Weather forecasters and forecasting methods

Abstract

The weather forecasting method includes the step of generating a radiance difference as the difference between the radiance measured from the satellite and the forecast satellite radiance generated by the forecasted state profile output by the radiative transfer model and the numerical weather forecasting (NWP) model. If the radiance difference exceeds a noise threshold, the method includes the step of generating an updated state profile by generating a radiance sensitivity using the Jacobian model and the forecasted state profile, constructing a Kalman gain matrix from the background error covariance (BEC) matrix and the radiance sensitivity, generating a filtered state profile change from the Kalman gain matrix and the radiance difference, updating the state profile by adding the filtered state profile change to the forecasted state profile, and generating an updated state profile.

Description

[Background technology]

[0001] (Cross-reference of related applications) This application claims the benefit of U.S. Provisional Application No. 63 / 453,403, filed on March 20, 2023, whose disclosure is incorporated herein by reference in its entirety.

[0002] (Government rights) This invention was developed with government support under U.S. Air Force Phase II SBIR Contract No. FA9453-20-C-0713. The Government has certain rights in this invention.

[0003] (background) Weather forecasting models currently have limited ability to predict short-term and small-scale weather effects within a 0-12 hour range by effectively utilizing real-time data from passive microwave radiometers, due to their inability to adequately utilize hydrological intrinsic features in the data on the required 15-minute timescale. This leaves a range of market opportunities unresolved for improving industry- and region-specific forecasts to address a range of weather and climate change challenges. Improving forecast accuracy can enhance emergency response operations, inform critical financial and risk management industry decision-making, optimize evacuations, and be used to improve aircraft and ship route planning and weather impact mitigation. Highly localized weather forecasts are also critical for military operations on a global scale. [Overview of the project] [Means for solving the problem]

[0004] (Summary of the embodiment) This infeasibility leaves a range of market opportunities for improving industry- and region-specific forecasts unresolved in order to address a range of weather and climate change challenges. Improving the accuracy of forecasts can be used to improve emergency responder operations, inform decision-making in important financial and risk management industries, optimize evacuations, and improve aircraft and ship route planning and mitigation of weather impacts. Highly local weather forecasts are also decisive for military activities worldwide.

[0005] In a first aspect, a weather prediction method is disclosed. The method includes generating a radiance difference as the difference between measured radiance from a satellite, and a predicted satellite radiance generated by a radiative transfer model and a predicted state profile output by a numerical weather prediction (NWP) model. When the radiance difference exceeds a noise threshold, the method uses a Jacobian model and the predicted state profile to generate a radiance sensitivity, constructs a Kalman gain matrix from a background error covariance (BEC) matrix and the radiance sensitivity, generates a filtered state profile change from the Kalman gain matrix and the radiance difference, and updates the state profile by adding the filtered state profile change to the predicted state profile, thereby generating an updated state profile.

[0006] In a second aspect, a weather predictor includes a processor and a memory. The memory stores machine-readable instructions that control the processor to execute the weather prediction method of the first aspect when executed by the processor.

Brief Description of the Drawings

[0007] [Figure 1] FIG. 1 shows a basic data assimilation procedure implemented by an embodiment of the weather predictor disclosed in this specification.

[0008] [Figure 2] Figure 2 is a schematic diagram of a weather forecaster in one embodiment.

[0009] [Figure 3] Figure 3 is a functional block diagram of an exemplary data assimilation implemented by the weather forecaster in Figure 2 in one embodiment.

[0010] [Figure 4] Figure 4 is a flowchart illustrating a weather forecasting method that can be implemented using the weather forecaster shown in Figure 2, in one embodiment.

[0011] [Figure 5A] Figures 5A-5D illustrate the respective hydrological profiles partitioned into K=20 clusters over four time frames, according to one embodiment of the weather forecaster shown in Figure 2. [Figure 5B] Same as above. [Figure 5C] Same as above. [Figure 5D] Same as above.

[0012] [Figure 6A] Figures 6A-6D illustrate the hydrological profiles partitioned into K=10 clusters over four time frames, according to one embodiment of the weather forecaster shown in Figure 2. [Figure 6B] Same as above. [Figure 6C] Same as above. [Figure 6D] Same as above.

[0013] [Figure 7] Figure 7 graphically illustrates the Jacobian function generated using the linear radiative transfer model by the weather forecaster shown in Figure 2.

[0014] [Figure 8] Figure 8 graphically illustrates the vertical hydrological distribution for precipitation profiles used for hydrological Jacobian tests.

[0015] [Figure 9] Figure 9 graphically illustrates the Jacobian functions obtained using a linear radiative transfer model for the observed rain-phase hydrological and hydrological profiles for Hurricane Sandy on October 29, 2012.

[0016] [Figure 10] Figure 10 graphically illustrates the Jacobian functions obtained using a linear radiative transfer model for the cloud-ice phase hydrological and hydrological profiles observed for Hurricane Sandy on October 29, 2012.

[0017] [Figure 11] Figure 11 shows the physical location of cluster #3 within the first frame of the Midwest storm.

[0018] [Figure 12] Figure 12 shows the correlation matrix for cluster #3. [Modes for carrying out the invention]

[0019] (Detailed description of the embodiment) 1. Overview The ability to lock hydrological conditions into satellite data for numerical weather forecasting (NWP) models and provide quantitative predictions of such events requires that the data be sampled at intervals less than the correlation time for hydrological variables within precipitation events. This also requires that the satellite data be sensitive to the presence of hydrological events directly below the cloud tops, which is a crucial advantage of passive microwave sounding and imaging channels for infrared channels. This further requires that the data have a spatial resolution comparable to, or close to, the horizontal spatial scale of convective events of about 10–15 kilometers. All of these features can be designed in the systems and methods disclosed herein.

[0020] 2. Weather forecasters and methods Figure 1 shows an exemplary basic data assimilation procedure 100 implemented by embodiments disclosed herein. Generally, any of several NWP models that incorporate explicit cloud and precipitation microvisual models, such as the Global Aeronautical and Land Meteorological Utilization Model (GALWEM), the Global Forecasting System (GFS), and the Weather Research and Forecasting (WRF) model, may be used. The selection of a WRF model for deploying the weather forecaster 200 is based on its general reliability, open accessibility, numerical efficiency, scalability, and ability to be incorporated into loop-based assimilation schemes.

[0021] The WRF model was used to simulate combined hydrological and thermodynamic states of the atmosphere on a regional basis through the occurrence of "nature" and "assimilation" runs in several case studies. The "nature" run represents the "truth" for the test meteorological scenario, which is the expected and observed brightness temperature (T) from the GEMS satellite constellation. b This is used to generate simulated (i.e., artificial but realistic) state data that forms the basis of the data. In the assimilation run, the natural run is perturbed as observed by the GEMS constellation, and the simulated data and weather forecasting methods are used to converge, i.e., "lock" the NWP state vector onto the true weather state vector. This process is therefore an observational system simulation experiment (OSSE) for a GEMS passive microwave sensor constellation used in the Rapid Update All Weather 3D Variational (3Dvar) Extended Kalman Filter (XKF) assimilation scheme.

[0022] 2.1 Weather Forecaster Figure 2 is a schematic diagram of a weather forecaster 200 that outputs a converged state profile 229 from measured satellite radiance 204 and noise statistics 206. The weather forecaster 200 includes a processor 286 and memory 202. The noise statistics 206 include statistics on noise from the satellite that generates the measured satellite radiance 204.

[0023] In the embodiment, the forecasted state profile 319 includes, for each of a plurality of horizontal grid points within a geographically defined domain, at least one vertical state profile of air temperature, humidity, altitude, hydrological density, and hydrological size. The weather forecaster 200 may be communicatively coupled to a satellite constellation, such as the aforementioned GEMS satellite constellation. The measured satellite radiance 204 may be data measured by the satellite constellation.

[0024] The processor 286 represents any type of circuit or integrated circuit capable of performing logic, control, and input / output operations. For example, the processor 286 may include one or more of the following: a microprocessor with one or more central processing unit (CPU) cores, a graphics processing unit (GPU), a digital signal processor (DSP), a field-programmable gate array (FPGA), a system-on-a-chip (SoC), a microcontroller unit (MCU), and an application-specific integrated circuit (ASIC). The processor 286 may also include a memory controller, a bus controller, and other components that manage the data flow between the processor 286 and the memory 202.

[0025] Memory 202 may be transient and / or non-transient and may include one or both of volatile memory (e.g., SRAM, DRAM, compute RAM, other volatile memory, or any combination thereof) and non-volatile memory (e.g., FLASH®, ROM, magnetic media, optical media, other non-volatile memory, or any combination thereof). Part or all of memory 202 may be integrated into processor 286. Memory 202 stores software 220, which includes non-transient machine-readable instructions. When executed by processor 286, software 220 causes processor 286 to implement the functionality of weather forecaster 200 as described herein. Software 220 is or may include firmware.

