The black bars denote the observed values. The computation was done for (a) December during and (b) January during % of Correct forecasts Threat Score (%) (a) Dec 21-Jan Percent Correct (8.5) ANL_ (79.5) 21-Dec 211-Jan 211-Dec 212-Jan (b) 1 29-Dec 21-Jan 21-Dec (49.4) ANL_ (38.1) 211-Jan 211-Dec 212-Jan (c) Threat Score (d) Bias 1 2 (78.) (1.1) 9 ANL_ (76.2) ANL_ (.94) Dec 21-Jan 21-Dec 211-Jan Months 211-Dec 212-Jan False alarm rate (%) Bias Dec 21-Jan False alarm rate 21-Dec 211-Jan Months 211-Dec 212-Jan Figure 1. Skill scores of the analogue fog model driven by Climate Observation and Modelling Network (COMoN) data () and by the forecast from the Weather Research and Forecasting () model (ANL_) during December January (29 212): (a) percentage correct; (b) percentage of false alarm rate; (c) threat score as a percentage; (d) bias. The numbers in parentheses represent averages over the six periods (December 29 to January 212) for the respective cases.