20.9: Homework Exercises
- Page ID
- 10968
20.9.1. Broaden Knowledge & Comprehension
B1. Search the web for info about each of the following operational weather forecast centers. Describe the full title, location, computers that they use, and models that they run. Also answer any special questions indicated below for these forecast centers.
- CMC (and list the branches of CMC).
- NCEP (and list the centers that make up NCEP)
- ECMWF
- FNMOC
- UK Met Office
B2. Search the web for the government forecast centers in Germany, Japan, China, Australia, or any other country specified by your instructor. Describe the NWP models they run.
B3. Based on web searches for each of the numerical models listed below:
- Define the full title.
- At which centers or universities are they run?
- Find the max forecast duration for each run.
- Find the domain (i.e., world, N. Hem. N. America, Canada, Oklahoma, etc.).
- What is the finest horizontal grid resolution?
- How many model layers are in the vertical?
- Which type of grid arrangement (A, B, etc.) is used?
- What order spatial and time differencing schemes are used?
GEM (Canada)
NAM
GFS
AVN
NAVGEM
ECMWF IFS
MC2
UW-NMS
MM5
RAMS
ARPS
WRF-NMM
WRF-ARW
FV3
B4. Search the web for models in addition to those listed in the previous exercise, which are being run operationally. Describe the basic characteristics of these models.
B5. Search the web for a discussion of MOS. What is it, and why is it useful to forecasting?
B6. Find on the web different forecast models that produce precipitation forecasts for Vancouver, Canada (or other city specified by your instructor). Do this for as many models as possible that are valid at the same time and place. Specify the date/time for your discussion. Try to pick an interesting day when precipitation is starting or ending, or a storm is passing. Compare the forecasts from the different models, and if possible search the web for observation data of precipitation against which to validate the forecasts.
B7. At which web sites can you find forecast sea states (e.g., wave height, etc.)?
B8. Based on results of a web search, discuss different ways that ensemble forecasts can be presented via images and graphs.
B9. What types of publicly available daily forecasts are being made by a university (not a government operational center) closest to your location?
B10. What are the broad categories of observation data that are used to create the analyses (the starting point for all forecasts). Hint, see the ECMWF data coverage web site, or similar sites from NCEP or the Japanese forecast agency
B11. Search the web for verification scores for the national weather forecast center that forecasts for your location. How have the scores changed by season, by year? How do the anomaly correlation scores vary with forecast day, compared to the results from ECMWF shown in this chapter?
B12. Find a web site that shows plots of the Lorenz “butterfly”, similar to Fig. 20.18. Even better, search the web for a 3-D animation, showing how the solution chaotically shifts from wing to wing.
B13. Search the web for other equations that have different strange attractors. Discuss how the equations and attractors differ from those of Lorenz.
B14. Examine from the web the forecast maps that are produced by various forecast centers. Instead of looking at the quality of the forecasts, look at the quality of the weather map images that are served on the web. Which forecast centers produce the maps that are most attractive? Which are easiest to understand? Which are most useful? What geographic map projections are used for your favorite maps?
B15. Search the web for a meteogram of the weather forecast for your town (or for a town near you, or a town specified by the instructor). What are the advantages and disadvantages of using meteograms to present weather forecasts, rather than weather maps?
B16. Search the web for a summary of different options that are used for physics parameterizations in WRF, or other model selected by your instructor.
B17. Search the web for images/photos of some of the earliest computers used in weather forecasting, such as the ENIAC computer. Discuss how computers have changed.
B18. Find examples of probabilistic forecasts on the web. Print a few examples and discuss.
B19. Computational fluid dynamics (CFD) is the name for numerical methods used to solve fluid dynamics equations in engineering. Search the web for images of CFD forecasts for the finest grid resolution that you can find. What flow situation is it solving? What are the grid spacings and time steps?