[0026] Figure 3 is a functional block diagram of a data assimilar 300 that may be implemented by an embodiment of the weather forecaster 200. The functional elements of the data assimilar 300 may be stored as software 220, while data accessible to and / or generated by the data assimilar 300 may be stored in memory 202. The functional elements of the data assimilar 300 include a radiative transfer model 320, a Jacobian model 322, a convergence checker 325, a comparator 330, an NWP forecast model 310, a classifier 340, a BEC calculator 350, a Kalman filter 360, an iterative scaler 364, an iterative state profile updater 366, and a constraint checker 370. The data accessible to and / or generated by the data assimilarizer 300 includes measured satellite radiance 204, noise statistics 206, radiance difference 339, converged state profile 229, forecast satellite radiance 324, forecasted state profile 319, radiance sensitivity 326, filtered state profile change 362, scaled state profile change 365, iterated state profile 368, background error covariance (BEC) matrix 359, Kalman gain matrix 336, cluster database 341, clustered state profile 349, and updated state profile 379. The data assimilarizer 300 includes a Kalman filter cycle 308 that includes the aforementioned functional elements and some of the data. The data assimilarizer 300 may run the Kalman filter cycle 308 as an iterative loop.

[0027] The following describes exemplary operation of one embodiment of a data assimilarizer 300, as implemented by one embodiment of the weather forecaster 200. The weather forecaster 200 receives satellite radiances 204 and processes them with high precision by a radiation metric calibration method. These radiances, also known as radiance temperatures, are provided by the satellite at one or above several different frequencies. These frequencies are associated with important natural features of the Earth's radiance spectrum and include microwave frequencies at various gas resonance frequencies, as well as frequencies between these resonances that allow transmission to the surface through the atmosphere. The frequencies may also include infrared and optical bands in addition to microwave frequencies. The satellite data may be provided in a data cube, which is a stack of two-dimensional images, with one such image at each frequency. Other formats are also possible.

[0028] The comparator 330 compares the satellite radiance 204 with an equal set of forecast satellite radiances 324 predicted for each specific satellite using the radiative transfer model 320. The initial forecast radiances 324 may be obtained and processed by the radiative transfer model 320 from estimates of the forecast state profile 319. These initial estimates may be obtained from any other numerical weather forecasting (NWP) model, which is operated by other actual atmospheric observations, for example, by government services or using satellites, radar, weather balloons, or other instruments.

[0029] The forecast radiance 324 is subtracted from the satellite radiance 204 using a comparator 330 to form a radiance difference 339. The radiance difference 339 is used to correct the state profile and produce a converged state profile 229, which is further used to forecast atmospheric conditions using a numerical weather forecasting model 310. Several forecasting models, for example, one of the Weather Research and Forecast (WRF) models, may be used by the NWP model 310.

[0030] When the radiance difference 339 is statistically indistinguishable from the noise statistic 206, no further information can be derived from the radiance difference 339. The noise statistic 206 may include the square root sum of noise matrices from (1) the satellite, (2) the radiative transfer model 320, and (3) the NWP model 310, all of which produce the measured satellite radiance 204. This square root sum comparison test is performed by a convergence checker 325, which tests for algorithmic convergence by comparing the radiance difference 339 with the noise statistic 206. The noise statistic 206 may also be the square root sum statistical noise covariance at all frequencies and in 2D image pixels, as shown on the right-hand side of equation (1). The convergence test is described using equation (2.1).

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[0031] In the case of successful convergence, the data assimilar 300 proceeds with the step of returning the current forecast state profile 319 to the NWP model 310 through copying and transmission, for example, to the NWP model 310 as a converged state profile 229 of these profiles. The NWP model 310 then subsequently forecasts appropriate atmospheric conditions for the next satellite observation. The data assimilar 300 may use enough satellites so that the time interval between consecutive observations is typically within a few minutes, but can be as long as three hours, depending on the number of satellites available.

[0032] Because the atmosphere is likely to develop until the time of the next satellite observation, the convergence checker 325 is likely to show non-convergence. In this case, the information within the radiance difference 339 is used to correct the atmospheric state profile through the associated processes of the Kalman filter 360 and the Kalman filter cycle, providing the Kalman filter 360 with important information.

[0033] The filtering algorithm proceeds as follows: The Jacobian model 322 calculates the radiance sensitivity 326 from the forecasted state profile 319. The radiance sensitivity 326 effectively provides sensitivity to all radiance temperatures for small variations within each atmospheric state profile variable. A fast and accurate means of calculating these sensitivities for all types of atmospheric conditions (clear, cloudy, stratiform, convective precipitation, etc.) is important for the calculations of the filtering algorithm. The Jacobian model 322 may include an algorithm that provides a solution to the radiance transfer equation (for example, used by the radiance transfer model 320), and may also include a library, a lookup table, and machine learning and artificial intelligence algorithms for calculating these radiance sensitivities 326.

[0034] The radiance sensitivity 326 is used in conjunction with a preferred set of background error covariance (BEC) matrices 359 to construct a Kalman gain matrix 336. The number of BEC matrices in the set may be determined, at least in part, by the number required to cover a set of meteorological conditions over a given area of ​​the Earth. In embodiments, the BEC matrix 359 includes 20 to 25 BEC matrices.

[0035] In the embodiment, each pixel or subset of pixels in the radiance difference data cube or data volume (included within the radiance difference 339) may have a distinctly different Kalman gain matrix constructed, so that the Kalman gain matrix 336 may include several Kalman gain matrices. The Kalman filter 360 uses the Kalman gain matrix 336 and the radiance difference 339 to determine a set of filtered state profile changes 362 by matrix multiplication. The filtered state profile changes 362 are defined in equation (2.2).

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[0036] The filtered state profile changes 362 undergo at least one of the processing steps (scaling by the iterative scaler 364 and imposing physical-based constraints by the iterative state profile updater 366) before they are used as updated state profiles 379. These steps may be used as necessary to maintain the physical suitability of the iterative state profiles 368 prior to their use within the Kalman filter cycle 308.

[0037] The processing step includes determining an iterative scaling factor 364F based on a measure of expected accuracy of the Jacobian model 322, which provides only a linear relationship between radiance and state profile quantities. The iterative scaler 364 determines this accuracy for each state profile based on the degree of linearity of the relationship between the following state profile quantities, namely temperature, humidity, and one or more of the cloud and rain parameters, as a function of radiance (satellite brightness temperature). The scaling factor 364F may be zero to 1 (including them).

[0038] Linearity is estimated within the Jacobian model 322 once the sensitivity is calculated. Generally, a scaling factor 364F, which is different for each pixel or group of pixels, is applied to scale the state profile change, resulting in a scaled state profile change 365, which is defined in equation (2.3).

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[0039] The iterative state profile updater 366 predicts the state profile 319 ( ) in the scaled state profile change 365.

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[0040] In the embodiment, before the iterated profiles 368 are adopted, they are checked for consistency with the conservation laws of energy, thrust, and mass by mutual comparison with the constraint checker 370. Only those state profiles 368 that conform to the conservation laws are propagated as corrections to produce the updated state profile 379, which is further used within the Kalman filter cycle 308. Scaling, iterative updating, and conservation constraint elements (scaler 364, updater 366, and constraint checker 370), applied to all state profile variables including temperature, humidity, cloud and rain, and surface parameters, are key features of the embodiment of the data assimilation 300.

[0041] Between each cycle of the Kalman filter cycle 308, the updated state profile 379 is used to determine the updated set of BEC matrices 359. The BEC matrix 359 is dynamically dependent on the atmospheric conditions and the state profile

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[0042] However, during dynamically evolving atmospheric conditions such as those occurring in convective rain, the BEC matrix 359 is likely to introduce large errors within the cloud and rain cell profile parameters; therefore, the focus is on correcting the Kalman gain matrix 336 for parameters such as cloud and rain density. Thus, the generation of the BEC matrix leads to the use of measured satellite radiance 204 and the NWP model 310, taking advantage of high-frequency satellite radiance observations.

[0043] The BEC calculator 350 rapidly calculates the BEC matrix 359 using the state profile database 341, which is fed into the classifier 340. Database 341 contains many atmospheric state profiles. This ensemble of profiles in database 341 can represent the most likely atmospheric state profiles that may occur in a given region of the Earth over a given season of the year.

[0044] In the embodiment, the classifier 340 uses standard data clustering methods such as the K-means algorithm to facilitate the classification of the updated state profile 379 into a few possible atmospheric conditions, which are then categorized by the BEC calculator 350 into clustered state profiles 349. In the embodiment, each clustered state profile 349 has a specific pre-calculated BEC matrix that captures statistics for these possible atmospheric conditions. These atmospheric state profile statistics may be obtained from either (a) a long series of NWP model simulations of weather over a given season in a given region, or (b) actual atmospheric observations made using satellites, radar, weather balloons, or other instruments.

