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Considerable use has been m ade o f t he geometric-op tical propagation thco ry to describe lon g wavelength terrestrial radio waves between t he ionosphere and t he ground b~1 BrclIlill cr [1 949], Wait and Murphy [1 957], Wai t [H)61 ], Wait a nd Co nda [1 961], and J o hler [196 J , [1 962]. The physical in terpretation of pulse signa ls propagated a round t he te rrest ria l sphere as a n application of t he t heory has been describcd by J ohle r [1 962, 1963a, 1963b]. Indeed , t he use of t he Lora n-C radio navigation-timi ng sy stem to s t ud y the p ropagatio ll of 1,F ionos phe ric waves has given f ur the r impet us to s uch a st ud y . T he geometric se ries from which t he geomet ric-optica l ray limi t is derived li ke the full wave residue se ries 0 1' the se ri es of zonal ha rmonics [Johler a nd Berry, I D62] is ri go ro us . H oweve r, the com pa rat ively gross app rox imations usually em ployed in the gco l1w trie-op t ical ray li mit a re wo r t hy of ca reful sc ru tin y at t his tim e s in ce the ri go rous resid ue se ries me t hods have bee n wo rked o ut in some detail fo r la rge scale com pute rs [Johler a nd Berry , 1!J62, 1963]. It is appa re n t from t he rigo rous t heory t hat the geom etric-op t ical rays do not me rely diff ract aro und t he c urvature of the t e rres t rial sp here b u t at considerable d istance beyond t he geometric-op tical horizo n a re built up to s tronger fie lds by addi t iona l ionosp heric reflections of t he nature of a de tac hed mode of propagation 1I 0t o rd ina rily ta ken in to accoun t by the saddle poin t m e thod o f thc ray li mit.

Concerning limitations and further corrections to geometric-optical theory for LF, VLF propagation between ionosphere and the ground