A method to determine relative stroke detection efficiencies from multiplicity distributions

Transcription

1 A ethod to deterine relative stroke detection eiciencies ro ultiplicity distributions Schulz W. and Cuins K. 2. Austrian Lightning Detection and Inoration Syste (ALDIS), Kahlenberger Str.2A, 90 Vienna, Austria, 2. Institute o Atospheric Physics, University o Arizona, Tucson, USA.. Introduction The perorance o a lightning location syste (LLS) can vary with tie as a result o changes in the location and perorance o the sensors that coprise the network. The ost coon changes are () the addition or reoval o sensors and (2) updated sensor technology. Changes in LLS perorance can lead to signiicant changes in estiated lightning paraeters (e.g. peak current and lash ultiplicity statistics - Cuins and Bardo, 2004). Given these acts, it is iportant to be able to quantiy these network changes in ters o the relative detection eiciency () between various network conigurations. A ethod to deterine the overall relative stroke using peak current distributions has been presented by Cuins and Bardo [2004], and is described in detail in an upcoing CIGRE report by Task Force C4.404A. This approach has the liitation that any changes in network coniguration that alter the individual stroke peak current estiates will introduce errors in the calculation. Further, it does not provide relative lash without additional calculations involving ultiplicity easureents. The ethod described in this paper is not subject to the liitations noted above, although we show that it is sensitive to other LLS-derived paraeters. The basis or this detection eiciency () correction using ultiplicity distributions is the work by Rubinstein [995], where he shows the relation between lash and stroke. 2. Theory 2. Relation between Flash and Stroke Rubinstein [995] shows that the detected lash ultiplicity distribution N can be calculated ro the actual lash ultiplicity distribution a N using a () n= N () = F(n,)N (n) =,2,..., where F(n,) is the probability to detect an n-stroke lash as -stroke lash. With the assuption that individual stroke (deined as P) is independent o stroke order, F(n,) can be calculated according to Eq. (2).

2 F(n,) = n P ( P) n (2) In this exaple F(n,) is copletely speciied by the value o P. In our work we use two soewhat ore general odels or deterining F(n,): a) Dierent but constant stroke s or dierent stroke orders are allowed. b) In addition to condition a), irst-stroke is allowed to depend on ultiplicity. For both cases we calculate F(n,) using a recursive unction (see Eq. 3) where p(i) is the o the ith stroke order and q(i)=-p(i). I the real ultiplicity (n) is increased, then F(n+,) ay be calculated as F(n+,) = q(n+) * F(n,) + p(n+) * F(n,-) (3) The irst product ter in Eq. 3 handles the case where the additional stroke is not detected but other strokes are, and the second ter handles the case where the additional stroke is detected and one o the other strokes is not detected. The two extree ters o the recursive unction F(n,0) and F(,) are deined in Eq. 4 and Eq. 5. n F(n,0) = q(i) (4) i= F(,) = p(i) (5) i= Each colun in the atrix F is related to a dierent ultiplicity value. The echanis to ipleent dierent irst stroke s is to calculate all eleents o atrix F related to a speciic ultiplicity using the related irst stroke. 2.2 Relative Correction Our new ethod to deterine stroke corrections relies on describing the proble in ters o relative s. I we are interested in the o a LLS or a certain period with bad (N ) relative to a period with good (N high ) we siply have to rewrite Eq.() as N () = F(n,)N (n) =,2,..., high (6) n= In the case where one assues only two dierent stroke s, one or irst and one or subsequent strokes (p irst and p sub ), then each F(n,) ter in Eq. (6) contains those two stroke s as unknowns. With a 2

3 nonlinear least square algorith (see Appendix A) it is possible to deterine those two unknowns (starting with an initial guess ), and thereore deterine estiates o the relative stroke s or the network. Once the stroke s and related F(n,) are coputed, the relative lash or the low condition can be calculated. This is done with Eq. 7 where the total nuber o lashes or the low condition (Nuerator) is related to the total nuber o lashes or the high condition (Denoinator). lash = ( n= n= F(n,) N n= N high (n) high (n)) =,2,..., (7) 3. Test o the algorith In order to veriy the calculation o the relative, we have tested the algorith in several ways: We assued a irst-stroke, a subsequent-stroke and a ultiplicity distribution N high, and used those values to calculate the ultiplicity distribution N using Eq. (6). We then used the two ultiplicity distributions in the new algorith to estiate the stroke s. This calculation procedure resulted in stroke s identical to the assued ones. We assued a real ultiplicity distribution and calculated distributions or two conditions o the network -- N high (with p irst =0.9 and p sub =0.7) and N (with p irst =0.75 and p sub =0.5). We then used those two ultiplicity distributions to calculate the relative s. Since we can directly copute the relative lash between both distributions N high and N we were able to copare with the resulting lash obtained using the correction algorith. Also in this case the algorith converged to the correct value. We tested the algorith with real Austrian data to evaluate the eect o dierent axiu ultiplicities or the distributions (up to a axiu o 20). We ound that there is no signiicant dierence in the result i we ignore lashes with ultiplicity greater than 4. Only the calculation tie increases signiicantly. 4. Results with real data We used data ro the NLDN or this analysis because corrections (using the correction algorith described in the CIGRE report ro Task Force C4.404A) were already available. These existing corrections provide an overall stroke or the individual 2x2 degree regions shown in Figure 4.. For our evaluation we have chosen a region close to Tucson (region 69) and a region in Florida (region 20). In those two regions we copare the perorance o the NLDN between 999 and 2004 (ater the upgrade o the network). 3