[0045] These statistics are used by the BEC computer 350 to represent the statistics of a specific state profile 379 for the Kalman filter cycle 308. The BEC computer 350 generates a BEC matrix 359 by using the calculation of the covariance of the clustered state profile 349 with respect to the updated state profile 379. Classification of the updated state profile 379 allows for either interpolation from the clustered state profile 349, library lookup, or machine learning generation of the BEC matrix. The BEC computer 350 may include a neural network trained on the clustered state profile 349 to generate the BEC matrix 359.

[0046] The deployment of the clustered state profile database 341 may occur offline, producing a static database. However, the database 341 and profiles 349 may be periodically revised and updated as new information about possible atmospheric state profiles becomes available, and in accordance with either seasonal or climate changes occurring on both slow and fast time scales. For example, new cluster data may be incorporated as the Earth transitions between steady El Niño and La Niña states.

[0047] Between each iteration of the Kalman filter cycle 308, the data assimilar 300 recalculates the forecast satellite radiances 324, captures their differences 339, and checks for convergence using the convergence checker 325. Convergence may occur either by reducing the differences 339 to the level of the noise statistic 206, or after a fixed number of iterations of the Kalman filter cycle 308 (whichever occurs first). A fixed number of iterations may be used if new satellite radiances arrive before the Kalman filter loop reaches convergence by adjusting the state profile. In response to convergence, the data assimilar 300 receives the latest forecast state profile 319 as the converged state profile 229, and then proceeds to the step of forecasting the state profile (as state profile 319(k+1)) and adapting to the arrival of the next set of satellite radiances 204.

[0048] When (a) satellite radiance 204 is observed and assimilated at a sufficiently high frequency, (b) state profile 379 is updated sufficiently frequently, and (c) Jacobian model 322 and BEC error covariance calculator 350 are used between each update to provide accurate error covariance, respectively, the data assimilarizer 300 is locked to continuously track small variations in the actual atmospheric state profile.

[0049] One of the key technical benefits of the data assimilar 300 is its ability to lock onto rapidly evolving atmospheric condition profile changes associated with clouds and rain in dynamically developing storm conditions. When the data assimilar 300 is locked, it uses satellite radiance responses to all atmospheric conditions to track evolving conditions, particularly dynamically evolving conditions of rain and weather front events, which would otherwise lead to highly inaccurate forecast radiance. A key feature of the data assimilar 300 is that, when locked, it uses high-frequency satellite radiance updates to eliminate the need for four-dimensional variational assimilation of data into the NWP model (which, at best, is known to have limited accuracy due to large changes in atmospheric conditions that can occur during dynamic conditions).

[0050] 2.2 Weather Forecasting Methods Figure 4 is a flowchart illustrating weather forecasting method 400. Method 400 may be implemented, at least in part, within the software 220 of the weather forecaster 200 in Figure 2. Method 400 includes at least one of steps 420, 430, 440, 450, 460, and 460. Method 400 may also include at least one of steps 410, 432, 445, and 480.

[0051] Step 420 includes generating a radiance difference as the difference between the radiance measured from the satellite and the forecast satellite radiance generated by the radiative transfer model and the forecasted state profile output by the numerical weather forecasting (NWP) model. In one embodiment of step 420, the comparator 330 generates a radiance difference 339(k) as the difference between the measured radiance 204 and the forecast satellite radiance 324(k-1) generated by the radiative transfer model 320 and the forecasted state profile 319(k-1). Hereinafter, the subscript k indicates the k-th iteration of the data assimilation unit 300. In embodiments, each iteration begins with the generation of the radiance difference 339.

[0052] Method 400 may include step 410, which includes generating forecast satellite radiance using a radiative transfer model before generating a radiance difference. In one embodiment of step 410, the radiative transfer model 320 generates forecast satellite radiance 324(k-1).

[0053] Step 430 is a decision. If the radiance difference exceeds the noise threshold, method 400 proceeds to steps 440, 450, 460, and 460, which produce an updated state profile. For example, in iteration k, if the radiance difference 339(k) exceeds the noise threshold based on the noise statistic 206, the data assimilarizer 300 produces an updated state profile 379(k). The noise statistic 206 may include noise associated with the radiative transfer model 320. The noise threshold may be the root mean square (RMS) of noise from the instrument (e.g., satellite) and the radiative transfer model 320.

[0054] Step 440 includes the step of generating a radiance sensitivity using a Jacobian model and a predicted state profile. In an embodiment of step 440, the Jacobian model 322 generates a radiance sensitivity 326 from a predicted state profile 319(k-1).

[0055] Step 450 includes the step of constructing a Kalman gain matrix from the background error covariance (BEC) matrix and the radiance sensitivity. In one embodiment of step 450, the data assimilarizer 300 constructs a Kalman gain matrix 336 from the BEC matrix 359 and the radiance sensitivity 326.

[0056] Method 400 may include step 445. Step 445 includes generating a background error covariance (BEC) matrix by interpolating updated state profiles across a library of clustered state profiles obtained by the classification method. Embodiments of the classification method include k-mean clustering. In one embodiment of step 445, a BEC calculator 350 generates a BEC matrix 359 by interpolating updated state profiles 379 across a library of clustered state profiles 349. This library may include BEC matrices. The classifier 340 may generate clustered state profiles 349 from a cluster database 341.

[0057] Step 460 includes generating a filtered state profile change from the Kalman gain matrix and the radiance difference. In an embodiment of step 460, the Kalman filter 360 generates a filtered state profile change 362 from the Kalman gain matrix 336 and the radiance difference 339. The filtered state profile change may be the product of the Kalman gain matrix and the radiance difference.

[0058] Step 460 may include step 462, wherein the filtered state profile change to be generated is an unscaled filtered state profile change. Step 462 includes, for example, scaling the unscaled filtered state profile change by an iterative scaling factor 364F to produce a filtered state profile change.

[0059] Step 470 includes updating the state profile by adding filtered state profile changes to the predicted state profile, thereby producing an updated state profile. In one embodiment of step 470, an iterative state profile updater 366 adds a scaled state profile change 365 to the predicted state profile 319(k-1), producing an iterative state profile 368(k). Step 470 may also include step 472, which includes removing updated state profiles that are not physically feasible from the updated state profile. In an embodiment of step 472, a constraint checker 370 removes physically unfeasible state profiles from the iterative state profile 368(k), producing an updated state profile 379.

[0060] If the radiance difference exceeds a noise threshold, method 400 may also include step 480. Step 480 is a step of repeating the step of generating a radiance difference, wherein the updated state profile replaces the predicted state profile. In one embodiment of step 480, the data assimilar 300 replaces the updated state profile 379(k) with the predicted state profile 319(k), generates the predicted state profile 319(k), and repeats step 420 using the predicted state profile 319(k). Subsequent steps of method 400 may be repeated after step 420 has been repeated.

[0061] If the radiance difference output from step 420 does not exceed the noise threshold, the embodiment of method 400 proceeds to steps 432 and 434. Step 432 includes generating a subsequent predicted state profile and a predicted state profile as input thereto using an NWP model. In one embodiment of step 432, a convergence checker 325 outputs a converged state profile 229 from which the NWP forecasting model 310 generates a predicted state profile 319(k+1). The converged state profile 229 may be equal to the predicted state profile 319(k). Step 434 includes repeating step 420, wherein the subsequent predicted state profile replaces the predicted state profile. In step 434, the predicted state profile 319(k+1) output from step 432 is an embodiment of the subsequent predicted state profile.

[0062] In embodiments, the forecasted state profile includes a vertical state profile of multiple state profile variables for each of multiple horizontal grid points within a geographically defined domain. The multiple state profile variables may include air temperature, humidity, and at least one or more additional state profile variables. Examples of additional state profile variables include altitude, hydrological density, hydrological size, vapor density, cloud density, rain density, ice density, snow density, graupel density, mean rain particle size, mean ice particle size, and mean hydrological size. In embodiments, the multiple state profile variables may include both air temperature and humidity, and one or more of the aforementioned additional state profile variables. Each of the multiple state profile variables has one for each of multiple correlation times.

[0063] In such embodiments, step 470 may include updating each state profile variable of the vertical state profile of the predicted state profile 319. Also in such embodiments, when constructing the Kalman gain matrix, each BEC matrix (e.g., BEC matrix 359) contains the covariance between each state profile variable of the multiple state profile variables.

[0064] The duration between consecutive runs of step 470 (e.g., iterations k and k+1) may be less than the shortest correlation time of multiple correlation times. The duration may be adjusted in real time depending on whether the atmosphere is dynamic, in particular, over a given region (e.g., a tornado or frontal boundary). The duration may be between 1 minute and 3 hours, depending on environmental conditions. Exemplary durations include 1 to 5 minutes for tornado supercells, derechos, or hailstorms; 5 to 15 minutes for frontal convection, clouds producing microbursts, or hurricane rainbands; 15 to 30 minutes for downdrafts such as foehns; 30 to 60 minutes for stratiform precipitation or snowfall events; and 2 to 3 hours for cloudy non-precipitation clouds.