20.9.2. Apply
A1. An NWP model has a bottom hydrostatic pressure of 95 kPa over the mountains, and a top pressure of 5 kPa. Find the sigma coordinate value for a height where the pressure (kPa) is:
a. 90 | b. 85 | c. 80 | d. 75 | e. 70 | f. 65 | g. 60 | h. 55 |
i. 50 | j. 45 | k. 40 | m. 35 | n. 30 | o. 25 | p 20 |
A2. For a polar stereographic map projection with a reference latitude of 60°, find the (x, y) coordinates on the map that corresponds to the lat & lon at:
a. Montreal | b. Boston | c. New York City | d. Philadelphia | e. Baltimore | f. Washington DC |
g. Atlanta | h. Miami | i. Toronto | j. Chicago | k. St. Louis | m. New Orleans |
n. Minneapolis | o. Kansas City | p. Oklahoma City | q. Dallas | r. Denver | s. Phoenix |
t. Vancouver | u. Seattle | v. San Francisco | w. Los Angeles | x. A location specified by your instructor |
A3. For a polar stereographic map projection with a reference latitude of 90°, find the map factors m_{o}, m_{x}, and m_{y} for latitudes (°) of:
a. 90 | b. 85 | c. 80 | d. 75 | e. 70 | f. 65 | g. 60 | h. 55 |
i. 50 | j. 45 | k. 40 | m. 35 | n. 30 | o. 25 | p 20 |
A4. Estimate subgrid-scale cloud coverage of low and high clouds for a grid-average RH (%) of:
a. 70 | b. 75 | c. 80 | d. 85 | e. 90 | f. 95 | g. 100 |
A5. Given the following temperature T values (°C) at the indicated grid-point indices (i) for ∆X = 10 km:
i: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
T: | 30 | 27 | 26 | 28 | 24 | 20 | 18 | 23 | 25 | 25 |
Find the gradient ∆T/∆x for upwind first-order difference, centered second-order difference, and centered fourth-order difference, at grid index i:
a. 3 | b. 4 | c. 5 | d. 6 | e. 7 | f. 8 |
A6. Same as the previous exercise, except find the advection term –U·∆T/∆x for a C grid with U values (m s^{–1}) of:
i: | 1.5 | 2.5 | 3.5 | 4.5 | 5.5 | 6.5 | 7.5 | 8.5 | 9.5 |
U: | 3 | 3 | 4 | 5 | 7 | 10 | 14 | 19 | 25 |
A7(§). Suppose that an equation of motion in some strange universe has the form \(\ \Delta U / \Delta t=U \cdot t^{2} / \tau_{o}^{3}\) where constant \(\ \tau_{o}=1\) min, and variable wind U = 1 m s–1 initially (t = 0). With time steps of ∆t = 1 min, use the (a) Euler method, (b) leapfrog method, and (c) fourth-order Runge-Kutta method to forecast the value of U at t = 5 min. Compare your results to the analytical solution of \(\ U=U_{o} \cdot \exp \left[(1 / 3) \cdot\left(t / \tau_{o}\right)^{3}\right]\) where \(\ U_{o}=1 \mathrm{m} \mathrm{s}^{-1}\) is the initial condition. [Hint: for the leapfrog method, you will need to use the Euler forward method for the first step.] Show your work and the results at each time step.
A8(§). Given a 1-D array consisting of 12 grid points in the x-direction with the following initial temperatures (°C). \(\ T_{i}(t=0)=\) 20 24 20 16 20 24 20 16 20 24 20 16 20 for i = 1 to 12. Assume that the lateral boundaries are cyclic, so that the number sequence repeats outside this primary domain. Grid spacing is 5 km, and wind speed is from the west at 10 m s^{–1}. Use the leapfrog time-step method (except for the first time step) and 4^{th}-order spatial differencing. Make enough time steps to forecast out to t = 5000 s. Use time steps of size ∆t (s) = a. 100 b. 200 c. 300 d. 400 e. 500 f. 600
Plot the temperature graph at each time step, and comment on the numerical stability
A9. Given the following pairs of [grid spacings (km), wind speeds (m s^{–1})], find the largest time step that satisfies the CFL criterion.
a. [0.1, 50] | b. [0.2, 30] | c. [0.5, 75] | d. [1.0, 50] |
e. [2, 40] | f. [3, 80] | g. [5, 50] | h. [10, 100] |
i. [15, 75] | j. [20, 20] | k. [33, 50] | m. [50, 75] |
A10. A program has 3 subprograms that each take 1/3 of the running time of the whole program. If the first subprogram is sped up 10 times, the second subprogram is sped up 40%, and the third one is sped up as indicated, what is the total speedup of the program?
