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Towards Probabilistic Quantitative Precipitation Estimation:

Towards Probabilistic Quantitative Precipitation Estimation: Modeling Radar-Rainfall Error Structure. Witold F. Krajewski, Grzegorz J. Ciach, and Gabriele Villarini.

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Towards Probabilistic Quantitative Precipitation Estimation:

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  1. Towards Probabilistic Quantitative Precipitation Estimation: Modeling Radar-Rainfall Error Structure Witold F. Krajewski, Grzegorz J. Ciach, and Gabriele Villarini

  2. “First I shall make some experiments before I proceed further, because my intention is to consult experience first and then by means of reasoning show why such experiment is bound to work in such a way. And this is the true rule by which those who analyze natural effect must proceed; and although nature begins with the cause and ends with the experience, we must follow the opposite course, namely, […] begin with the experience and by means of it investigate the cause.” Leonardo da Vinci

  3. Uncertainty Propagation Key Concept: Input+ Uncertainty Transformation: (hydrologic prediction model) Deterministic or Stochastic Output + Uncertainty

  4. Product-Error Driven Approach • Collect reliable data on the relation between different RR products and the corresponding True Rainfall; • Create a flexible model of this relation and apply it to the PQPE product generator; • Develop empirically based generalizations of the model for different situations. Combined effect of all error sources!

  5. Definitions • True Rainfall: Amount of rain-water falling on a specified area in a specified interval • Radar Rainfall (RR): An approximation of the True Rainfall based on radar data • RR Uncertainties: All discrepancies between RR and the corresponding True Rainfall • Ground Reference (GR): Approximation of True Rainfall, based on rain-gauge measurements, used to evaluate RR

  6. Mathematical Apparatus Describe family of bivariate frequency distributions (“verification distributions“): (Rr , Ra)A,T,d with A,T,d indexing space, time scales, and radar range, Ra isTrue Rainfall

  7. Mathematical Apparatus Bivariate distribution (X1 , X2) can be expressed in two equivalent ways thorough the relationships: Rr = h1 (Ra , ε1) physical meaning Ra = h2 (Rr , ε2) good for PQPE hi - deterministic factor εi - independent random variable

  8. Ground Reference Errors • The errors in GR based on single rain-gauge are large. They can dominate the radar-gauge comparisons and lead to confusing results • The GR errors should not be ignored • Two ways to deal with the problem: • Building more accurate GR systems; • Filtering GR errors from the radar-gauge verification samples

  9. Oklahoma DataResults(after considerable QC/QA)

  10. Range Effect Analysis Zone IV ARS Micronet Zone I

  11. Overall Bias

  12. Cold (NDJFM) Warm (AMO) Conditional Gauge Mean (mm) All Hot (JJAS) Radar-Rainfall (mm)

  13. Cold (NDJFM) Warm (AMO) Additive error Random error standard deviation, se (mm) Hot (JJAS) All Radar-Rainfall (mm)

  14. Cold (NDJFM) Warm (AMO) Multiplicative error Random error standard deviation, se Hot (JJAS) All Radar-Rainfall (mm)

  15. Cold (NDJFM) Random error quantiles Radar-Rainfall (mm)

  16. Warm (AMO) Random error quantiles Radar-Rainfall (mm)

  17. Hot (JJAS) Random error quantiles Radar-Rainfall (mm)

  18. Cold (NDJFM) Spatial correlation of the random error, re Separation lag (km)

  19. Warm (AMO) Spatial correlation of the random error, re Separation lag (km)

  20. Hot (JJAS) Spatial correlation of the random error, re Separation lag (km)

  21. Cold (NDJFM) Warm (AMO) Temporal correlation of the random error, re Hot (JJAS) All Time lag (minutes)

  22. Modeling Results

  23. Zone I Cold (NDJFM) Warm (AMO) Conditional Gauge Mean (mm) Hot (JJAS) All Radar-Rainfall (mm)

  24. Zone II Cold (NDJFM) Warm (AMO) Conditional Gauge Mean (mm) Hot (JJAS) All Radar-Rainfall (mm)

  25. Zone III Cold (NDJFM) Warm (AMO) Conditional Gauge Mean (mm) Hot (JJAS) All Radar-Rainfall (mm)

  26. Zone IV Cold (NDJFM) Warm (AMO) Conditional Gauge Mean (mm) Hot (JJAS) All Radar-Rainfall (mm)

  27. Zone V Cold (NDJFM) Warm (AMO) Conditional Gauge Mean (mm) Hot (JJAS) All Radar-Rainfall (mm)

  28. Zone I Cold (NDJFM) Warm (AMO) Random error standard deviation, se Hot (JJAS) All Radar-Rainfall (mm)

  29. Zone II Cold (NDJFM) Warm (AMO) Random error standard deviation, se Hot (JJAS) All Radar-Rainfall (mm)

  30. Zone III Cold (NDJFM) Warm (AMO) Random error standard deviation, se Hot (JJAS) All Radar-Rainfall (mm)

  31. Zone IV Cold (NDJFM) Warm (AMO) Random error standard deviation, se Hot (JJAS) All Radar-Rainfall (mm)

  32. Zone V Cold (NDJFM) Warm (AMO) Random error standard deviation, se Hot (JJAS) All Radar-Rainfall (mm)

  33. Zone I Cold (NDJFM) Warm (AMO) Spatial correlation of the random component, re Hot (JJAS) All Separation distance (km)

  34. Zone II Cold (NDJFM) Warm (AMO) Spatial correlation of the random component, re Hot (JJAS) All Separation distance (km)

  35. Zone III Cold (NDJFM) Warm (AMO) Spatial correlation of the random component, re Hot (JJAS) All Separation distance (km)

  36. Zone IV Cold (NDJFM) Warm (AMO) Spatial correlation of the random component, re Hot (JJAS) All Separation distance (km)

  37. Zone V Cold (NDJFM) Warm (AMO) Spatial correlation of the random component, re Hot (JJAS) All Separation distance (km)

  38. Zone I Cold (NDJFM) Warm (AMO) Temporal correlation of the random component, re Hot (JJAS) All Time lag (minutes)

  39. Zone II Cold (NDJFM) Warm (AMO) Temporal correlation of the random component, re Hot (JJAS) All Time lag (minutes)

  40. Zone III Cold (NDJFM) Warm (AMO) Temporal correlation of the random component, re Hot (JJAS) All Time lag (minutes)

  41. Zone IV Cold (NDJFM) Warm (AMO) Temporal correlation of the random component, re Hot (JJAS) All Time lag (mintes)

  42. Zone V Cold (NDJFM) Warm (AMO) Temporal correlation of the random component, re Hot (JJAS) All Time lag (minutes)

  43. Cold Warm Coefficient a (standard deviation) All Hot Time scale (hours)

  44. Cold Warm Coefficient b (standard deviation) All Hot Time scale (hours)

  45. Cold Warm Exponent c (standard deviation) All Hot Time scale (hours)

  46. GR Error Filtering • Assume that, for given spatio-temporal resolution (A,T) and radar-range (d), we have available: • Large sample of corresponding (Rr ,Rg) pairs; • Detailed information about spatial rainfall variability in this sample. • Can we retrieve a good estimate of the verification distribution (Rr , Ra)?

  47. PicoNet

  48. Oklahoma PicoNet

  49. One Hour

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