[0065] 2.3 Spontaneous execution OSSE uses XKF data assimilation, performed by weather forecaster 200, to investigate the forecasting impact of the passive microwave satellite GEMS constellation. The satellite has sounding channels in the major 118, 183, and 50-58, as well as the 424 GHz microwave sounding band. The forecasting impact of GEMS is assessed by considering the reduction of (primarily) 1-12 hour forecast anomalies resulting from the use of simulated GEMS data to yield an NWP model in which its state vector is perturbed so that it returns true with respect to all thermodynamic and hydrological variables. OSSE thus provides a means for quantitatively assessing the data impact on short-term precipitation forecasts, which is of both strategic and commercial interest.

[0066] 3.0 Methods, Assumptions, and Procedures 3.1 UMRT Forward Radiation Transfer Model Radiation transfer model 320 may use, or include, the unified microwave radiation transfer (UMRT v4) model [1] as the forward radiation transfer model used to calculate upwelling polarized radiance in the upper atmosphere. UMRT is a coupled multi-flow bipolarized (V and H) scattering-based RT model that uses a slab-duplexing engine inherited from the inherently stable discrete longitudinal tangent linear radiation transfer (DOTLRT) model [2]. DOTLRT is an early T model from which the new UMRTv4 model has been refined in operation. b It was used to compute the simulation. In the embodiment, this supports a rapid Jacobian calculation that provides all radiance derivatives with respect to arbitrary profile emission parameters, for example, by Jacobian model 322.

[0067] This rapid Jacobian calculation may follow the perturbation method implemented within the DOTLRT model. Table 1 lists the profile parameters provided to UMRT from the WRF model. UMRT was originally formulated to use exponential size distributions for each of the five hydrological species (rain clouds, liquid water, cloud ice, snow, and graupel) based on observed particle size spectra. The UMRT v4 model may be modified to use arbitrary size distributions. In embodiments, the size distribution is exponential because it is the simplest from a mathematical standpoint and because it captures a considerable amount of physical absorption and scattering from the polydispersity of naturally occurring hydrological phenomena.

[0068] The use of exponential distributions is justified to validate embodiments of weather forecaster 200 because they are performed in a closed environment and, in principle, can employ any reasonable physics. To characterize embodiments of weather forecaster 200, WRF microphysical hydrophysical parameterization is used, providing mass densities of cloud liquid, rain, ice, and graupel; however, only the number densities of rain and ice are available to provide a complete and unique exponential size distribution. For cloud liquid and graupel, the average particle size is obtained using an empirical model. In addition to hydrophysical data, UMRT uses vertical profiles of temperature and humidity from the WRF model.

[0069] UMRT uses a millimeter-wave propagation model (MPM) [3] to calculate gas absorption from water vapor, oxygen, and nitrogen, and uses Mie scattering theory to calculate the absorption and scattering coefficients and phase matrix elements of the hydrological phenomena. Uniform liquid or frozen water spheres are therefore used to approximate all hydrological phenomena. The calculation of Mie absorption and scattering coefficients for exponential spherical hydrological distributions proceeds rapidly using a tabular library [1]. However, the calculation of the Mie phase matrix is ​​not yet tabular and therefore requires a significant amount of computer time (typically about 100 to 1,000 times that required for real-time operation implementation on a similar machine) to be completed on the University of Wisconsin's multicore machine. As a means of achieving speed improvements, the computationally simpler Henyey-Greenstein (HG) phase function [4] may be substituted for the Mie phase matrix in the radiative transfer model 320. This substitution has helped to achieve an improvement of almost double the calculation time. Reducing the number of radiant streams from 16 to 8 resulted in a 4x acceleration per profile. [Table 1]

[0070] Predicted multiflow radiances using UMRT from natural storm runs in the Midwest reveal the expected characteristics of radiant emissions from scattering and absorbing clouds and rain cells. These radiance fields are significantly dependent on background absorption of microwave gas, and therefore on microwave channel frequencies. High-frequency channels near the 183.31 GHz water vapor resonance exhibit expected cooling and marginal attenuation of radiance at high flow incidence angles, across regions of moist upper air. These shallow-angle flows are particularly sensitive to high convective clouds, while at steeper flow angles (i.e., closer to the zenith), they exhibit only moderate path-integrated scattering. In all cases, clouds and rain cells are more prominent at high frequencies than in lower microwave frequency bands and have high signal-to-noise ratio characteristics. In embodiments, Jacobian model 322 explains these characteristics in the calculation of the Jacobian for precipitation regions, which is a function of occupied area observation angle, polarization, and channel subband center frequency.

[0071] The surface background radiation metric emission model used in UMRT is for a simple specular surface, where land is assumed to be at a fixed emission rate of 95%, and water (both seas and lakes) is assumed to be at a emission rate of 55%. For the natural run of storm systems in the Midwest, the background is largely land, which makes the weighting function, which peaks low at 118–183 GHz, relatively insensitive to lower tropospheric temperatures and water vapor. However, clouds and rain cells in the mid-to-upper troposphere provide strong radiometric cooling, due to both absorption by liquid hydrophysics and scattering by cloud ice. Precipitation is therefore readily observed in the microwave image, which forms the basis for the innovative variables (e.g., radiance difference 339) used in the Kalman filter cycle 308 in embodiments.

[0072] 3.2 Constellation Simulator Weather forecaster 200 is a GEMS satellite T a Simulated upwelling T, derived from NR geophysical state data to simulate observations. bData may be used. The simulated T a data is artificial and is calculated from the UMRT T b model output. In an embodiment, the spatial resolution, temporal resolution, and polarization of the simulated T a data are the same as those of the calibrated T a values expected from the planned GEMS constellation. The data assimilator 300 may calculate the T a values that a typical GEMS satellite would observe. In an embodiment, the innovation (e.g., radiance difference 339) used in the assimilation process is thus effectively calculated as the main beam T a difference.

[0073] Each GEMS satellite strip is about 2,000 km wide or ±1,000 km with respect to the ground track. Each cross-track raster has 83 overlapping footprint samples. With a 500 km orbital altitude, the sample footprint has a 3 dB radius of about 20 km at nadir, increasing to 40 km near the periphery of the Earth, which makes the cross-track footprint raster coverage a slightly hourglass-shaped point track. All sample footprints are mapped onto the WRF latitude and longitude grid within the natural run for all satellites and interpolated in time and linear polarization between boundary analysis times. This mapping implements numerical quadrature with respect to the simulated upwelling T b values and provides a set of weights used to obtain the simulated main beam T a strips. The coefficients of this linear algorithm are essential for forming the measurement Jacobian and calculating the full Jacobian (using the Jacobian model 322) and the Kalman gain matrix 336 in the data assimilator 300 to generate the updated state profile 379.

[0074] 3.3 Unsupervised Hydrometeor State Clustering In the embodiment, a key component of the Kalman gain matrix required for the Kalman filter cycle 308 is a flow-dependent BEC matrix (e.g., as one of the BEC matrices 359) that is relevant with respect to all state vector profile parameters. The relevant parameters generally include not only the assimilated variables of temperature and humidity, but also all NWP model hydrological parameters. The BEC with respect to these hydrological parameters is highly state (or flow)-dependent and therefore must be generated based on the local state vector of the atmosphere. Thus, the Kalman gain may be calculated when the model state is close to the actual state (i.e., when the data assimilarizer 300 is "locked"). The use of the Kalman gain matrix 336 leads to the most accurate use of the innovation, which in turn serves to keep the model state locked.

[0075] Embodiments of the data assimilarizer 300 enable the rapid generation of a flow-dependent BEC matrix (e.g., BEC matrix 359) for temperature, humidity, and all hydrological variables (a total of eight parameters using a selected WRF hydrological microphysical scheme). In embodiments, the BEC matrix 359 may combine the vector errors of these eight parameters and estimate Gaussian error statistics. While it is well known that atmospheric variables are non-Gaussian over large movement ranges, Gaussian statistics are nevertheless reasonable to assume for small deviations of these variables from true. In practice, such small deviations are achievable conditions when the model is locked.

[0076] However, the BEC matrix 359 can also combine errors over a horizontal range of sufficient distance to statistically estimate these parameter deviations (ideally in its most complete form), using only a small number of overlapping occupied areas of observed satellite data at all vertical levels. That is, the horizontal range described by the BEC matrix 359 can describe the horizontal relative distance of hydrological parameters, covering satellite occupied areas and varying according to the type of cloud or precipitation event. In embodiments, stratiform precipitation may require a BEC matrix (of the BEC matrix 359) that combines errors horizontally over a distance of perhaps up to about 100 km, or over some microwave occupied area, while convective precipitation may only combine errors up to a few kilometers, or at best within a microwave occupied area. In embodiments not applicable to such horizontal range limitations, the sizes of the BEC matrix 359 and the Kalman gain matrix 336 would increase to an unacceptably and unnecessarily large size.