a. 10% | b. 50% | c. 75% | d. 100% | e. 3 times |
f. 5 times | g. 10 times | h. 20 times | i. 50 times |
A11. A surface weather station reports a temperature of 20°C with an observation error of σ_{T} = 1°C. An NWP model forecasts temperature of 24°C at the same point. For optimum interpolation, find the analysis temperature and the cost function if the NWP forecast error (°C) is
a. 0.2 | b. 0.4 | c. 0.6 | d. 0.8 | e. 1.0 | f. 1.2 |
g. 1.5 | h. 2.0 | i. 3 | j. 4 | k. 5 |
A12. Suppose the first guess pressure in an optimum interpolation is 100 kPa, with an error of 0.2 kPa. Find the analysis pressure if an observation of P = 102 kPa was observed by:
a. surface weather station | b. ship | c. Southern Hemisphere manual analysis |
A13(§). A surface weather station at P = 100 kPa reports a dew-point temperature of T_{d} = 10°C with an observation error of σ_{Td} = 3°C. An NWP model forecasts a mixing ratio of r = 12 g kg^{–1} at the same point. For variational data assimilation, find the analysis mixing ratio (in g kg^{–1}, and plot the variation of the cost function with mixing ratio) if the NWP mixingratio forecast error (g kg^{–1}) is:
a. 0.1 | b. 0.2 | c. 0.4 | d. 0.5 | e. 0.7 | f. 1.0 | |
g. 1.2 | h. 1.5 | i. 2 | j. 2.5 | k. 3 | m. 4 | n. 5 |
[Hint: Use (4.15b) from the Water Vapor chapter as your “H” function to convert from r to T_{d}.]
A14. Using Fig. 20.14 estimate at what forecast range (days) do we lose the ability to forecast:
a. tornadoes | b. hurricanes |
c. fronts | d. cyclones |
e. Rossby waves | f. thunderstorms |
g. Boras | h. lenticular clouds |
A15. Given the following pairs of x, y values. Use linear regression to find the slope and intercept of the best-fit straight line.
x= | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
a. | y= | 0.1 | 0.4 | 0.2 | 0.6 | 0.3 | 0.3 | 0.5 | 0.8 | 0.7 |
b. | y= | 2 | 4 | 7 | 7 | 10 | 12 | 16 | 14 | 18 |
c. | y= | 7 | 9 | 12 | 12 | 15 | 17 | 21 | 19 | 23 |
d. | y= | –3 | –1 | 2 | 2 | 5 | 7 | 11 | 9 | 13 |
e. | y= | 10 | 7 | 9 | 8 | 6 | 3 | 3 | 3 | 2 |
f. | y= | –20 | –25 | –30 | –35 | –40 | –45 | –50 | –55 | –60 |
A16. Given the forecast bias input (y, thin line) in Fig. 20.G, plot the Kalman filter estimate x of tomorrow’s bias for error variance ratios (r) of:
a. 0.001 | b. 0.002 | c. 0.005 | d. 0.01 | e. 0.03 | |
f. 0.04 | g. 0.05 | h. 0.07 | i. 0.08 | j. 0.09 | k. 0.1 |
A17. Using the MOS regression from the Sample Application in this chapter, calculate the predictand if each of the predictors based on forecast-model output increased by
a. 1% | b. 2% | c. 3% | d. 4% | e. 5% |
f. 6% | g. 7% | h. 8% | i. 9% | j. 10% |
A18.(§) For the Lorenz equations, with the same parameters and initial conditions as used in this chapter, reproduce the results similar to the first Sample Application, except for all 1000 time steps. Also:
a. Plot L and C on the same graph vs. time.
b. Plot M vs. L
c. Plot L vs C.
A19.(§) Given the following fields of 50-kPa height (km). Find the:
a. mean forecast error
b. mean persistence error
c. mean absolute forecast error
d. mean squared forecast error
e. mean squared climatology error
f. mean squared forecast error skill score
g. RMS forecast error
h. correlation coefficient between forecast and verification
i. forecast anomaly correlation
j. persistence anomaly correlation
k. Draw height contours by hand for each field, to show locations of ridges and troughs.