4 Fig. 4.: Red nubered regions will be used or coparison In the initial analysis we assued that all stroke s are independent o ultiplicity, and that irst and subsequent strokes can have dierent (but constant) values. The results or the two regions and or dierent data sets are copared to the original correction ade with the algorith described in the CIGRE report and are given in Tables and 2. Estiated values were obtained or three conditions, as described below. The overall stroke (irst colun) or the new ethod is coputed using a variation o Eqs. () and (3) in Cuins and Bardo [2004]. The derivation is given in Appendix B (see Eq. B6). It can be seen that i all data are used ( All Data condition in Tables -2) an unrealistically sall irst stroke (Region 69: 3%; Region 20: 0%) is obtained. This sall irst-stroke is unlikely, given that irst strokes norally exhibit peak currents greater than subsequent strokes, which ean that irststroke should not be lower than subsequent stroke. 4

5 Table : Region 69 - irst stroke is constant Stroke st Stroke Sub. Stroke Flash correction according to CIGRE All data Flashes with irst stroke peak current >0kA Flashes with irst stroke peak current >5kA Table 2: Region 20 - irst stroke is constant Stroke st Stroke Sub. Stroke Flash correction according to CIGRE All data Flashes with irst stroke peak current >0kA Flashes with irst stroke peak current >5kA The cause o this sall irst-stroke could be a cobination o several actors, e.g. isclassiied cloud strokes, strokes with a bad position (outlier) that are not grouped with other strokes in the lash, lashes with ultiplicity higher than 5 which are split in a lash with 5 strokes and a lash with the reaining strokes, and the assuption that irst stroke is independent o ultiplicity (low ultiplicity is associated with low-current irst strokes Orville et al., 2002; Schulz et al., 2005). All o these possible causes will tend to produce excess low-peak-current single-stroke lashes which lead to low irst stroke s. To test the inluence o an excess nuber o single-stroke lashes, we explored the sensitivity o the estiated to sall increases in the raction o single-stroke lashes. We eployed the second siulation test condition described in Section 3 to create true ultiplicity distributions, and then the raction o single-stroke lashes or the high condition was artiicially increased. When an additional % o the lashes were orced to be single-stroke, the estiated irst-stroke decreased ro 83.8% to 65.4% and the lash decreased ro and 89.4% to 77.%. When an additional 0% o the lashes were orced to be single-stroke, the irst-stroke and lash s decreased urther to 2.7% and 46.%, respectively. Clearly irst-stroke and lash estiates are very sensitive to errors in the raction o single-stroke lashes. We note that the estiated subsequent stroke varied less than % ro the true value or both siulated conditions. To test the inluence o low-peak-current lashes on estiates derived ro easured LLS data, we calculated the s or conditions where we excluded lashes with irst-stroke peak current less than 0 ka 5

6 and less than 5 ka. As can be seen ro Tables and 2, both the irst-stroke and lash values increase when lashes with low-peak-current irst strokes are excluded. However, or region 69 the irststroke is still lower than the subsequent-stroke. Note also that the overall stroke values or lashes with irst-stroke peak current greater than 0 ka are very siilar to the overall stroke according to the ethod published at CIGRE. A correction or region 20 and irst-stroke peak currents greater than 5 ka was not possible because the algorith did not converge to a reasonable result. In order to evaluate the iportance o the dependence o irst-stroke peak current on ultiplicity, we have augented the odel to include this relationship. Prior studies indicate that there is a actor o about 2 dierence between the aplitude o a single-stroke lash and the aplitude o a irst stroke in a lash with ultiplicity 0, with a onotonic increase in peak current as a unction o ultiplicity [Orville et al., 2002; Schulz et al., 2005]. We assue a linear relationship between the irst-stroke and the ultiplicity. To explore this odel, we also assued a variety o speciic values or the slope o this relationship, and then we estiated both the intercept o this irst-stroke-:ultiplicity relationship and the subsequent-stroke. We have tested the assuption with data ro region 20 or dierent slopes and ound the lowest squared-error value o the optiization or a slope o 0.05, yielding a actor o about 6 between a single stroke lash and a lash with ultiplicity 0, as shown in Fig The squared-error value in this case is even lower than the squared-error value or the calculation without dependence o irst-stroke on ultiplicity. First stroke Multiplicity [ ] Fig. 4.2: First stroke as a unction o ultiplicity (slope = 0.05). For this slope we coputed a subsequent-stroke o 66% and a lash o 56%. It is interesting to note that the lash did not increase signiicantly. We also perored this calculation or the ultiplicity distribution o lashes excluding irst-stroke peak currents less than 0 ka. This condition had the sallest 6