[0077] In the case of a Midwest storm, the size of the background error covariance matrix across the entire domain is approximately 5 × 10⁻⁶ when both thermodynamic and hydrological state variables in a 3D WRF grid are all considered simultaneously across a mesoscale weather system. 7 ×5×10 7 This is the scale. However, restricting the matrix to combine errors across only a 5x5 grid of the profile (for example) results in approximately 1.5 x 10⁻¹⁶. 4 ×1.5×10 4 This yields an invertible matrix of size [size].

[0078] In the embodiment, a parameterized vertical-only flow-dependent approach is initially employed to constrain the BEC matrix 359 to a manageable size. With respect to the ensemble of cloud vertical profiles available in the WRF output, each vertical profile may be defined as a single column of M elements relating to both thermodynamic and cloud-hydrological state variables at all vertical levels (e.g., M = 518 for a 74-level WRF profile). Based on a simple but well-established altitude-density model [5] for cloud and rain cells, which is a finite-parameter precipitation cell model, each vertical hydrological profile can instead be reasonably well represented by a reduced-dimension vector H with only 15 parameters.

number

number

number

[0079] Using the parameterization described above, an ensemble of vertical hydrological profiles may subsequently be classified into multiple hydrological modes (e.g., by classifier 340) using a clustering analysis method such as the K-means algorithm. Using the K-means method, the vertical profiles described using high-density representations are optimally partitioned into K clusters (e.g., clustered state profiles 349) such that the metric distance between the set of vertical profiles and the assigned cluster mean is minimized.

[0080] Each cluster should therefore contain several precipitation profiles corresponding to rain cell "modes" and be larger than the size of the BEC matrix (e.g., BEC matrix 359) to exclude rank missings in this matrix. Within each cluster, the covariance of the reduced-scale state variable vectors may be estimated using a standard unbiased covariance matrix estimator (e.g., BEC calculator 350). Assuming that rain cells are classified as being in their particular mode, and the mean of the reduced-scale set of profiles for those within a given cluster is treated as the true state with respect to the corresponding rain cell mode, the covariance matrix estimation provides a simple scaled measurement of the BEC matrix with respect to the one-dimensional vertical cloud state vectors.

[0081] For rain cell vectors located at arbitrary locations within a reduced-scale space, multidimensional (15-dimensional) interpolation between cluster covariance matrices is performed to determine the scaled BEC matrix relevant to their specific profile. This clustering analysis method was applied to a WRF-based Midwestern storm natural run using atmospheric state vectors at 15-minute time intervals between 12:00:00 on May 26, 2019 and 12:00:00 on May 28, 2019 (i.e., 48 hours). The state vectors for each time frame provided a 381×498 cloud profile with 74 vertical levels. A total of 193 frames were generated from the state vectors for clustering analysis, but only a subset of the frames could be run within a single ensemble.

[0082] The clustering analysis proceeds in two stages: (1) cloud profiles across individual time frames are partitioned into K clusters using the K-means algorithm (K=10 or 20); and then (2) all cloud profiles within the frame from 15:00:00 on May 27, 2019 to 12:00:00 on May 28, 2019 are partitioned into 20 clusters. The first part of the analysis identifies precipitation cell modes at fixed frame times. Adjusting the number of clusters helps, for example, determine the optimal value of K for use in the K-means algorithm used by classifier 340. Once the optimal number of clusters for the Midwest storm event is estimated, the second part of the analysis is needed to identify the dominant precipitation mode for the weather event as it develops over time. In this embodiment, within the Kalman filter cycle 308, a BEC matrix for each of the K clusters (K clustered state profiles 349) is subsequently estimated by the BEC calculator 350 based on the clustering analysis results for the entire meteorological event.

[0083] Representative results from clustering algorithms for K=10 or 20 in four distinctly different 12-hour timeframes are shown in Figures 5A-5D and 6A-6D. Note that the clusters are not ordered by rank in any specific color scheme and are therefore not visually inspected to track convective intensity. However, the images clearly reveal the potential for consistently separating similar hydrological profile types within similar rain cell types throughout the duration of the event. The locations of stratiform clouds and convective precipitation tend to appear in adjacent areas that develop together over time.

[0084] Initial performance results suggest an excellent means of grouping cloud and rain cell vertical modes generated by arbitrary NWP models into clusters that behave similarly in terms of water distribution and phase. Therefore, the evolution over time, and thus the stability and predictive power of such modes, may be incorporated into the Kalman gain matrix 336 and the iterative state profile updater 366 using this method. Since stability and predictive power are related to error covariance, it is expected that each of these clusters will have a background error covariance matrix that is somewhat unique, which can then be found by clustering followed by interpolation across a pre-calculated scaled library of covariances for the profiles within the specific cluster.

[0085] It should be noted that the application of a classification method for expanding clusters (e.g., cluster database 341) for a representative geophysical state vector ensemble may be an offline computation, requiring only one (or possibly occasional) step to build a covariance library associated with each of the clustered state profiles 349. In embodiments, a classifier 340 performs a classification method, which may be k-means clustering or include k-means clustering. Furthermore, these libraries are modest in size compared to the many data elements used in NWP modeling and forecasting. For example, interpolation applied to this library by a BEC calculator 350 to determine the optimal BEC (e.g., BEC matrix 359) may be a rapid computation that can be easily performed in real time.

[0086] 3.4 Background Error Covariance Modeling The rapid development of clouds and rain cells associated with frontal convection and hurricane rainbands is linked to microwave T2020 (T2020) fluctuations in temperature and water vapor. bThis corresponds to the superiority of the signature. Therefore, a dynamically fluctuating BEC matrix may be necessary to stabilize assimilation renewal in each assimilation cycle. The BEC matrix among the BEC matrix 359 may include statistical profile correlation information relevant with respect to various hydrological parameters and phases. In embodiments, this statistical information is relevant to specific atmospheric conditions and therefore fluctuates dynamically over the lifetime of a convective event and its geographical location. Importantly, error covariances with respect to temperature and humidity fluctuations in the hydrologically strongly scattering and absorbing environment can be artificially suppressed in the BEC matrix with respect to these fluctuations in order to remain stable in such scenarios during renewal.

[0087] This suppression can be achieved by either attenuating or zeroing the error covariance with respect to the thermodynamic (i.e., temperature and humidity) variables. If all state variables are purely Gaussian and the correct BEC statistics for both thermodynamic and hydrological variables are accurately known, then such artificial suppression would be unnecessary when applying XKF. However, since none of the state variable processes (except temperature) are Gaussian and the BEC statistics are not fully known, suppression of changes in thermodynamic state variables is necessary to stabilize the assimilation step.

[0088] In general, such artificial suppression is both applicable to and necessary for any state variable that has a signature masked by a strong signature of a convective event. Examples of parameters masked by convection include temperature and humidity fluctuations, sea surface wind vectors, surface temperature, snow cover, soil moisture, and surface ice, as scattering by ice in the upper part of the cell is strong. This is even more relevant because strong signals can change on a 15-minute basis as convection develops.

[0089] Two dynamically changing BEC matrix models, namely (1) a cluster-based BEC model and (2) a Brown BEC model, may be used within the data assimilar 300. The BEC calculator 350 may run either of these models or a combination thereof. Both of these models have the potential to enable rapid library-based computation of the BEC matrix, which is a continuous function of the hydrological state. The computation of the BEC matrix for each model may then be artificially suppressed to achieve temperature and humidity stability. However, the two methods are fundamentally different in the basis used to compute the BEC matrix. Section 3.6 describes the Brown BEC model method. The development of the cluster-based BEC model, which verifies the usefulness of the K-means clustering technique, is described below in Section 3.5.

[0090] 3.5 Cluster-Based BEC Model Validation Clustering of hydrological states using the K-means algorithm was computed in a reduced-dimensional hydrological state space. The WRF atmospheric profile natural run dataset used contained 499 × 382 grid points with 74 atmospheric levels at a horizontal resolution of 5 km. The analysis interval for natural run (i.e., archive or output frequency) was 15 minutes, stretching over 48 hours for 193 snapshot frames. To facilitate clustering, each profile containing a total of 900 vertical variables (T, q, and density and mean size for five hydrological phases at 74 levels) is reduced to a 15-dimensional hydrological space H, as described in equation (3.1).

[0091] Clustering was performed across 20 hypothetical cluster centers in this 15-dimensional space for all ensembles of 193 frames. The cluster centers were used to calculate individual covariance matrices in the 900-dimensional space of the original WRF, and the clustering process and cluster centers were evaluated for their physical consistency with known mesoscale convective behavior. We found that profiles in the same cluster are associated with similar regions of convective behavior in this diverse mesoscale hypothesis test. Clearly different profiles are classified into different clusters. The cluster constituent populations vary from a few to tens of thousands of profiles out of a total of 499 × 382 ~ 1.9E5 profiles, but the distinctions made using K-mean clustering clearly suggest meteorologically relevant classifications.