Each field (i.e., each weather map) below covers an area from North to South and West to East.
Analysis: | |||
5.2 5.3 5.4 5.5 5.6 |
5.3 5.4 5.5 5.6 5.7 |
5.4 5.5 5.6 5.7 5.8 |
5.3 5.4 5.5 5.6 5.7 |
Forecast: | |||
5.3 5.5 5.6 5.8 5.9 |
5.4 5.4 5.6 5.7 5.8 |
5.5 5.5 5.6 5.6 5.7 |
5.4 5.6 5.6 5.7 5.8 |
Verification: | |||
5.3 5.4 5.5 5.7 5.8 |
5.3 5.3 5.4 5.5 5.7 |
5.3 5.4 5.5 5.6 5.6 |
5.4 5.5 5.5 5.6 5.6 |
Climate: | |||
5.4 5.4 5.5 5.6 5.7 |
5.4 5.4 5.5 5.6 5.7 |
5.4 5.4 5.5 5.6 5.7 |
5.4 5.4 5.5 5.6 5.7 |
A20. Given the following contingency table, calculate all the binary verification statistics.
Observation | |||
Yes | No | ||
Forecast | Yes: | 150 | 65 |
No: | 50 | 100 |
A21. Given forecasts having the contingency table of exercise N20. Protective cost is $5k to avoid a loss of $50k. Climatological frequency of the event is 50%. (a) Find the value of the forecast. (b) If you can get probabilistic forecasts, then what probability would you want in order to decide to take protective action?
A22. Given the table below of k = 1 to 20 forecasts of probability p_{k} that 24-h accumulated precipitation will be above 25 mm, and the verification o_{k} = 1 if the observed precipitation was indeed above this threshold. (a) Find the Brier skill score. (b) For probability bins of width ∆p = 0.2, plot a reliability diagram, and (c) find the reliability Brier skill score.
k | p_{k} | o_{k} | k | p_{k} | o_{k} | |
1 2 3 4 5 6 7 8 9 10 |
0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 |
1 1 0 1 1 1 0 1 0 1 |
11 12 13 14 15 16 17 18 19 20 |
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.02 0 |
0 0 1 0 0 1 0 0 0 0 |
A23. For any one part of this exercise (Ex a to Ex d) of this problem, a 10-member ensemble forecast system forecasts probabilities that 24-h accumulated rainfall will exceed 5 mm. The observation flags (o) and forecast probabilities (p) are given in the table (in the next column) for a 30-day period. Calculate the hit rate and false-alarm rate for the full range of allowed probability thresholds, and plot the result as a ROC diagram. Also find the area under the ROC curve and find the ROC skill score.
Data for calculations of a ROC diagram:
(Ex a) | (Ex b) | (Ex c) | (Ex d) | (Ex a) | (Ex b) | (Ex c) | (Ex d) | |||||
Day | o | p(%) | p(%) | p(%) | p(%) | Day | o | p(%) | p(%) | p(%) | p(%) | |
1 | 1 | 50 | 10 | 100 | 0 | 16 | 0 | 60 | 30 | 20 | 40 | |
2 | 0 | 20 | 0 | 0 | 10 | 17 | 1 | 70 | 60 | 60 | 50 | |
3 | 1 | 20 | 30 | 90 | 20 | 18 | 1 | 90 | 70 | 60 | 60 | |
4 | 1 | 60 | 40 | 90 | 30 | 19 | 1 | 80 | 80 | 60 | 70 | |
5 | 0 | 50 | 30 | 0 | 40 | 20 | 0 | 70 | 70 | 30 | 80 | |
6 | 0 | 20 | 40 | 0 | 50 | 21 | 0 | 10 | 80 | 30 | 90 | |
7 | 0 | 30 | 50 | 10 | 60 | 22 | 0 | 10 | 90 | 30 | 100 | |
8 | 1 | 90 | 80 | 80 | 70 | 23 | 0 | 0 | 0 | 40 | 10 | |
9 | 0 | 40 | 70 | 10 | 80 | 24 | 0 | 0 | 10 | 40 | 20 | |
10 | 1 | 30 | 100 | 80 | 90 | 25 | 1 | 80 | 40 | 50 | 30 | |
11 | 1 | 100 | 100 | 70 | 100 | 26 | 0 | 0 | 30 | 40 | 40 | |
12 | 0 | 10 | 0 | 10 | 0 | 27 | 0 | 0 | 40 | 0 | 50 | |
13 | 0 | 0 | 0 | 20 | 10 | 28 | 1 | 100 | 70 | 50 | 60 | |
14 | 0 | 10 | 10 | 20 | 20 | 29 | 0 | 10 | 60 | 0 | 70 | |
15 | 1 | 80 | 40 | 70 | 30 | 30 | 1 | 90 | 10 | 50 | 0 |
20.9.3. Evaluate & Analyze
E1. Use the meteogram of Fig. 20.1.