7 squared-error when we eployed a saller slope, which is reasonable considering that the relation between irst-stroke peak currents and ultiplicity will be weaker when lashes with low-current irst strokes are excluded. The results were siilar to the values given in Table 2 or the >0 ka condition. 5. Suary/Discussion In this paper we introduced a new ethod to deterine relative changes in detection eiciency o a LLS based on ultiplicity distributions. Contrary to earlier ethods that eploy peak current distributions, this new ethod provides direct estiates o irst stroke, subsequent stroke and lash. We have shown that the odel with the lowest squared-error (o those tested) is a odel that allows or dierent irst- and subsequent-stroke s, and also allows irst stroke to depend on lash ultiplicity. By applying this new ethod we iplicitly assue that the underlying true ultiplicity distribution is the sae or both periods, and that no other actors contributed to the easured ultiplicity values. It is urther iportant to note that the stroke to lash grouping algorith and its coniguration should be the sae or both periods. The new ethod sees to be quite sensitive to additional (erroneous) single-strokes lashes. These ay occur as a result o isclassiied cloud strokes, outliers, liits in the stroke-to-lash grouping algorith, and liits to the odel that relates the two ultiplicity distribution to each other. It is clear ro this work, as well as other work related to odels, that odelling allows us to identiy and evaluate anoalies in real-world lightning location data. Further investigations are necessary to ully understand the interaction between easureent errors and odelling errors. We also plan to explore the use o a weighting atrix in the optiization algorith, allowing us to accoodate dierent errors associated with dierent ultiplicity values. 7

8 Appendix A: Nonlinear Least Square Optiization The nonlinear Least Square analysis used to get the two unknown uses the ollowing procedure: ) Estiate start values p 0 irst, p 0 sub. Written as Matrix the p 0 0 irst p = 0 (A) psub 2) Calculate atrix A, which is the gradient o Eq. (6) with respect to the p values:. irst irst () irst () (2). sub sub A =.. (A2) () sub () (2) 3) Solve the ollowing equation or the change in the p vector: L= [ N - N ] A T *A*p = A T *L n n- (A3) 4) Correct previous p vector or the nth iteration using p n = p n- + p. 5) Iterate until p n p n- <, where is suiciently sall so that the gradient approaches zero. Appendix B: The overall stroke We deine M as the true average lash ultiplicity. This value is deined as the total nuber o strokes in a dataset (S T ), divided by the total nuber o lashes (F T ): S F T M = (B) T Given an iperect easureent syste, not all strokes or lashes will be detected. We thereore deine the easured average ultiplicity () as the total nuber o detected strokes in a dataset divided by the total nuber o detected lashes. The detected quantities are siply the true quantities ultiplied by their respected detection eiciency ractions ( s and ), thereore: S T s = (B2) F T 8

9 Given the deinition o M in Eq. (B), we can rewrite this equation as M s = (B3) An alternate expression or the total nuber o detected strokes is the raction o detected irst strokes plus the raction o subsequent strokes. This is accoplished by deining a irst-stroke ( ) and a subsequent-stroke ( su ), and recognizing that the nuber o subsequent strokes in a dataset is siply the nuber o lashes ultiplied by (M-). Thereore the total nuber o strokes can be expressed as F T + F M ) = F ( + ( M ) ) (B4) T ( su T su Substituting Eq. (B4) into Eq. (B2) yields M + ( ) su = (B5) Note that Eqs. (B3) and (B5) are dierent ors o the sae equation. Equating the right-hand-side nuerators o these equations and re-arranging ters yields Eq. B6. s + ( M ) su = (B6) Reerences: CIGRE Task Force C4.404A: Cloud-to-Ground Lightning Paraeters Derived ro Lightning Location Systes: The Eects o Syste Perorance (in inal review, Noveber 2007). Cuins K.L. and E.A. Bardo: On the relationship between lightning detection network perorance and easured lightning paraeters, paper presented at the st International Conerence on Lightning Physics and Eects, Sao Paulo, Brazil, Noveber 7-, Orville R.E., G.R. Huines, W.R. Burrows, R.L. Holle, K.L.Cuins: The North Aerican Lightning Detection Network (NALDN) First Results: , Monthly Weather Review, Vol. 30, Rubinstein M.: On the deterination o the lash detection eiciency o lightning location systes given their stroke detection eiciency. EMC, 995. W. Schulz, K. Cuins, G. Diendorer, and M. Dorninger: Cloud-to-Ground Lightning in Austria: A 0- years study using data ro a Lightning Location Syste, Journal o Geophysical research, vol. 0,