[0092] The key to enabling the development of a state-dependent and dynamically fluctuating background error covariance matrix is ​​precisely this K-means classification ability. This matrix can be computed by optimal interpolation across a fixed library of error covariance matrices defined in a reduced hydrological profile space of 15D. Under the proposed scheme, each 15×5 error covariance matrix in the library is assumed to be proportional to the hydrological covariance matrix for its cluster. In embodiments, unbiased weighted interpolation in 15D space using a unified set of Euclidean distance metrics and weights (similar to those routinely performed in typical 2D Kriging) provides the most relevant error covariance matrix for use in computed Kalman gains.

[0093] 3.6 Brown Background Error Covariance Modeling In addition to cluster-based BEC models, steps have been taken to develop the proposed Brown-based background error covariance model. The Brown BEC model background error covariance model is based on the “NMC method,” where the NWP model state variable increments are small enough to be assumed to be Brown random processes in which both hydrological and thermodynamic state variables have their error covariances increasing linearly over time. Under the Brown assumption, the covariance matrix is ​​developed from increments in the forecast state variables, which are therefore Gaussian together over sufficiently short time periods. This study focuses on determining appropriate time differences, in particular, appropriate scaling factors between time difference increments and the resulting background error covariance matrix, which should be used to justify the Brown assumptions for both hydrological (e.g., precipitation, cloud liquid water, cloud ice, snow, and graupel density) and thermodynamic variables (e.g., temperature, water vapor) within the framework of meteorological research and forecasting (WRF) models.

[0094] Assimilation of satellite data observed across clouds and precipitation requires second-moment costatistics of hydrological parameters, particularly the error covariance of the number density and other distribution parameters of the five hydrological microphysical phases (rain, cloud liquid water, cloud ice, snow, and graupel) [6]. The time evolution of partial water density and distribution parameters of hydrological phenomena descending, advection, convection, and developing within clouds is statistically analogous to the well-known Brownian motion process [7]. Brownian motion is the continuous random movement of small particles suspended in a dissipative heat medium under the thermodynamic influence of all surrounding molecules [8].

[0095] Key features of Brownian behavior include a linearly increasing variance of state variables over time and statistically independent increments of state variables over non-overlapping time intervals. The development of NWP model errors for cloud and water phenomena and their associated thermodynamic variables (temperature, relative humidity, wind) can be analogously modeled as a stochastic Brownian process over limited timescales, attributable to inherently noisy and virtually unobservable meteorological phenomena that jointly influence these variables.[9]

[0096] The background error covariance matrix is ​​defined as follows:

number

number

[10]

[11] . Parrish and Derber

[12] illustrate this method using the following ensemble averaging process to estimate the BEC matrix.

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[0097] In the above, x 48 and x 24 These are valid 48-hour and 24-hour forecasts, respectively, for the same forecast time, but starting with an analysis at 24-hour intervals. Function M β←α (·) is the forward time operator of the NWP model from analysis time β to α, and x a (t) is the state variable vector at analysis time t.

[0098] arbitrary analysis time t i Considering this, the updated (or analyzed) state vector x a (ti ) is time t i This is the most available knowledge of the true state at a later time t. j The associated forecasts and forecast errors are therefore as follows:

number

[0099] Assuming that both errors η are Brownian processes, their covariance matrix increases linearly over time.

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[0100] The key aspects of the above method are the assumption of Brownian error growth and uncorrelated forecast errors. Note that the identification of the time offset variable involves nonlinear function minimization rather than pure linear regression.

[0101] 3.8 Parallelization of Radiation Transfer Models In the embodiment, the radiative transfer model 320 has one or more of the following properties: 1. Ability to accurately model the scattering and absorption characteristics of the five major hydrological phases—rain, cloud liquid water, glaupel, cloud ice, and snow—at frequencies up to approximately 500 GHz. 2. Ability to accurately calculate the upwelling upper-plane parallel radiance for up to at least eight Gaussian direct AC angles under arbitrary single-scatter albedo conditions. 3. Rapid geophysical and radiative Jacobian calculations for all hydrological parameters, in addition to temperature and humidity. 4. Ability to perform forward transfer and Jacobian calculations on a moderately parallel processing machine (i.e., 16–256 cores) with an average time of approximately 0.1 milliseconds per profile for 4.74 level profiles and up to approximately 40 channel frequencies.

[0102] 3.9 Estimation of Hydrological Profile Parameters A data assimilar 300, for example, a classifier 340, may utilize clustering to reduce the number of hydrological variables to be estimated. The WRF model is configured to use 74 atmospheric levels with 12 unknown state variables (density, mean size, and temperature and humidity) at each level, resulting in 900 potential unknowns per profile. The number of microwave spectral channels using GEMS-2 may be 24, which, without proper adjustment, leads to the potential for unstabilized XKF inversion. To stabilize this inversion, the reduction of each profile to a three-dimensional hydrological vector, including total (integrated) column mass, peak height, and height standard deviation, has been studied.

[0103] In the embodiment, the data assimilar 300 constrains the number of parameters to be estimated by separately estimating the cloud vertical structure, which includes small cloud liquid water (CLW) and ice particles, and the precipitation vertical structure, which includes larger rain, snow, and graupel particles. Freezing hydrophysics is always homogeneous in terms of nucleation altitude (h H h H It exists above ) and the liquid water phenomenon has a melting level (h) at 0°C.M It is located below ). The level of homogeneous nucleation h H It is typically at a temperature of -40°C. M and h H Between these two points, the mass fraction of liquid versus frozen hydrology varies linearly

[13] . This height density model distributes both the phase and mass of precipitation according to a physically meaningful parameterized model. It is assumed that a mixed layer with both liquid and frozen hydrology lies within the height range between freezing and homogeneous nucleation.

[0104] The ab initio approximation to the altitude density model assumes that cloud liquid and precipitation hydrophenosities (i.e., small hydrophenosities) are distributed within the given approximate vertical distribution, as well as separate profiles of rain, graupel, and snow (i.e., large hydrophenosities) similarly distributed. Thus, this approximate altitude density (AAD) model has a total of six hydrophenostic parameters for each profile, namely the total column mass, peak height, and height standard deviation for cloud and precipitation hydrophenosities respectively, thus reducing the number of degrees of freedom in each profile by (74 × 10) / 6 ≈ 123 times. This reduction in dimensionality constrains inversion and, in this embodiment, results in the requirement of a 6 × 6 BEC matrix.

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[0105] The update study of hydrological state vectors will focus on three forms of basis functions: B: Gaussian, cubic B-spline, and quadratic. All three represent continuous functions with continuous derivatives, and therefore allow for the association of AAD model derivatives with Jacobians, connecting radiance observations and hydrological densities using the chain law with respect to the phases of small and large hydrological phenomena. WRF profiles and radar observations suggest that clouds and precipitation truncate the nominal vertical distribution and therefore favor either the quadratic or cubic B-spline basis function. Any extraction or data assimilation is D tot , z peak And, as part of an optimization scheme for estimating Δz, the Jacobian is requested. Therefore, the chain law may be used to compute the requested Jacobian.

number

[0106] Regarding precipitation, the AAD model slightly overestimated rain below freezing levels and underestimated snow and graupel above freezing levels. The estimated precipitation is z peakThe profile peaks near the surface at 0.4 km. The estimated profile also misses the lower snow and graupel densities above 4 km due to the vertical constraints imposed by the basis set.

[0107] 3.10 Jacobian calculation and assessment Radiative transfer model 320 may use the DOTLRT forward RT model to deploy initial whole-weather microwave assimilation demonstrations. The DOTLRT model may incorporate fast Jacobian calculations, which are currently being assessed and improved for both speed and interoperability with WRF for operational use. Assessment includes a step of determining the accuracy of Jacobian calculations with respect to temperature, water vapor, and precipitation parameter derivatives.

[0108] Figure 7 shows the vertical temperature and water vapor Jacobians for a typical clear-sky WRF profile from natural convection run in the Midwest. Jacobian functions were obtained using DOTLRT for selected microwave temperature and water vapor sounding channels and for a typical clear-sky WRF profile ((left) temperature Jacobian, and (right) water vapor Jacobian). A representative set of temperature and water vapor sounding channels was selected for these plots, which is a flow propagation angle of 37.6° and a non-reflective blackbody surface. In both cases, the expected performance is shown, and the incremental response function is positive with respect to temperature and negative with respect to water vapor.

[0109] A more demanding test is the calculation of the Jacobian for scattering and absorbing hydrophysics, which is also coded into the DOTLRT algorithm. This capability was studied using precipitation hydrophysics profiles with vertical density for all hydrophysics phases, as shown in Figure 8. Figure 8 graphically illustrates the vertical hydrophysics distribution for precipitation profiles used for the hydrophysics Jacobian test. Figure 8 includes curves 801–805 for cloud water (curve 801), precipitation (curve 802), cloud ice (curve 803), graupel (curve 804), and snow (curve 805). Although several coding divisions that result in pseudobehavior are still corrected, the general behavior of the Jacobian (for precipitation and cloud ice, Figures 9 and 10, respectively) shows the expected incremental response indications and spectral trends. In Figures 9 and 10, the zenith angle of propagation is 37.6°.