a. After 20 Feb, when does the low pass closest to Des Moines, Iowa?
b. During which days does it rain, and which does it snow?
c. During which days is there cold-air advection?
d. Based on the wind direction, does the low center pass north or south of Des Moines.
e. After 20 Feb, when does the cold front pass Des Moines?
f. What is the total amount of precipitation that fell during the midweek storm?
g. How does this forecast, which was initialized with data from 19 Feb, compare with the actual observations (refer to a previous chapter)?
E2(§). Reproduce the polar stereographic map from the Sample Application for map projections. Then add:
a. Greenland | b. Europe | c. Asia | d. your location if in the N. Hemisphere. |
E3. If Moore’s law continues to hold, and if forecast skill continues to improve as it has in the past, then estimate the transistor count on an integrated circuit, and the skillful forecast period (days) for the year: a. 2010 b. 2015 c. 2020 d. 2025 e. 2030 Also, comment on what factors might cause errors in your estimate.
E4. Speculate on the capability of weather forecasting if digital computers had not been invented.
E5. Critique the validity of a statement that “A variable mesh grid is analogous to a number of discrete nested grids.”
E6. What procedure (i.e., what equations and how they are manipulated) would you use to create the 4^{th}-order centered difference of eq. (20.BA6).
E7. If NWP Corollaries 1 and 2 did not exist, describe tricks that you could use to increase the speed of numerical weather forecasts.
E8. Write a finite-difference equation similar to eq. (20.13) for the shaded grid cell of Fig. 20.9, but for:
a. vertical advection
b. advection in the y-direction
E9. Draw the stencil of grid points used for computing horizontal advection, but for the ___ grid for 2^{nd}-order spatial differencing.
a. A | b. B | c. D |
E10. If the atmosphere is balanced, and if observations of the atmosphere are perfectly accurate, why would numerical models of the atmosphere start out imbalanced?
E11. For the case study forecast of Figs. 20.15, first photocopy the figures. Then, on each map
a. Draw the likely location for fronts.
b. Indicate the locations of low centers
c. Comment on the forecast accuracy for fronts, cyclones, and the large-scale flow for this case.
E12. Suppose you are making weather forecasts for Pittsburgh, Pennsylvania, which is close to the intersection of the 40°N parallel and 80°W meridian, shown by the intersection of latitude and longitude lines in Figs. 20.15 just south of Lake Erie. During the 7.5 days prior to 00 UTC 24 Feb 94, your temperature forecasts for 00 UTC 24 Feb would likely change as you received newer updated forecast maps.
What is your temperature forecast for 00 UTC 24 Feb, if you made it ___ days in advance from the ECMWF forecast charts of Fig. 20.15?
a. 7.5 | b. 5.5 | c. 3.5 | d. 1.5 | e. and which forecast was closest to the actual verifying analysis? |
E13.(§) Suppose the Lorenz equations were modified by assuming that C = L. For the second two Lorenz equations, replace every C with L, and recalculate for the first 1000 time steps.
a. Plot L and M vs. time on the same graph.
b. Plot M vs. L.
Note that the solution converges to a steady-state solution. On the graph of M vs. L, this is called a fixed point. This fixed point is an attractor, but not a strange attractor.
c. Describe what type of physical circulation is associated with this solution.
E14. Experiment with the Lorenz equations on a spreadsheet. Over what range of values of the parameters σ, b, and r, do the solutions still exhibit chaotic solutions similar to that shown in Fig. 20.18?