[0110] 3.11 Radiation Transfer Model Acceleration The radiative transfer model 320 may include and / or utilize the DOTLRT model. DOTLRT incorporates relevant physical properties to estimate hydrological distributions but performs full Mie series calculations, which are currently impractical for operational data assimilation. This strategy focuses on (i) integrating Mie library lookup tables [1] and (ii) reducing the number of atmospheric layers at altitudes where no hydrological features are undoubtedly present.

[0111] With respect to the first of these strategies, when implemented by the radiative transfer model 320, DOTLRT may perform a Mie scattering routine for all layers and all hydrological types, typically accounting for more than half of the total forward RT calculation execution time. However, this calculation is deterministic using only three inputs: frequency, hydrological density, and temperature. Since the calculation is only required at a discrete set of frequencies associated with the midband frequencies of each microwave channel, a set of lookup tables (one for each channel) and bilinear interpolation can be performed for less than 1% of the time required by the full Mie routine, to obtain the absorption and scattering coefficients κ (respectively) as functions of hydrological density and temperature. a and κ s This is implemented to calculate the asymmetric parameter g of the phase matrix.

[0112] Regarding the second strategy, the WRF vertical grid extends well into the stratosphere, but convection is almost nonexistent above approximately 20 km altitude. Processing the non-scattered stratospheric layer over most of the convection increases run time but has little effect on the Kalman gain for the hydrological state vector component. Preferred conditions based on atmospheric hydrological state vectors are being sought to determine when scattering above dynamically variable cloud top altitudes can be safely ignored in Jacobian (and therefore Kalman gain) calculations. For channels and state vectors when this condition is met, the lower atmosphere DOTLRT model may be used in conjunction with a faster stratospheric model component to reduce the overall profile run time by an expected 25% to 75% on average.

[0113] In this embodiment, the Kalman filter equation for a single cycle of the Kalman filter cycle 308 is obtained by equations (3.14)-(3.16)

[14] . Analysis increment:

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[0114] In equations 14-16, x a This is the analyzed (updated) state vector, and x f is the background forecast state vector, and y is the multispectral GEMS satellite T, which is simulated from naturally occurring data. a The observed vector is , and h is the predicted model vector using forward radiative transfer (including instrument antenna pattern convolution, UMRTv4).

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[0115] These equations provide a framework for integrating flow-dependent BEC, full Jacobian, and simulated GEMS observations into a planned GEMS satellite constellation update cycle (e.g., 15 minutes). Matrix, matrix storage, inversion, and multiplication operations can all be supported by a medium-sized multiprocessor, e.g., a 286 processor. These equations can be run in real time over a medium-sized domain (typically 1,500 × 1,500 km) using a WRF model or its operating equivalent and used for short-term convection forecasts at 15-minute update intervals.

[0116] In the embodiment, the forecasted state profile 319 includes a parameterized hydrological state vector. This state vector may be augmented to include adjacent parameterized profiles in a simple extension of the method that enables clustering of spatially correlated hydrological profiles. Calculations using these extensions are the focus of future internal development efforts to commercialize this weather forecasting method and are based on iterative sweeps across domains of interest within each XKF update, but require increasing the dimensionality of the BEC by (typically) 3 × 3 = 9 times to acknowledge meaningful results. To study longer horizontal correlation lengths, the number of vertical levels and / or vertical resolutions may need to be reduced, possibly using either principal component analysis (PCA) rank reduction, upper-level truncation, and / or performing updates in a reduced-dimensional high-density model space.

[0117] The scaling of covariance within an arbitrarily selected set of K clusters for estimating the BEC matrix (for example, the BEC matrix 359) is worth discussing. To achieve this goal, we consider the determinants of the covariance and subcovariance matrices for various K rain cell modes. Since the determinant scaling is approximately 1 / K, obviously, the larger the number of clusters K, the smaller each of these determinants will be. To demonstrate this, we consider the entire reduced-dimensional rain cell signal energy, which is distributed as a constant, but when clustered, is distributed across the K sets. Therefore, assuming that this energy is distributed approximately uniformly across the various clusters, in this embodiment, the interpolated covariance matrix (BEC matrix 359) itself is initially scaled by 1 / K, which is an arbitrary number that will increase as the diversity of rain cell vectors in any given training set increases.

[0118] The main problem of scaling can be circumvented by recognizing that the interpolated covariance matrix (assuming K is chosen to be large enough to provide a nearly continuous discretization of the rain cell modes) can be arbitrarily scaled, and thus (1) represents the correlation between rain cell parameter variations that is deterministic to retain within any Kalman filter update (via the Kalman filter cycle 308), and (2) needs to be scaled by arbitrary adjustment coefficients when used as the BEC matrix 359 within the Kalman filter cycle 308. However, since the update can be carried out using an extended Kalman filter, preferably small, scaled updates are required for both the convergence and stability of the XKF cycle. Thus, method 400 may include the use of a single adjustment coefficient or a small number of such coefficients (e.g., one each for temperature, water vapor, and rain cell parameter covariances) in addition to the interpolated BEC matrix.

[0119] The scaling relationship between error covariance and state covariance may be important to ensure the stability of thermodynamic state variable updates within cloud and rain cells. Thus, this may provide an effective means of implementing whole-weather radiance assimilation, where the sensitivity of upwelling radiance to hydrophysics is dominant over that of temperature and humidity; however, these less sensitive variables still require to be treated as they would be under clear-sky conditions, despite involving much smaller Kalman gains, in order to ensure their stability.

[0120] 3.12 Prototype results from cluster-based background error covariance method During the prototyping process, clusters #3 and #6 from natural storm runs in the Midwest were selected as sources for several chosen visualizations. Note that the cluster numbering is entirely arbitrary.

[0121] Figure 11 shows the physical location of cluster #3 within the first frame of the Midwest storm. The ring-like distribution indicates moderate to severe convective activity. These rings surround the center of the storm. Other clusters inside the rings exhibit even more extreme weather, while clusters outside the rings exhibit milder weather.

[0122] Figure 12 is the correlation matrix for cluster #3, which can be calculated from the BEC matrix 359 for cluster #3. There are 11 important parameters, tracked in each of the 74 vertical atmospheric layers for a total of 814 variables. The overlaid red grid defines the boundaries of subblocks of correlations for specific parameters across the vertical atmospheric profiles. The matrix is ​​laid out so that moving to the right or downward within a subblock corresponds to referring to parameter correlations at higher altitudes. The dark blue stripes through the center of the matrix reflect the fact that hydrological phenomena do not exist above certain altitudes, and therefore the correlation data for these points has no physical significance.

[0123] This matrix condenses many atmospheric features into a single data block, and by examining, for example, the top-left subblock, it represents temperature-to-temperature correlations across different altitudes. The tropopause appears as the boundary between positive and negative correlations.

[0124] As a check to determine whether the clusters represent meaningful meteorological events, the high-energy modes of the covariance matrix were examined. First, diagonal subblocks of the larger covariance matrix (i.e., blocks containing information about the correlation of single-parameter pairs themselves across different altitudes) were extracted, and eigenvector decomposition was performed for each subblock.

[0125] Figure 12 does not represent a visualization of all data features captured by the complete covariance matrix, but it still provides a diagram of the dominant behavior of this particular cluster. Such plots were generated and examined for each of the 21 clusters, all of which exhibit meaningful features regarding their respective physical locations within the storm.

[0126] When data for a new atmospheric profile is input into the weather forecaster 200, one of the first steps is to calculate a covariance matrix for that particular profile. To achieve this goal, the modified Euclidean mean between the profile and each center of mass of the clusters in the covariance library is calculated. The covariance matrix for the incoming profile (e.g., the BEC matrix 359) may be calculated, for example, as the weighted mean of all the covariance matrices in the library of clustered state profiles 349. The weights used in the weighted mean may be equal to zero if there is no relevance.

[0127] References [1] Tian, M, and AJ Gasiewski (2013), A Unified Microwave Radiative Transfer Model for General Planar Stratified Media, IEEE Transactions on Geoscience and Remote Sensing, Vol. 51, No 7, 4103-4118. [2] Voronovich, AG, AJ Gasiewski, and BL Weber (2004), A Fast Multistream Scattering-Based Jacobian for Microwave Radiance Assimilation, IEEE Transactions on Geoscience and Remote Sensing, Vol. 42, No. 8, 1749. [3] Liebe, Hans J. (1989), MPM - An atmospheric millimeter-wave propagation model, International Journal of Infrared and Millimeter Waves, Vol 10, 631-650. [4] Sandeep, S, and AJ Gasiewski (2012), Fast Jacobian Mie Library for Terrestrial Hydrometeors, IEEE Transactions on Geoscience and Remote Sensing, Vol. 50, No. 3, 747. [5] Janssen, Michael A., AJ Gasiewski, et al. (1993), Atmospheric Remote Sensing By Microwave Radiometry, Wiley Series In Remote Sensing, Chapter 3. [6] Reisner, J., R. M. Rasmussen, and R. T. Bruintjes. (1998). “Explicit Forecasting of Supercooled Liquid Water in Winter Storms Using The MM5 Mesoscale Model.” Quarterly Journal of The Royal Meteorological Society 124 548: 1071-1107. https: / / doi.org / 10.1002 / qj.49712454804. [7] A brief account of microscopical observations made in The months of June, July and August 1827, on The particles contained in The pollen of plants, and on The general existence of active molecules in organic and inorganic bodies. https: / / www.math.utah.edu / davar / REU-2002 / notes / lec8.pdf [8] Georg A. Grell, Dezso Devenyi. A generalized approach to parameterizing convection combining ensemble and data assimilation techniques. Geophysical research letters, Vol. 29, NO. 14, 1693, 10.1029 / 2002GL015311, (2002). [9] Shutts G. (2005). A kinetic energy backscatter algorithm for use in ensemble prediction systems. Q. J. R. Meteorol. Soc. 131: 3079-3102.

[10] Buehner M, Gauthier P, Liu Z. (2005). Evaluation of new estimates of background and observation error covariances for variational assimilation. Q. J. R. Meteorol. Soc. 131: 3373-3384.

[11] Ingleby NB. (2001). The statistical structure of forecast errors and its representation in The Met Office global three-dimensional variational data assimilation system. Q. J. R. Meteorol. Soc. 127: 209-231.

[12] Parrish and Derber. The National Meteorological Center’s Spectral Statistical-interpolation analysis system. Monthly weather review,1992: Vol120:1747-1763

[13] Gasiewski, A.J. “Numerical Sensitivity Analysis of Passive EHF and SMMW Channels to Tropospheric Water Vapor, Clouds, and Precipitation,” IEEE Trans. Geosci. Remote Sensing, vol. 30, no. 5, pp. 859 870, (September 1992).

[14] Kalnay, E “4-D-Var or ensemble Kalmar filter?” DOI: 10.1111 / j.1600-0870.2007.00261.x

[0128]

Table 2

[0129] With regard to the terms "and / or" and "at least one of" in this specification, for example, in the cases of "A and / or B" and "at least one of A and B", such phrasing encompasses the choice of (1) "A only", (ii) "B only", or (iii) "both A and B". In the cases of "A, B, and / or C" and "at least one of A, B, and C," such phrasing encompasses the choices of (1) "A only," (ii) "B only," (iii) "C only," (iv) "A and B only," (v) "A and C only," (vi) "B and C only," or (vii) "each of A, B, and C." This can be extended to the number of items listed.

[0130] Modifications may be made in the above methods and systems without departing from the scope of these embodiments. Therefore, it should be noted that the subject matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not restrictive. In this specification, unless otherwise indicated, the phrase "in embodiments" is equivalent to the phrase "in certain embodiments" and does not refer to all embodiments. The following claims are intended to encompass all general and specific features described herein, as well as all descriptions of the scope of the methods and systems, which, as a matter of wording, may be said to be intermediate between them.

Claims

1. A weather forecasting method, wherein the method is The radiance difference is generated as the difference between the radiance measured from the satellite and the forecast satellite radiance generated by the radiative transfer model and the predicted state profile output by the numerical weather prediction (NWP) model. When the radiance difference exceeds the noise threshold, Using the Jacobian model and the predicted state profile, a radiance sensitivity is generated, The Kalman gain matrix is ​​constructed from the background error covariance (BEC) matrix and the radiance sensitivity, From the Kalman gain matrix and the radiance difference, a filtered state profile change is generated, By adding the filtered state profile changes to the predicted state profile, the state profile is updated, thereby generating an updated state profile. This generates the updated state profile. Methods that include...

2. The method according to claim 1, further comprising: (i) repeating the step of generating an updated radiance difference by generating a radiance difference, wherein the updated state profile replaces the predicted state profile; and (ii) repeating the step of generating an updated state profile when the updated radiance difference exceeds the noise threshold.

3. The method is performed when the radiance difference is less than a noise threshold based on noise from the satellite, the radiative transfer model, and the NWP model, The input is to generate a subsequent predicted state profile using the NWP model and the predicted state profile, The process involves repeating the step of generating a radiance difference, wherein the subsequent predicted state profile replaces the previous predicted state profile. The method according to claim 1, further comprising performing the following:

4. The method according to claim 1, wherein the forecasted state profile includes a vertical state profile of a plurality of state profile variables for each of a plurality of horizontal grid points within a geographically defined domain, the plurality of state profile variables are selected from the group including air temperature, humidity, altitude, hydrological density, hydrological size, vapor density, cloud density, rain density, ice density, snow density, graupel density, mean rain particle size, and mean ice particle size.

5. The method according to claim 4, wherein updating the state profile includes updating each state profile variable of the vertical state profile.

6. The method according to claim 4, wherein the step of generating an updated radiance difference is repeated, and if the updated radiance difference exceeds the noise threshold, the step of generating the updated state profile is repeated.

7. The method according to claim 6, wherein each of the plurality of state profile variables has a correlation time for each of the plurality of correlation times, and the duration between generating the updated state profile and repeating the step of generating the updated state profile is less than the shortest correlation time among the plurality of correlation times.

8. The method according to claim 4, wherein when the Kalman gain matrix is ​​constructed, each BEC matrix of the BEC matrix includes the covariance between each of the state profile variables of the plurality of state profile variables.

9. The method according to claim 8, wherein the plurality of state profile variables include air temperature, humidity, and at least one of altitude, hydrological density, hydrological size, vapor density, cloud density, rain density, ice density, snow density, graupel density, mean rain particle size, and mean ice particle size.

10. The method according to claim 1, further comprising generating the forecast satellite radiance using the radiance transfer model before generating the radiance difference.

11. The method according to claim 1, further comprising generating the background error covariance (BEC) matrix by interpolating the updated state profiles across a library of clustered state profiles obtained by a classification method.

12. The method according to claim 1, wherein generating the filtered state profile change further comprises (i) generating an unscaled filtered state profile change, and (ii) generating the filtered state profile change by scaling the unscaled filtered state profile change.

13. The method according to claim 1, wherein updating the state profile further includes removing from the updated state profile any updated state profiles that are not physically achievable.

14. The method according to claim 1, wherein the filtered state profile change is the product of the Kalman gain matrix and the radiance difference.

15. A weather forecaster, wherein the weather forecaster is Processor and memory and Equipped with, The aforementioned memory is A machine-readable instruction, when executed by the processor, The radiance difference is generated as the difference between the radiance measured from the satellite and the forecast satellite radiance generated by the radiative transfer model and the predicted state profile output by the numerical weather prediction (NWP) model. When the radiance difference exceeds the noise threshold, Using the Jacobian model and the predicted state profile, a radiance sensitivity is generated, The Kalman gain matrix is ​​constructed from the background error covariance (BEC) matrix and the radiance sensitivity, To generate a state profile change filtered from the Kalman gain matrix and the radiance difference, By adding the filtered state profile changes to the predicted state profile, the state profile is updated, resulting in an updated state profile. This generates the updated state profile. Machine-readable instructions to control the processor to perform the following actions A weather forecaster that memorizes data.

16. The aforementioned memory is A machine-readable instruction, when executed by the processor, The process involves repeating the step of generating an updated radiance difference by generating a radiance difference, wherein the updated state profile replaces the predicted state profile. If the updated radiance difference exceeds the noise threshold, the step of generating an updated state profile is repeated. Machine-readable instructions to control the processor to perform the following actions The weather forecaster according to claim 15, further storing the following.

17. The aforementioned memory is A machine-readable instruction, when executed by the processor, such that the radiance difference is less than a noise threshold based on noise from the satellite, the radiative transfer model, and the NWP model, The input is to generate a subsequent predicted state profile using the NWP model and the predicted state profile, The process involves repeating the step of generating a radiance difference, wherein the subsequent predicted state profile replaces the previous predicted state profile. Machine-readable instructions to control the processor to perform the following actions The weather forecaster according to claim 15, further storing the following.

18. The weather forecaster according to claim 15, wherein the forecasted state profile includes a vertical state profile of a plurality of state profile variables for each of a plurality of horizontal grid points within a geographically defined domain, the plurality of state profile variables being selected from the group including air temperature, humidity, altitude, hydrological density, hydrological size, vapor density, cloud density, rain density, ice density, snow density, graupel density, mean rain particle size, and mean ice particle size.

19. The aforementioned memory is A machine-readable instruction, when executed by the processor, The background error covariance (BEC) matrix is ​​generated by interpolating the updated state profile across a library of clustered state profiles obtained by the classification method. Machine-readable instructions to control the processor to perform the following actions The weather forecaster according to claim 15, further storing the following.

20. The weather forecaster according to claim 15, wherein the filtered state profile change is the product of the Kalman gain matrix and the radiance difference.

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