E15. A pendulum swings with a regular oscillation.
a. Plot the position of the pendulum vs. time.
b. Plot the velocity of the pendulum vs. time.
c. Plot the position vs. velocity. This is a called a phase diagram according to chaos theory. How does it differ from the phase diagram (i.e., the butterfly) of the Lorenz strange attractor?
E16. Use the ensemble forecast for Des Moines in Fig. 20.19.
a. What 85-kPa temperature forecast, and with what reliability, would you make for forecast day: 1, 3, 5, 7, and 9 ?
b. Which temperature ranges would you be confident to forecast would NOT occur, for day: 1, 3, 5, 7, and 9 ?
c. In spite of the forecast uncertainty, are you confident about the general trends in temperature?
d. Could you confidently forecast when rain is most likely? If so, how much rain would you predict?
E17. Fit an exponential curve to the persistence data of Fig. 20.22. What is the e-folding time?
E18. In what order are weather maps presented in the weather briefing given by your favorite local TV meteorologist? What are the advantages and disadvantages of this approach compared to the order suggested in this chapter?
20.9.4. Synthesize
S1. Learn what an analog computer is, and how it differs from a digital computer. If automated weather forecasts were made with analog rather than digital computers, how would forecasts be different, if at all?
S2. a. Suppose that there were no weather observations in the western half of N. America. How would the forecast quality over Washington, DC, and Ottawa, Canada, be different, if at all? Given that national legislators live in those cities, speculate on the changes that they would require of the national weather services in the USA and Canada in order to improve the forecasts.
b. Extending the discussion from part (a), suppose that weather observations are back to normal in N. America, but the seats of government were moved to Seattle and Vancouver. Given what you know about the Pacific data void, speculate on the changes that they would require of the national weather services in the USA and Canada in order to improve the forecasts.
S3. How many grid points are needed to forecast over the whole world with roughly 1-m grid spacing? When do you anticipate computer power will have the capability to do such a forecast? What, if any, are the advantages to such a forecast?
S4. Design a grid arrangement different from that in Fig. 20.9, but which is more efficient (i.e., involves fewer calculations) or utilizes a smaller stencil.
S5. a. Suppose one person developed and ran a NWP model that gave daily forecasts with twice the skill as those produced by any other NWP model run operationally around the world. What power and wealth could that person accumulate, and how would they do it? What would be the consequences, and who would suffer?
b. Same question as part (a), but for one country rather than one person.
S6. Look up the Runge-Kutta finite-difference method in a book or internet site on numerical methods. Find equations for a Runge-Kutta method that is higher order than fourth-order. Can you implement this method on a computer spreadsheet? Try it.
S7. Suppose that there was not a CFL numerical stability criterion that restricted the time step that can be used for NWP. How would NWP be different, if at all? Even without a numerical stability criterion, would there be any other restrictions on the time step? If so, discuss.
S8. Speculate on the ability of national forecast centers to make timely weather forecasts if a computer hacker destroyed the internet and other world-wide data networks.
S9. If greater spread of ensemble members in an ensemble forecast means greater uncertainty, then is greater spread desirable or undesirable in an ensemble forecast?
S10. Which would likely give more-accurate forecasts: a categorical model with very fine grid spacing, or an average of ensemble runs where each ensemble member has coarse grid spacing? Why?
S11. Same as the previous question, but specifically for over steep mountainous terrain. Discuss.
S12. Finite-difference equations are approximations to the full, differential equations that describe the real atmosphere. However, such finite-difference equations can also be thought of as exact representations of a numerical atmosphere that behaves according to different physics. How is this numerical atmosphere different from the real atmosphere? How would physical laws differ for this numerical atmosphere, if at all? When the NWP model runs for a long time, if it approaches steady state, is this state equal to the real climate or to the “model” climate?
S13. Suppose that electricity did not exist. How would you make numerical weather forecasts? Also, how would you disseminate the results to customers?
S14. How good must a numerical forecast be, to be good enough? Should the quality and value of a weather forecast be determined by meteorologists, computer scientists, or end users? Discuss.
S15. Lorenz suggested that there is a limit to predictability. Is that a “hard” limit, or might it be possible to make skillful forecasts beyond that limit? Discuss.
S16. Comment on the interconnectivity of the atmosphere, as expressed by NWP corollary 1.